Previous Year Paper

DSSSB TGT Maths (Male) 4 Sep 2021 Shift 1 (Previous Year Paper)

Held in 2021

200 questions120 minutes
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General Ability (20 questions)

Question 1

The following question is based on the five three-digit numbers given below.

625 463 875 327 136

If the second and third digits of each of the numbers are added, then which of the following numbers will not have the resultant sum exactly divisible by 3?

  1. AOption (a)

    625

  2. BOption (b)

    463

  3. COption (c)

    875

  4. DOption (d)

    327

View answer & explanation

Correct answer

Answer: (a)A.

625

Why this answer is correct

Given:

625 463 875 327 136

according to the given conditions, 

Add second and third digit of each number.

So, 

625 = add 2 and 5 = (2 + 5) = 7 (not divisible by 3)

463 = add 6 and 3 = (6 + 3) = 9 (divisible by 3)

875 = add 7 and 5 = (7 + 5) = 12 (divisible by 3)

327 = add 2 and 7 = (2 + 7) = 9 (divisible by 3)

136 = add 3 and 6 = (3 + 6) = 9 (divisible by 3)

Except "625" all the given number's second and third digit's sum is divisible by 3.

Hence, the correct answer is "625".

Question 2

Which of the following expression is true if the expression M > N ≤ D = S < F > I < U is definitely true?

  1. AOption (a)

    F > N

  2. BOption (b)

    S < U

  3. COption (c)

    M > S

  4. DOption (d)

    F > M

View answer & explanation

Correct answer

Answer: (a)A.

F > N

Why this answer is correct

Given:

M > N ≤ D = S < F > I < U 

Option (1). F > N (True, as N ≤ D = S < F is given in the series, This is definitely true)

Option (2). S < U (False, as S < F > I < U is given in the series, in which > and < both the symbol has been used. So, there is no definite relation can be found in between S and U).

Option (3). M > S (False, as M > N ≤ D = S is given in the series, in which > and < both the symbol has been used. So, there is no definite relation can be found in between M and S).

Option (4). F > M (False, M > N ≤ D = S < F is given in the series, in which > and < both the symbol has been used. So, there is no definite relation can be found in between M and F).

Hence, the correct answer is "F > N".

Question 3

How many such pairs of letters are there in the word "NORM" each of which has as many letters between them in the word (in both forward and backward directions), as they have between them in the English alphabetical series?

  1. AOption (a)

    Two

  2. BOption (b)

    One

  3. COption (c)

    Four

  4. DOption (d)

    Three

View answer & explanation

Correct answer

Answer: (a)A.

Two

Why this answer is correct

Hence, there are two pairs possible in the word "NORM".

Question 4

Six persons A, B, C, D, E and F are sitting in a row facing North. B sits to the left of D who sits to the immediate right of E. F sits to the immediate left of C who sits to the right of D. A sits at the left most position. Which of the following option is correct about the arrangement?

  1. AOption (a)

    C is sitting fifth to the right of B

  2. BOption (b)

    E sits third to the left of C.

  3. COption (c)

    A is sitting fifth to the right of F

  4. DOption (d)

    F is fourth to the left of B

View answer & explanation

Correct answer

Answer: (b)B.

E sits third to the left of C.

Why this answer is correct

According to the given conditions, 

Six persons A, B, C, D, E and F are sitting in a row facing North.

  • B sits to the left of D who sits to the immediate right of E.

  • F sits to the immediate left of C who sits to the right of D.

  • A sits at the left most position. And the final arrangement as follows:

​Checking the options:

Option1: C is sitting fifth to the right of B. (not correct - as C is sitting fourth to the right of B)

Option2: E sits third to the left of C. (correct)

Option3: A is sitting fifth to the right of F. (not correct - as A sits fourth to the left of F)

Option4: F is sitting fifth to the right of B. (not correct - as F sits third to the right of B)

Hence, the correct answer is "E sits third to the left of C".

Question 5

The loss incurred by selling an article for Rs. 11596 is equal to the profit incurred when the same article is sold for Rs. 16040. What should be the selling price to make 50% profit?

  1. AOption (a)

    Rs. 20727

  2. BOption (b)

    Rs. 20027

  3. COption (c)

    Rs. 13818

  4. DOption (d)

    Rs. 20277

View answer & explanation

Correct answer

Answer: (a)A.

Rs. 20727

Why this answer is correct

Given:

The loss incurred by selling an article for Rs. 11596 is equal to the profit incurred when the same article is sold for Rs. 16040.

Formula used:

Loss = cost price - selling price

Profit = selling price - cost price

Selling price = cost price × 100+profit percentage100\dfrac{100+profit \ percentage}{100}

Calculations:

Let the profit earned when an article is sold for Rs. 16040 be x.

So, cost price of an article = 16040 - x (eqn1)

According to the problem,

The loss earned when an article is sold for Rs. 11596 be x.

Cost price of an article = 11596 + x (eqn2)

From eqn1 and eqn2,

16040 - x = 11596 + x

2x = 4444

x = 2222

From eqn1,

Cost price of an article = 16040 - 2222

= Rs. 13818

Now, the selling price of an article when profit is 50%.

Selling price = 13818 × (100 + 50%)/100

= 13818 × 150100×100\dfrac{150}{100 × 100}

= 2072700100\dfrac{2072700}{100}

= Rs. 20727

The answer is Rs. 20727.

Question 6

Out of the given options, three are similar in a certain manner. However, one option is NOT like the other three. Select the option which is different from the rest.

  1. AOption (a)

    84 - 24

  2. BOption (b)

    21 - 6

  3. COption (c)

    42 - 12

  4. DOption (d)

    46 - 16

View answer & explanation

Correct answer

Answer: (d)D.

46 - 16

Why this answer is correct

The logic follows here is,

(first number) - (second number)

The digits of the first number are interchanged and then divided by 2 gives the second number.

1) 84 - 24

→ 48 ÷ 2 = 24 = Second number.

2) 21 - 6

→ 12 ÷ 2 = 6 = Second number.

3) 42 - 12

→ 24 ÷ 2 = 12 = Second number.

4) 46 - 16

→ 64 ÷ 2 = 32 ≠ 16

Hence, the correct answer is "46 - 16".

Question 7

Out of the given options, three are similar in a certain manner. However, one option is NOT like the other three. Select the option which is different from the rest.

  1. AOption (a)

    E3H

  2. BOption (b)

    D1F

  3. COption (c)

    A3D

  4. DOption (d)

    F9O

View answer & explanation

Correct answer

Answer: (b)B.

D1F

Why this answer is correct

The logic follows here is:

The sum of the positional value of the first letter and the number is equal to the positional value of the second letter.

  1. E3H

2) D1f

  1. A3D

  2. F9O

Thus, the odd term is 'D1F'.

Hence, the correct answer is "D1F".

Question 8

Which of the following cube in the answer figure cannot be made based on the unfolded cube in the question figure?

  1. AOption (a)

  2. BOption (b)

  3. COption (c)

  4. DOption (d)

View answer & explanation

Correct answer

Answer: (d)D.

Why this answer is correct

On folding the cube, the following faces are opposite to each other:

  1. 1 is opposite to 3.
  2. 2 is opposite to 4.
  3. 5 is opposite to 6.

As we know, no two opposite surfaces are adjacent to each other.

Option 1: No opposite surfaces are adjacent to each other.

So, the cube in option 1 can be based on the given unfolded cube in the question figure.

Option 2: No opposite surfaces are adjacent to each other.

So, the cube in option 2 can be based on the given unfolded cube in the question figure.

Option 3: No opposite surfaces are adjacent to each other.

So, the cube in option 3 can be based on the given unfolded cube in the question figure.

Option 4: 1 and 3 are adjacent to each other.

So, the cube in option 4 cannot be based on the given unfolded cube in the question figure, as two opposite sides are adjacent to each other.

Hence, the correct answer is "option 4".

Question 9

Find the wrong term in the given series.

0, -6, -24, -65, -120, -210

  1. AOption (a)

    -24

  2. BOption (b)

    -210

  3. COption (c)

    -65

  4. DOption (d)

    -6

View answer & explanation

Correct answer

Answer: (c)C.

-65

Why this answer is correct

The logic follows here is :

Given: 0, -6, -24, -65, -120, -210

1 - (1)3  = 1 - 1 = 0

2 - (2)3 = 2 - 8 = -6

3 - (3)3 = 3 - 27 = -24

4 - (4)3 = 4 - 64 = -60 ≠ -65

5 - (5)3 = 5 - 125 = -120

6 - (6)3 = 6 - 216 = -210

Thus, the wrong term is '-65'.

Hence, the correct answer is "-65".

Question 10

Read the following statements and answer the question on their basis.

A @ B means that A is the mother of B.

A # B means that A is the father of B.

A $ B means that A is B 's brother.

A % B means that A is sister of B.

In expression A $ B # C @ D, how is B related to D?

  1. AOption (a)

    Brother

  2. BOption (b)

    Maternal uncle

  3. COption (c)

    Maternal Grandfather

  4. DOption (d)

    Father

View answer & explanation

Correct answer

Answer: (c)C.

Maternal Grandfather

Why this answer is correct

Blood relation table:

Logic:

Drawing the family tree,

B is the maternal grandfather of D.

Hence, the correct answer is "maternal grandfather."

Question 11

Following are the conditions for selecting a candidate for MBA in a top college

The candidate must-

i) Be not more than 26 years of age on 1 July 2021.

ii) Be a 10th pass with minimum 90% marks.

iii) Be a 12th pass in any discipline with minimum 90% marks.

iv) Have secured minimum 70% in Group Discussion.

v) Have secured minimum 75% in Personal Interview.

In the case of candidate who fulfils all the conditions except

a) At iii) above, but has secured at least 85% in class 12th with science stream ,his/her case is to be referred to department head of MBA.

b) At v) above, but has secured at least 70% in Personal Interview, his/her case is to be referred to admission committee.

In question below, details of one candidate are provided. You have to take one of the following courses of action based on the conditions given above and mark the number of that course of action as your answer. All these cases are given to you as on 1st July 2021.

Neha scored 87% in 12th in Science stream and 92% in 10th class. She scored 75% in personal interview and 75% in Group Discussion. She was born on 24th April 1997.

  1. AOption (a)

    The case is referred to department head of MBA.

  2. BOption (b)

    The case is referred to Admission Committee.

  3. COption (c)

    The candidate is selected.

  4. DOption (d)

    The candidate is not selected.

View answer & explanation

Correct answer

Answer: (a)A.

The case is referred to department head of MBA.

Why this answer is correct

As per the given statements,

Neha scored 87% in 12th standard in science stream, criteria to be fulfilled was to get 85% and she got 2% more fulfilling the criteria.

Scored 92% in class 10 fulfilling the criteria as she has to score 90%.

She scored 75% in personal interview and 75% in group discussion where she has to score 70% as per criteria.

Born on 24 april 1997 and her age on 1 july 2021 will be 24 years not more than 26 years and fulfilling the criteria.

So, as per the criterions, she is to be referred to the department head of MBA.

Hence, the correct answer is "option 1".

Question 12

Complete the following letter series:

NPY, KNX, HLW, _ _

  1. AOption (a)

    EJV

  2. BOption (b)

    FKV

  3. COption (c)

    EKV

  4. DOption (d)

    FJV

View answer & explanation

Correct answer

Answer: (a)A.

EJV

Why this answer is correct

The logic follows here is:

Hence, the correct answer is "EJV".

Question 13

Out of the given options, three are similar in a certain manner. However, one option is NOT like the other three. Select the option which is different from the rest.

  1. AOption (a)

  2. BOption (b)

  3. COption (c)

  4. DOption (d)

View answer & explanation

Correct answer

Answer: (c)C.

Why this answer is correct

The logic follows here is,

All the given options have a square which has five shapes two triangles (one blank and one filled), a square, a circle, and a rectangle inside it except option(3).

option (3)

The square in option(3) has five shapes, three triangles (two blank and one filled), and two squares. Thus, it is different from the others.

Hence, the correct answer is "option (3)".

Question 14

For selection in IT companies through campus placement certain eligibility criteria has been set up.

Following are some requirements as IT companies are looking for some decent candidates for recruitment.

i) Candidate should have at least 8 CGPA as average of all the semester exams. CGPA is the score which a student get in semester exams out of 10.

ii) Candidate should not have any backlog in the semester exams.

iii) Candidate should have done Bachelor of Technology in computer science.

iv) Candidate should not be less than 21 years of age and also not more than 24 years of age.

In the case of a student fulfilling all the conditions except

a) At ii) above, a candidate has CGPA of more than 8.2, then he would be eligible for interview.

b) At i) above, a candidate has managed to score at least 7.5 CGPA and has no backlog, then he will be eligible for written test.

In question below, details of one candidate are provided. You have to take one of the following course of actions based on the conditions given above and the information provided in the question and mark the number of that course of action as your answer.

Nishu, a 22-year-old girl has an average CGPA of 8 in all semesters and has one backlog. She is a graduate in Bachelor of Technology in Mechanical Engineering.

  1. AOption (a)

    She will be eligible for written test.

  2. BOption (b)

    She will be eligible for interview.

  3. COption (c)

    She will not be selected.

  4. DOption (d)

    She will be selected.

View answer & explanation

Correct answer

Answer: (c)C.

She will not be selected.

Why this answer is correct

As per the given statements:

Nishu was a 22-year-old girl fulfilling the criteria as the student must be of more than 21 years of age.

She has an average of 8 in all semesters but has one backlog not fulfilling the criteria as the student should not have any backlog.

She is a graduate in Bachelor of Technology in Mechanical Engineering, but the student has done Bachelor of Technology in computer science, not fulfilling the criteria.

Hence, the correct answer  is  "She will not be selected".

Question 15

From the given answer figures, select the one in which the question figure is hidden/embedded.

  1. AOption (a)

  2. BOption (b)

  3. COption (c)

  4. DOption (d)

View answer & explanation

Correct answer

Answer: (d)D.

Why this answer is correct

The logic follows here is :

On close observation, we find that the problem figure is embedded in option (4) as shown below

Hence, the correct answer is "option (4)".

Question 16

If a mirror is placed on the line AB, then which of the answer figure is the correct mirror image of the given figure?

  1. AOption (a)

  2. BOption (b)

  3. COption (c)

  4. DOption (d)

View answer & explanation

Correct answer

Answer: (b)B.

Why this answer is correct

Given:

Mirror image of the given figure,

Hence, the correct answer is "option 2".

Question 17

In the following Venn Diagram, the triangle represents 'poets', the circle represent s 'teachers' and the rectangle represent s 'writers'. The number given in the diagram represents the number of persons of that particular category.

How many teachers are poets but not writers?

  1. AOption (a)

    7

  2. BOption (b)

    9

  3. COption (c)

    5

  4. DOption (d)

    3

View answer & explanation

Correct answer

Answer: (d)D.

3

Why this answer is correct

Given:

The triangle represents 'poets', the circle represent s 'teachers' and the rectangle represent s 'writers'.

3

Hence, "3" represent the teachers who are poets but not writers.

Question 18

Select the option that is related to the third term on the same basis as the second term is related to the first term.

FM : IJ :: EP : ?

  1. AOption (a)

    HL

  2. BOption (b)

    GM

  3. COption (c)

    HM

  4. DOption (d)

    MH

View answer & explanation

Correct answer

Answer: (c)C.

HM

Why this answer is correct

The logic followed here is,

In the first pair, 3 is added to the position value of the first element and subtract 3 from the position value of the second element to get the second pair.

Similarly, 

In the third pair, add 3 to the positional value of the first element and subtract 3 from the position value of the second element to get the fourth pair.

According to the above logic,

The fourth pair will be "HM".

Question 19

Find the missing number in the given series:

125, 102, 81, 62, 45, --

  1. AOption (a)

    32

  2. BOption (b)

    30

  3. COption (c)

    31

  4. DOption (d)

    33

View answer & explanation

Correct answer

Answer: (b)B.

30

Why this answer is correct

The logic followed here is, 

Subtraction of odd numbers from the previous number to get the next number, starting from 23.

So, the pattern is shown as,

Hence, the next term will be "30".

Question 20

There are six players Rubi, July, Mitu, Prita, Aman, and Rohan in a team. Prita is shorter than only 3 players. Aman is taller than Rohan but not the tallest. Rohan is taller than Prita. Rubi is taller than July. July is not the shortest. How many players are taller than July?

  1. AOption (a)

    4

  2. BOption (b)

    1

  3. COption (c)

    2

  4. DOption (d)

    3

View answer & explanation

Correct answer

Answer: (a)A.

4

Why this answer is correct

According to the given information,

  • There are six players Rubi, July, Mitu, Prita, Aman, and Rohan in a team. Prita is shorter than only 3 players.

  • Aman is taller than Rohan but not the tallest. Rohan is taller than Prita.

  • Rubi is taller than July. July is not the shortest.

  • Final arrangement:

From the arrangement, number of players taller than July = 4.

Hence, the correct answer is 4.

General Awareness (20 questions)

Question 21

Mohiniyattam dance form belongs to which state of India?

  1. AOption (a)

    Kerala

  2. BOption (b)

    Maharashtra

  3. COption (c)

    Gujrat

  4. DOption (d)

    Punjab

View answer & explanation

Correct answer

Answer: (a)A.

Kerala

Why this answer is correct

The correct answer is Kerala.

Key Points

  • Mohiniyattam is the classical dance form of Kerala.
  • Mohiniyattam
  • ​It is literally interpreted as the dance of ‘Mohini’.
  • It is one of the two classical dance forms that originated in Kerala, the other one being Kathakali.
  • It is based on Natya Shastra’s lasya style.
  • It has delicate movements and more feminine facial expressions.
  • The repertoire of Mohiniyattam includes music in the Carnatic style, singing and acting a play through the dance, where the recitation may be either by a separate vocalist or the dancer themselves.
  • The song is typically in Malayalam-Sanskrit hybrid called Manipravalam
  • The movements are gentle and glide-like. They do not have terse rhythmic steps.
  • More emphasis is on the facial expressions and hand gestures.

Question 22

Which is the first anti-satellite (ASAT) missile has tested by India?

  1. AOption (a)

    Mission Ganga

  2. BOption (b)

    Mission Shanti

  3. COption (c)

    Mission Shakti

  4. DOption (d)

    Mission Bhakti

View answer & explanation

Correct answer

Answer: (c)C.

Mission Shakti

Why this answer is correct

The correct answer is Mission Shakti

Key Points

  • ‘Mission Shakti’ was the country’s first-ever Anti-Satellite (ASAT) Missile Test successfully conducted on 27th March 2019 from Dr AP J Abdul Kalam Island in Odisha, where a fast-moving Indian orbiting target satellite in Low Earth Orbit (LEO) was neutralized with pinpoint accuracy.
  • This was a highly complex mission, conducted at extremely high speed with remarkable precision.
  • The successful conduct of Mission Shakti made India the fourth nation in the world with the capability to defend its assets in outer space.

Question 23

Current (As on 22 May 2020) Repo Rate is:

  1. AOption (a)

    0.045

  2. BOption (b)

    0.03

  3. COption (c)

    0.08

  4. DOption (d)

    0.04

View answer & explanation

Correct answer

Answer: (d)D.

0.04

Why this answer is correct

The correct answer is 0.04

Key Points

  • Reverse repo rate is decided by RBI's Monetary Policy Committee led by the Governor of RBI.
  • Reserve rate operation absorbs the liquidity in the system.
  • With Reverse repo rate Bank deposit their surplus funds in RBI, RBI uses this tool when it feels there is much money floating in the market.
  • Reverse repo rate is an operation in which RBI borrows money from banks by lending securities.
  • In Monetary Policy Committee there are six members, three from the RBI and three from the Government of India

Question 24

Who was the Chairman of the Drafting Committee of the Indian Constitution?

  1. AOption (a)

    Dr. Sachchidananda Sinha

  2. BOption (b)

    Dr. Rajesh Prasad

  3. COption (c)

    Dr. B. R. Ambedkar

  4. DOption (d)

    Dr. Rajendra Prasad

View answer & explanation

Correct answer

Answer: (c)C.

Dr. B. R. Ambedkar

Why this answer is correct

The correct answer is Dr. B. R. Ambedkar

Key Points

  1. A Constitution is defined as a document that lays down the framework and provides the fundamental principles on which the government of a country is based. The Draft of the Constitution was prepared by a Drafting Committee consisting of 7 members, under the Chairmanship of Dr. Babasaheb Ambedkar.
  2. Rajendra Prasad was the first President of Independent India who held office from 1950 to 1962. He was elected as the permanent Chairman of the Constituent Assembly set up under the Cabinet Mission Plan, in 1946.
  3. C. Rajagopalachari was the last Governor-General of India and the first Indian to occupy the post. He held office from June 1948 to 26 January 1950, when India became a republic.
  4. Tej Bahadur Sapru was a famous freedom fighter and lawyer. The Constitutional Proposals of the Sapru Committee Report was published in 1945 to resolve issues pertaining to minorities. It is believed that this report had an influence on the process of making the Indian Constitution.

Question 25

Select the right location of World's longest river.

  1. AOption (a)

    Asia

  2. BOption (b)

    Europe

  3. COption (c)

    America

  4. DOption (d)

    Africa

View answer & explanation

Correct answer

Answer: (d)D.

Africa

Why this answer is correct

The correct answer is Africa

Key Points

  • The Nile River is the longest river in the world.
  • It is approximately 6,853 Km long, but its exact length is a matter of debate.
  • It flows northward through the tropical climate of eastern Africa and into the Mediterranean Sea.
  • Eleven countries, namely, Tanzania, Uganda, Rwanda, Burundi, the Democratic Republic of the Congo, Kenya, Ethiopia, Eritrea, South Sudan, Sudan, and Egypt share their water.

Question 26

According to researchers from Spain, which Vitamin supplements can decrease the Covid-19 admission rate in ICU?

  1. AOption (a)

    C

  2. BOption (b)

    B12

  3. COption (c)

    D

  4. DOption (d)

    A

View answer & explanation

Correct answer

Answer: (c)C.

D

Why this answer is correct

The correct answer is vitamin D.

Key Points

  • A study in Spain has shown that high doses of calcifediol, an activated form of vitamin D, significantly reduced the need for ICU treatment of hospitalized COVID-19 patients.
  • According to the study published in ScienceDirect, 50 of 76 patients hospitalized with COVID-19 were treated with calcifediol, and only one was required to be admitted to ICU.
  • All were discharged without complications.

Question 27

In the Bafta TV Awards 2021 who won the best actress (drama) award?

  1. AOption (a)

    Molly Windsor

  2. BOption (b)

    Billie Piper

  3. COption (c)

    Michaela Coel

  4. DOption (d)

    Hayley Squires

View answer & explanation

Correct answer

Answer: (c)C.

Michaela Coel

Why this answer is correct

The correct answer is Michaela Coel

Key Points

  • British Academy of Film and Television Arts(BAFTA):
  • ​Baftas is the UK's largest film award.
  • They are managed by the British Academy of Film and Television.
  • BAFTA was founded in 1947 by a group of British filmmakers.

Additional Information

List of winners at BAFTA (2021):

CategoryWinner
Best FilmNomadland
Best DocumentaryMy Octopus Teacher
Best DirectorChloe Zhao(Nomadland)
Best Leading ActorAnthony Hopkins(The Father)
Best Leading ActressFrances McDormand, (Nomadland)
Rising Star AwardBukky Bakray
Best Short FilmThe Present (Farah Nabulsi)
Best Animated FilmSoul

Question 28

Bihar is popular for which of the following style of paintings?

  1. AOption (a)

    Madhubani

  2. BOption (b)

    Kalamkari

  3. COption (c)

    Warli

  4. DOption (d)

    Gond

View answer & explanation

Correct answer

Answer: (a)A.

Madhubani

Why this answer is correct

The correct answer is Madhubani

Key Points

  • Mithila painting, popularly known as Madhubani art.
  • It is from the district of the same name in Bihar and is now well-known all over the world.
  • The colors used in Madhubani paintings are usually derived from plants and other natural sources.
  • Women usually paint their homes to celebrate festivals and the theme of the painting can be varied from nature to myths.
  • Women decorate the nuptial chamber and the inner walls of their homes to celebrate festivals.
  • The return of Ram from exile and Krishna playing with gopis are the preferred subject matter.
  • Artists often show scenes of nature, an abundant harvest, tantric images of snake worship, and even city scenes if they have visited one.
  • The United Nations in India decided to display Indian folk paintings for the eighth Millenium Development Goals programme.
  • For Madhubani artists Satya Narain and Moti Karn created a beautiful expression of prevention of child mortality.
  • They showed it by showing how elephants and other animals protect their young ones.

​

Thus, Madhubani is a traditional art form named after a village in the state of Bihar.

Additional Information

Other states and their traditional art forms:

StatesTraditional artform
OdishaPatachitra
RajasthanKavad art
MaharashtraWarli art
Tamil NaduTanjore painting
West BengalJharnapatachitra
Andhra PradeshKalamkari painting

Question 29

The WHO has declared this country as Malaria free. Which country is this?

  1. AOption (a)

    Japan

  2. BOption (b)

    Nepal

  3. COption (c)

    China

  4. DOption (d)

    India

View answer & explanation

Correct answer

Answer: (c)C.

China

Why this answer is correct

The correct answer is China

Key Points

  • The World Health Organization (WHO) has declared China as “malaria-free”.
  • Western Pacific Region: China is the first country in the WHO Western Pacific Region to be awarded a malaria-free certification in more than 3 decades.

Other Countries

  • In Western Pacific region the countries that have achieved this status include Australia (1981), Singapore (1982) and Brunei Darussalam (1987.
  • Global Status: Globally, 40 countries and territories have been granted a malaria-free certification from WHO – including, most recently, El Salvador (2021), Algeria (2019), Argentina (2019), Paraguay (2018) and Uzbekistan (2018).

Additional Information

  • Malaria is a life threatening mosquito borne blood disease caused by plasmodium parasites.
  • It is predominantly found in the tropical and subtropical areas of Africa, South America as well as Asia.
  • The parasites spread through the bites of infected female Anopheles mosquitoes.
  • After entering the human body, parasites initially multiply within the liver cells and then attack the Red Blood Cells (RBCs) resulting in their rupture.
  • There are 5 parasite species that cause malaria in humans, and 2 of these species – Plasmodium falciparum and Plasmodium vivax – pose the greatest threat.
  • Symptoms of malaria include fever and flu-like illness, including shaking chills, headache, muscle aches, and tiredness.

Question 30

In 1985, Olympics (National Games) were held in which city of India?

  1. AOption (a)

    Surat

  2. BOption (b)

    Kochi

  3. COption (c)

    Bombay

  4. DOption (d)

    New Delhi

View answer & explanation

Correct answer

Answer: (d)D.

New Delhi

Why this answer is correct

The correct answer is New delhi

Key Points

  • India participated for the first time at the Olympics in 1920, and the first Indian Olympic Games was conducted in the year 1924 in Lahore to select athletes from India to participate in the 1924 Olympic Games.
  • The games were organized by the Indian Olympic Association (IOA), established in the same year.
  • Since then, the games have been held every two years and were renamed as National Games during the 9th edition of the Games, organized in Bombay (Mumbai) in 1940.
  • The Games were conducted 12 times under British Rule till 1946.
  • Initially planned to be organized after a gap of every two years, the major hiccup occurred in organizing the 25th National Games which was held in Hyderabad after a span of nine years.
  • The first modern National Games were held in 1985 in New Delhi on the lines of the Olympic Games which were held with great pomp and show.

Confusion Points

  • The First National Games were organized in Bombay in 1940.
  • National Games comprises various disciplines in which sportsmen from the different states of India participate against each other.
  • The duration and the regulations of the National Games are entirely within the jurisdiction of the Indian Olympic Association.

Question 31

Which female won the Nobel Peace Prize in 2014?

  1. AOption (a)

    Malala Yousafzai

  2. BOption (b)

    Mother Teresa

  3. COption (c)

    Marie Curie

  4. DOption (d)

    Louise Glück

View answer & explanation

Correct answer

Answer: (a)A.

Malala Yousafzai

Why this answer is correct

The correct answer is Malala Yousafzai.

  • The 2014 Nobel Peace Prize was shared in two equal parts between Kailash Satyarthi and Malala Yousafzai

Key Points

  • Satyarthi is from India, the seventh person from his country to win the Nobel Prize and the second person after Mother Teresa to win the Peace Prize.
  • Yousafzai is a Muslim from Pakistan, the second Nobel laureate from his country after Abdus Salam.

Important Points

  • The Nobel Prize award was first conferred in 1901.
  • It is given in the name of Alfred Nobel who was a Swedish inventor.
  • The first Nobel Prizes in Physics, Chemistry, Physiology or Medicine and Literature were first awarded in Stockholm, Sweden and the Peace Prize in Kristiania (now Oslo), Norway.
  • Sveriges Riksbank (Sweden's central bank) established Prize in Economic Sciences in Memory of Sir Alfred Nobel in 1968.

Question 32

Sixteen great powers (Mahajanpadas) existed in which century?

  1. AOption (a)

    1st and early 2nd centuries BC.

  2. BOption (b)

    7th and early 6th centuries BC.

  3. COption (c)

    3rd and early 4th centuries BC.

  4. DOption (d)

    8th and early 9th centuries BC.

View answer & explanation

Correct answer

Answer: (b)B.

7th and early 6th centuries BC.

Why this answer is correct

The correct answer is the 7th and early 6th centuries BC.

Key Points

​Bhagavati Sutra is a sutra of Jainism, which gives a list of sixteen Mahajanapadas.

Mahajanapada Period (600 BC-325 BC):

  • Many Janapadas sprung up in the 6th century BC, the larger of which was called Mahajanapadas.
  • Buddhist literature like Anguttara Nikaya, Mahavastu, and Jain literature like Bhagavati Sutra presents a list of 16 Mahajanapadas with minor variations of names.

Mahajanapada were two types:

  • Non-monarchial/republican states:
  • The republicans, unlike the monarchies, were ruled by tribal oligarchies and Brahmanas had no place.
  • Lichchhavis is said to be the oldest Republican in the world.

Question 33

The ozone layer helps to protect life from which harmful radiation?

  1. AOption (a)

    Beta radiation

  2. BOption (b)

    Ultrared radiation

  3. COption (c)

    Ultraviolet radiation

  4. DOption (d)

    RF radiation

View answer & explanation

Correct answer

Answer: (c)C.

Ultraviolet radiation

Why this answer is correct

The correct answer is Ultraviolet radiation

Key Points

  • The ozone layer is one of the layers in the earth’s atmosphere inside the stratosphere layer.
  • It absorbs ultraviolet radiation and protects people, animals, plants, and marine life from the sun’s harmful rays.
  • It was discovered by Charles Fabry and Henri Buisson in 1913.
  • It can absorb 98% of the sun’s ultraviolet rays and almost 90% of ozone is located in the stratosphere layer at a distance of 15 to 30 km from the surface of Earth.
  • In 1989, the Montreal protocol came to protect the ozone layer by cutting the substances that deplete the ozone layer.

Additional Information

  • Ozone, or trioxygen, is an inorganic molecule with the chemical formula O ₃.
  • It is a pale blue gas with a distinctively pungent smell. It is an allotrope of oxygen that is much less stable than the diatomic allotrope O ₂, breaking down in the lower atmosphere to O ₂.
  • Ozone is a particularly effective absorber of UV radiation.
  • The ozone layer in the stratosphere absorbs a portion of the radiation from the sun, preventing it from reaching the planet's surface. Most importantly, it absorbs the portion of UV light called UVB.
  • UVB is a kind of ultraviolet light from the sun (and sun lamps) that has several harmful effects.

Question 34

The name of the National sports of Pakistan is:

  1. AOption (a)

    Football

  2. BOption (b)

    Volleyball

  3. COption (c)

    Hockey

  4. DOption (d)

    Cricket

View answer & explanation

Correct answer

Answer: (c)C.

Hockey

Why this answer is correct

The correct answer is Hockey.

Key Points

  • Field Hockey is the national sport of Pakistan.
  • Field Hockey is also the national sport of India.
  • Hockey is not a contact sport, so players may not trip, charge at, interfere with, touch, handle, or push any opponent.
  • A field hockey match has four quarters with a two-minute break between the first and second as well as the third and fourth quarters.

​

Additional Information

  • Football:
  • One of the most-watched games in the world.
  • It is a game consisting of 11 players each.
  • A team involves a goalkeeper and 10 field players.
  • It involves several fundamental skills, which are to be learned to be able to play football like:
  1. Shooting
  2. Passing
  3. Receiving, etc.
  • Cricket :
  • Cricket is a team game.
  • It is played between two teams.
  • It is governed by umpires and officials.
  • It consists of a batting and a bowling team.
  • A typical Cricket game has 11 players on each team.
  • The center of the field is a rectangular area of 22 meters called a pitch.

Question 35

The cooking gas acronym LPG stands for:

  1. AOption (a)

    Light petroleum gas

  2. BOption (b)

    Loose petroleum gas

  3. COption (c)

    Liquefied purpose gas

  4. DOption (d)

    Liquefied petroleum gas

View answer & explanation

Correct answer

Answer: (d)D.

Liquefied petroleum gas

Why this answer is correct

The correct answer is a Liquefied petroleum gas

Key Points

  • Liquefied petroleum gas (LPG), also called cooking gas, is a flammable mixture of hydrocarbon gases used as fuel in heating appliances, cooking equipment, and vehicles.
  • It is a mixture of 50% butane, 45% propane, and 5% pentane.

Additional Information

  • Butane is a colorless, odourless, gaseous hydrocarbon. Its chemical formula is C4H10.
  • Ethyne is also known as acetylene. It is an organic chemical compound with the chemical formula C2H2.
  • Propane is a flammable hydrocarbon gas that is liquefied through pressurization and is commonly used for fuel in heating, cooking, and vehicles.
  • Ethene is a gaseous by-product of the process of cracking long-chain hydrocarbons

Question 36

Which book the Greek writer Megasthenes wrote?

  1. AOption (a)

    Indica

  2. BOption (b)

    History of India

  3. COption (c)

    Indus Country

  4. DOption (d)

    Hindu Civilization

View answer & explanation

Correct answer

Answer: (a)A.

Indica

Why this answer is correct

The correct answer is Indica

Key Points

  • Megasthenes was a Greek diplomat, historian, and ethnographer whose extensive writings about the cultures of India gave an insight into the lives of ancient Indians during the rule of Chandragupta Maurya.
  • He was an ambassador of the Greek ruler of West Asia named Seleucus Nicator, who visited the court of Chandragupta Maurya.
  • He described Indian society and culture in his book Indica.
  • Though the original work has been lost, but it had been frequently quoted in the works of later writers.
  • Megasthenes also described India's two major rivers, Sindhu and Ganga.
  • He was the first person to describe ancient India, and for that reason, he has been called "The Father of Indian history”.

Question 37

As salt water, what percentage of the earth's water is found in oceans?

  1. AOption (a)

    97

  2. BOption (b)

    67

  3. COption (c)

    99

  4. DOption (d)

    57

View answer & explanation

Correct answer

Answer: (a)A.

97

Why this answer is correct

The correct answer is 97

Key Points 

  • The main source of salt water is the ocean.
  • All oceans and seas have salty water.
  • More than 97% of the Earth’s water is found in the oceans and is too salty for human use.
  • Don Juan Pond is the saltiest among all seas and oceans.
  • A dead sea in Israel has a salinity of 340 grams/liter of water.
  • Swimmers can float in the dead sea because the increased salt content makes it dense.
  • The seawater collected in shallow beds dug in the sand is allowed to dry in the sun and the salt remains on the ground is collected for commercial purposes.
  • All waters in nature contain dissolved mineral salts.
  • Salinity is the amount of salt in grams present in 1000 grams of water.
  • The average salinity of the oceans is 35 parts per thousand.

Question 38

Who is the Vice-president of India (As on August 2021)?

  1. AOption (a)

    Kocheril Raman Narayanan

  2. BOption (b)

    Venkaiah Naidu

  3. COption (c)

    Bhairon Singh Shekhawat

  4. DOption (d)

    Mohammad Hamid Ansari

View answer & explanation

Correct answer

Answer: (b)B.

Venkaiah Naidu

Why this answer is correct

The correct answer is Venkaiah Naidu

Key Points

  • Senior BJP leader & former Union minister M Venkaiah Naidu was elected as the 13th Vice-President of India.
  • He defeated Mahatma Gandhi’s grandson Gopalkrishna Gandhi by 272 votes. Naidu, the NDA candidate won by the largest margin in 30 years in a vice presidential election that had the highest polling ever.
  • Present Vice-president of India is Jagdeep Dhankhar

Question 39

The Head Office of the United Nations Conference on Trade and Development (UNCTAD) is located at:

  1. AOption (a)

    Geneva, Switzerland

  2. BOption (b)

    Washington, D.C., United States

  3. COption (c)

    Kathmandu, Nepal

  4. DOption (d)

    Shanghai, China

View answer & explanation

Correct answer

Answer: (a)A.

Geneva, Switzerland

Why this answer is correct

The correct answer is Geneva, Switzerland

Key Points

The United Nations Conference on Trade and Development is headquartered in Geneva, Switzerland.

  • UNCTAD was established on 30th December 1964 in response to developing countries trading better with developed ones.

  • It is an organ of the United Nations and deals with trade, investment, and development issues.

  • The Generalised System of Preferences (GSP) was developed by the organization.

  • It holds conferences every 4 years. In India, it was held in 1968 in New Delhi.

  • Raul Prebisch, an Argentine economist was the first Secretary-General and the founder. He held the post from 1964 to 1968.

Question 40

Who is the Indian with the most number of Oscar awards?

  1. AOption (a)

    Amir Khan

  2. BOption (b)

    A.R. Rahman

  3. COption (c)

    Ranveer Singh

  4. DOption (d)

    Bhanu Athaiya

View answer & explanation

Correct answer

Answer: (b)B.

A.R. Rahman

Why this answer is correct

The correct answer is A.R. Rahman

Key Points

  • Allah Rakha Rahman is an Indian film composer, record producer, singer, and, songwriter who works predominantly in Indian cinema.
  • Rahman is now a Golden Globe® winner for "Best Original Soundtrack" for SLUMDOG MILLIONAIRE and has been nominated for three Oscars for the film, for best original score and two for best original songs.
  • In 2010, the Indian government awarded him the Padma Bhushan, the nation's third-highest civilian award.
  • Jai Ho" is a song composed by A. R. Rahman for the 2008 film Slumdog Millionaire.

Additional Information

Oscar Awards:

  • They are considered the most prestigious and significant awards in the entertainment industry worldwide.
  • It is given annually by the Academy of Motion Picture Arts and Sciences (AMPAS).
  • It consists of a golden statuette as a trophy officially called the "Academy Award of Merit", which depicts a knight rendered in the Art Deco style.
  • The trophy was originally sculpted by George Stanley from a design sketch by Cedric Gibbons.
  • It was first presented in 1929 at a private dinner hosted by Douglas Fairbanks.
  • It was first broadcast by radio in 1930 and was televised in 1953 for the first time.
  • It is the oldest worldwide entertainment awards ceremony.
  • Since its inception, a total of 3,140 Oscar statuettes have been awarded.

Arithmetic Ability (20 questions)

Question 41

What percent 12 mm is of 18 cm?

  1. AOption (a)

    6.6%

  2. BOption (b)

    6.67%

  3. COption (c)

    6.7%

  4. DOption (d)

    7.0%

View answer & explanation

Correct answer

Answer: (b)B.

6.67%

Why this answer is correct

Given:

12 mm of 18 cm

Concept used: 

1 cm = 10 mm

Calculation:

⇒ 18 cm = 18 × 10 = 180 mm

Required Percentage = 12180\dfrac{12}{180 } × 100

⇒ 6.67%

∴  12 mm is 6.67% of 18 cm.

Question 42

Find the quotient when the biggest 4-digit number divisible by 16 is divided by 28.

  1. AOption (a)

    350

  2. BOption (b)

    356

  3. COption (c)

    375

  4. DOption (d)

    370

View answer & explanation

Correct answer

Answer: (b)B.

356

Why this answer is correct

Concept used: 

Quotient = Dividend / Divisor + Remainder

Calculation:

The largest 4-digit number is 9999.

The biggest 4-digit number divisible by 16 = 9984

⇒ Quotient = 9984/28 = 356 + remainder

∴ The quotient is 356.

Question 43

A man sold an article at Rs. 279 at a loss of 10%. What would be the gain percentage if he sells the same article at Rs. 372?

  1. AOption (a)

    20%

  2. BOption (b)

    21%

  3. COption (c)

    19%

  4. DOption (d)

    18%

View answer & explanation

Correct answer

Answer: (a)A.

20%

Why this answer is correct

Given:

SP = Rs. 279

Loss% = 10%

Concept used: 

CP = SP(1−Loss%100)\dfrac{SP}{(1 - {Loss\% \over 100})}

Gain% = GainCP\dfrac{Gain}{CP} × 100

Calculation:

⇒ CP = 279(1−10%100)\dfrac{279}{(1 - {10\% \over 100})}

⇒ CP = 2790.9\dfrac{279}{0.9} = Rs. 310

⇒ Gain = Rs. 372 - 310 = Rs. 62

⇒ Gain% = 62310\dfrac{62}{310} × 100

⇒ Gain% = 20%

∴ The Gain% is 20%.

Question 44

The sum of cube of 3 numbers that are in the ratio 4 ∶ 5 ∶ 6 is 50625 . What are the numbers?

  1. AOption (a)

    20, 30, 25

  2. BOption (b)

    20, 25, 30

  3. COption (c)

    25, 20, 30

  4. DOption (d)

    30, 25, 20

View answer & explanation

Correct answer

Answer: (b)B.

20, 25, 30

Why this answer is correct

Given:

The sum of cube of 3 numbers that are in the ratio 4 ∶ 5 ∶ 6 is 50625

Calculation:

Let, the three numbers be 4m, 5m and 6m.

According to the question,

⇒ (4m)3 + (5m)3 + (6m)3 = 50625

⇒ (64 + 125 + 216)m3 = 50625

⇒ m3 = 50625405\dfrac{50625}{405}

⇒ m3 = 125

⇒ m = 5

⇒ 4m = 20

⇒ 5m = 25

⇒ 6m = 30

∴ The numbers are 20, 25, 30.

Question 45

30% of a number is 12 more than 1/6 th of that number. What is that number?

  1. AOption (a)

    80

  2. BOption (b)

    60

  3. COption (c)

    90

  4. DOption (d)

    70

View answer & explanation

Correct answer

Answer: (c)C.

90

Why this answer is correct

Given:

30% of a number is 12 more than 1/6 th of that number.

Calculation:

Let, the number be m.

According to the question,

⇒ 30% × m = 12 + 16\dfrac{1}{6} × m

⇒ 0.3m = 12 + m6\dfrac{m}{6}

⇒ 1.8m = 72 + m

⇒ 0.8m = 72

⇒ m = 90

∴ The number is 90.

Question 46

Find the smallest 4 digit number when divided by 15, 24, 30 leaves a common remainder 9.

  1. AOption (a)

    1203

  2. BOption (b)

    1209

  3. COption (c)

    1200

  4. DOption (d)

    1201

View answer & explanation

Correct answer

Answer: (b)B.

1209

Why this answer is correct

Given:

The given numbers are 15, 24, 30.

Concept used: 

The smallest 4 digit number is 1000

The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of all of the numbers.

Calculation:

⇒ LCM(15, 24, 30) = 2 × 2 × 2 × 3 × 5 = 120

When we divide 1000 by 120, we get 40 as remainder.

⇒The smallest 4 digit number when divided by 15, 24, 30 leaves a common remainder 9 = 1000 - 40 + 120 + 9 =  1089

Since 1089 is not available in the option, we again add 120,

⇒ 1089 + 120 = 1209

∴ The smallest 4 digit number when divided by 15, 24, 30 leaves a common remainder 9 is 1209.

Question 47

There are three numbers given: 27, 42, 57. Find the largest number that divide and give a remainder of 10, 8 and 6.

  1. AOption (a)

    19

  2. BOption (b)

    11

  3. COption (c)

    17

  4. DOption (d)

    13

View answer & explanation

Correct answer

Answer: (c)C.

17

Why this answer is correct

Given:

There are three numbers given: 27, 42, 57

Concept used: 

HCF of two or more numbers is the largest number that can divide each of the numbers without leaving a remainder.

Calculation:

⇒ HCF of (27 - 10), (42 - 8) and (57 - 6)

⇒ HCF(17, 34, 51) = 17

∴ The largest number is 17.

Question 48

11 people went on a lunch together to a hotel. The expenses for 10 of them is Rs. 400 each. The expense of 11th person is Rs 100 more than the average of all the 11 people. What is the expense of the 11th person?

  1. AOption (a)

    490 rupees

  2. BOption (b)

    410 rupees

  3. COption (c)

    510 rupees

  4. DOption (d)

    500 rupees

View answer & explanation

Correct answer

Answer: (c)C.

510 rupees

Why this answer is correct

Given:

11 people went on a lunch together to a hotel.

The expenses for 10 of them is Rs. 400 each.

The expense of 10th person is Rs 100 more than the average of all the 11 people.

Concept used: 

Average = Sum of observation / Number of observations

Calculation:

⇒ Sum of the expense of 10 persons = 400 × 10 = Rs. 4000

According to the question,

⇒ m = 4000+m+10011\dfrac{4000 + m + 100}{11}

⇒ 11m = 4100 + m

⇒ 10m = 4100

⇒ m = 410

The expense of 11th person = m + 100 = Rs. 510

∴ The expense of 11th person is Rs. 510.

Question 49

In a school trip, the ratio of students to teachers are 16 ∶ 1. What is the number of teachers required in the trip if there are 112 students?

  1. AOption (a)

    6

  2. BOption (b)

    7

  3. COption (c)

    8

  4. DOption (d)

    5

View answer & explanation

Correct answer

Answer: (b)B.

7

Why this answer is correct

Given:

The ratio of students to teachers are 16 ∶ 1

Calculation:

Let, the number of teachers be m.

According to the question,

⇒ 161=112m\dfrac{16}{1} = \dfrac{112}{m}

⇒ m = 7

∴ The number of teachers required for the trip is 7.

Question 50

There are 3 numbers. If second number is 7 times the third number and second number is twice first number. The average of 3 numbers is 23 . What is the smallest number?

  1. AOption (a)

    6

  2. BOption (b)

    12

  3. COption (c)

    3

  4. DOption (d)

    21

View answer & explanation

Correct answer

Answer: (a)A.

6

Why this answer is correct

Given:

There are 3 numbers.

If the second number is 7 times the third number and second number is twice first number.

The average of 3 numbers is 23 .

Concept used: 

Average = Sum of observation / Number of observations

Calculation:

Let, the three numbers be m, n & s.

⇒ m+n+s3\dfrac{m + n + s}{3} = 23

⇒ m + n + s = 69

According to the question,

n = 7 × s = 2 × m

⇒ m = 3.5s

⇒ 3.5s + 7s + s = 69

⇒ 11.5s = 69

⇒ s = 6

⇒ m = 21

⇒ n = 42

∴ The smallest number is 6.

Question 51

A train is 135 m long. The speed of the train is 45 kmph. How many seconds will the train take to pass a pole?

  1. AOption (a)

    10 secs

  2. BOption (b)

    11 secs

  3. COption (c)

    10.8 secs

  4. DOption (d)

    11.8 secs

View answer & explanation

Correct answer

Answer: (c)C.

10.8 secs

Why this answer is correct

Given:

A train is 135 m long.

The speed of the train is 45 kmph.

Concept used: 

Distance = Speed × Time

1 km = 1000 m

1 hour = 60 minutes

1 minute = 60 seconds

Calculation:

⇒ Time = 13545×60×601000\dfrac{135}{45} \times \dfrac{60\times60}{1000}

⇒ Time = 10.8 seconds

∴ The time taken by train to pass the pole is 10.8 seconds.

Question 52

What is the single equivalent discount of 16%, 20% and 24% successive discounts?

  1. AOption (a)

    48.93%

  2. BOption (b)

    48.00%

  3. COption (c)

    48.39%

  4. DOption (d)

    48.90%

View answer & explanation

Correct answer

Answer: (a)A.

48.93%

Why this answer is correct

Concept used: 

Successive Discounts = D1 + D2 - D1×D2100\dfrac{D1\times D2}{100}

Calculation:

For 16% and 20%,

⇒ Successive discount = 16 + 20 - 16×20100\dfrac{16\times 20}{100}

⇒ 36 - 3.2 = 32.8%

For 16%, 20%, and 24%,

⇒ Successive discount = 32.8 + 24 - 32.8×24100\dfrac{32.8 \times 24}{100} 

⇒ 56.8 - 7.872 = 48.93%

∴ A single equivalent discount is 48.93%.

Question 53

A sphere of diameter 18 cm is drawn into a wire of 1 cm radius. What is the length of the wire?

  1. AOption (a)

    792 cm

  2. BOption (b)

    729 cm

  3. COption (c)

    927 cm

  4. DOption (d)

    972 cm

View answer & explanation

Correct answer

Answer: (d)D.

972 cm

Why this answer is correct

Given:

A sphere of diameter 18 cm is drawn into a wire of 1 cm radius.

Concept used: 

The volume of the sphere = 434\over 3 πr3

The volume of the cylinder = πr2h 

Calculation: 

The volume will remain the same.

⇒ 43\dfrac{4}{3}πr3 = πr2h

⇒ 43\dfrac{4}{3} × 93 = 12 × h

⇒ 4 × 3 × 81 = h

⇒ h = 972 cm

∴ The length of the wire is 972 cm.

Question 54

The simple interest and compound interest on a certain sum of money is Rs. 80100 and Rs. 81902.25. The duration for which the interest was calculated is 2 years. Find the sum of money?

  1. AOption (a)

    Rs. 890000

  2. BOption (b)

    Rs. 980000

  3. COption (c)

    Rs. 900000

  4. DOption (d)

    Rs. 800000

View answer & explanation

Correct answer

Answer: (a)A.

Rs. 890000

Why this answer is correct

Given:

SI = Rs. 80100 

CI = Rs. 81902.25

Time = 2 years

Concept used: 

SI = PRT/100

Calculation:

Simple interest for 2 years = 80100

⇒ Simple interest for 1 year = 80100/2 = 40050

Difference between CI and SI = 81902.25 - 80100 = 1802.25

⇒ Interest Rate = 1802.2540050×100\dfrac{1802.25}{40050} \times 100 = 4.5%

On putting the value of SI, R and T in the formula of simple interest of 1 year,

⇒ 40050 = P×4.5×1100\dfrac{P\times4.5\times1}{100}

⇒ P = 40050004.5\dfrac{4005000}{4.5} = Rs. 890000

∴ The sum of the money is Rs. 8,90,000.

Question 55

What is the average number of students in maths subject?

  1. AOption (a)

    279

  2. BOption (b)

    297

  3. COption (c)

    290

  4. DOption (d)

    295

View answer & explanation

Correct answer

Answer: (b)B.

297

Why this answer is correct

Concept used: 

Average = Sum of observation / Number of observations

Calculation:

Total number of maths students = 300 + 288 + 303 = 891

Total number of schools = 3

⇒ Average = 8913\dfrac{891}{3} = 297

∴ The average number of students in maths is 297.

Question 56

If p/q = 7/4 then what is (p2 - q2) / (p2 + q2)?

  1. AOption (a)

    0.51

  2. BOption (b)

    0.50

  3. COption (c)

    0.49

  4. DOption (d)

    0.52

View answer & explanation

Correct answer

Answer: (a)A.

0.51

Why this answer is correct

Given:

p/q = 7/4

Calculation:

Let, p = 7m and q = 4m for some value m.

⇒ p2−q2p2+q2=(7m)2−(4m)2(7m)2+(4m)2\dfrac{p^2-q^2}{p^2+q^2} = \dfrac{(7m)^2-(4m)^2}{(7m)^2+(4m)^2}

⇒ 49m2−16m249m2+16m2\dfrac{49m^2 - 16m^2}{49m^2+16m^2}

⇒ 33m265m2\dfrac{33m^2}{65m^2}

⇒ 3365\dfrac{33}{65} = 0.51

∴ The required value is 0.51

Question 57

Convert the following into fractions:

2.0006, 4.0088

  1. AOption (a)

    10003/5000, 5011/1520

  2. BOption (b)

    10003/5000, 5000/1250

  3. COption (c)

    10003/2500, 5011/1250

  4. DOption (d)

    10003/5000, 5011/1250

View answer & explanation

Correct answer

Answer: (d)D.

10003/5000, 5011/1250

Why this answer is correct

Given:

2.0006, 4.0088

Concept used: 

To convert any terminating decimal number into a fraction, divide the number after removing the decimal by 10 to the power m, where m is the number of digits after the decimal.

Calculation:

⇒ 2.0006 = 2000610000\dfrac{20006}{10000} = 100035000\dfrac{10003}{5000}

⇒ 4.0088 = 4008810000\dfrac{40088}{10000} = 50111250\dfrac{5011}{1250}

∴ 2.0006, 4.0088 in fractional form is 10003/5000, 5011/1250.

Question 58

Suresh can do a piece of work in 9 days and Nimesh can do the same work in 18 days. They both work together for 4 days and Suresh left after that, how long will now Nimesh take to complete the work?

  1. AOption (a)

    5 days

  2. BOption (b)

    7 days

  3. COption (c)

    6 days

  4. DOption (d)

    4 days

View answer & explanation

Correct answer

Answer: (c)C.

6 days

Why this answer is correct

Given:

Suresh = 9 days

Nimesh = 18 days

Calculation:

Work done by Nimesh in 1 day = 118\dfrac{1}{18} units

Work done by Suresh in 1 day = 19\dfrac{1}{9} units

⇒ Work done by both in 1 day = 118+19\dfrac{1}{18} + \dfrac{1}{9} = 16\dfrac{1}{6} units

⇒ Work done in 4 days by both = 4 × 16\dfrac{1}{6} = 23\dfrac{2}{3} units

x=−b±b2−4ac2ax = {-b \pm \sqrt{b^2-4ac} \over 2a}

⇒ Work left to be done = 1 - 23=13\dfrac{2}{3} = \dfrac{1}{3} units

Time taken by Nimesh to finish the left work = 183\dfrac{18}{3} = 6 days

∴ Time taken by Nimesh to complete the work is 6 days.

Question 59

A man sold two tables for Rs. 55450 each. On one he gains 18% and on the other he loses 18%. What is the gain or loss percentage in the entire transaction?

  1. AOption (a)

    3.2% loss

  2. BOption (b)

    3.4% gain

  3. COption (c)

    3.24% loss

  4. DOption (d)

    3.42% gain

View answer & explanation

Correct answer

Answer: (c)C.

3.24% loss

Why this answer is correct

Given:

A man sold two tables for Rs. 55450 each. On one he gains 18% and on the other he loses 18%.

Concept used: 

Loss = CP - SP

Loss% = Loss/CP x 100

Calculation:

⇒ Total SP = 55450 × 2 = 110900

CP of first table = 55450 × 100118\dfrac{100}{118} = 46991.52

CP of second table = 55450 × 10082\dfrac{100}{82} = 67621.95

Total CP = 46991.52 + 67621.95 = 114613.47

⇒ Loss = 114613.47 - 110900 = 3713.47

Loss% = 3713.47114613.47×100\dfrac{3713.47}{114613.47} \times 100 = 3.24%

∴ The Loss% is 3.24%.

Question 60

The graph shows the production of cars in different countries over two year. In how many countries the production of car has increased by less than 10% in the year 2020 over the last year?

  1. AOption (a)

    4

  2. BOption (b)

    3

  3. COption (c)

    2

  4. DOption (d)

    1

View answer & explanation

Correct answer

Answer: (d)D.

1

Why this answer is correct

Calculation:

Difference of Production in India = 840 - 754 = 86

⇒ Change in Production = 86754×100\dfrac{86}{754} \times 100 = 11.4%

Difference of Production in China = 848 - 736 = 112

⇒ Change in Production = 112736×\dfrac{112}{736} \times  100 = 15.2%

Difference of Production in Nepal = 834 - 744 = 90

⇒ Change in Production = 90744×\dfrac{90}{744} \times 100 = 12.09%

Difference of Production in Pakistan = 774 - 770 = 4

⇒ Change in Production = 4770×\dfrac{4}{770} \times 100 = 0.519%

Difference of Production in Sri Lanka = 838 - 744 = 94

⇒ Change in Production = 94744×\dfrac{94}{744} \times 100 =12.63%

∴ In 1 country the production of cars has increased by less than 10% in the year 2020 over the last year.

General English (20 questions)

Question 61

Fill in the blank with the most appropriate choice.

They offered ___ and manifestly crazy productions to their amazed contemporaries.

  1. AOption (a)

    moony

  2. BOption (b)

    lackadaisical

  3. COption (c)

    inert

  4. DOption (d)

    incomprehensive

View answer & explanation

Correct answer

Answer: (d)D.

incomprehensive

Why this answer is correct

The correct answer is "incomprehensive".

Key Points

  • In the above question we have to choose the most appropriate word to fill in the blanks**.**
  • The word "incomprehensive" means difficult to understand, which is an appropriate way to describe productions that are manifestly crazy.
  • The use of the word "incomprehensive" emphasizes that the productions are so strange and unusual that they are difficult to comprehend.
  • In this sentence, the speaker is describing a group of people who offered productions that were "incomprehensive" and "manifestly crazy."
  • The use of "incomprehensive" to describe the productions emphasizes their difficult-to-understand nature, while the use of "manifestly crazy" emphasizes their absurdity. The use of "amazed contemporaries" implies that these productions were not only unusual but also striking and captivating to the audience.
  • Thus, as per the above explanation, the most appropriate word is "incomprehensive"

Correct Sentence: They offered incomprehensive and manifestly crazy productions to their amazed contemporaries.

Additional Information

  • "Moony" means dreamy or fanciful.
  • "Lackadaisical" means lazy or apathetic, which does not match the tone of the sentence.
  • "Inert" means lacking in energy or movement, which is not appropriate to describe productions that are "manifestly crazy."

Question 62

Select the word segment that substitutes (replaces) the bracketed word segment correctly and completes the sentence meaningfully. Select the option 'no correction required' if the sentence is correct as given.

I'm keenly aware of (Principal of Priority, that states; you should) know the difference between what is urgent and what is important, and, you must do what 's important first.

  1. AOption (a)

    the Principle of Priority, which states; you must

  2. BOption (b)

    No correction required.

  3. COption (c)

    the Principal of Priority, which states; you must)

  4. DOption (d)

    that Principle of Priority, which states; you should)

View answer & explanation

Correct answer

Answer: (a)A.

the Principle of Priority, which states; you must

Why this answer is correct

The correct answer is "​the Principle of Priority, which states; you must".

Key Points

  • The word "Principal" is correctly replaced with "Principle" as "Principal" refers to the head of a school or organization, whereas "Principle" refers to a fundamental truth or rule.
  • The word "which" is used to introduce a non-restrictive clause which gives additional information about the Principle of Priority, it also replaces "that" which is less formal and more appropriate for the context.
  • "you must" is used to indicate that it is an imperative statement and it is necessary to know the difference between what is urgent and important and do the important things first.
  • Thus, the correct sentence would be, "I'm keenly aware of the Principle of Priority, which states you must know the difference between what is urgent and what is important, and, you must do what's important first.

Correct Sentence: I'm keenly aware of the Principle of Priority, which states you must know the difference between what is urgent and what is important, and, you must do what's important first.

Question 63

The sentence below has been divided into three parts. Select the part of the sentence that has an error. If the sentence has no error, select the option 'No Error'.

Only after such preparatory measures- if / necessarily spreading over a period of many years- will public / attention and public resolution be fully aroused.

  1. AOption (a)

    Only after such preparatory measures-if

  2. BOption (b)

    No Error

  3. COption (c)

    attention and public resolution be fully aroused.

  4. DOption (d)

    necessarily spreading over a period of many years- will public

View answer & explanation

Correct answer

Answer: (d)D.

necessarily spreading over a period of many years- will public

Why this answer is correct

The correct answer is "​necessarily spreading over a period of many years- will public".

Key Points

  • The error in the part of the sentence is that the word "if" and "necessarily" are not correctly used.
  • "if" is used to introduce a condition and "necessarily" means "in a necessary manner" both don't make sense in the context of the sentence. It is likely that the writer intended to use a different conjunction or adverb.
  • To fix this error we have to remove the word "necessarily" from this part and use "Spread" instead of "Spreading"

Correct Sentence: Only after such preparatory measures-if spread over a period of many years will public attention and public resolution be fully aroused.

Question 64

Four words are given, out of which only one word is spelt correctly. Choose the correctly spelt word.

  1. AOption (a)

    RECCESION

  2. BOption (b)

    RECESION

  3. COption (c)

    RECESSION

  4. DOption (d)

    RECCESSION

View answer & explanation

Correct answer

Answer: (c)C.

RECESSION

Why this answer is correct

The correct answer is "RECESSION"

Key Points

  • The correctly spelt word is "RECESSION".
  • A recession is a period of economic decline, typically characterized by a reduction in gross domestic product (GDP), increased unemployment, and a decline in the stock market. During a recession, consumer and business spending decreases, and this reduction in spending can cause businesses to lay off workers and reduce production.(मंदी)
  • For example, the 2008 global financial crisis and the following recession is one of the most significant examples of a recession in recent history. It was caused by the collapse of the housing market and the resulting decline in consumer spending. The crisis led to a significant decrease in GDP and a sharp increase in unemployment, with millions of people losing their jobs. The stock market also experienced a significant decline, with many investors losing a significant amount of money.
  • Thus, the correct spelling is "RECESSION".

Correct answer: RECESSION

Question 65

Select the most appropriate meaning for the word given below.

Torrent

  1. AOption (a)

    A violent or forceful flow of fluid.

  2. BOption (b)

    A surprise violent attack.

  3. COption (c)

    Single track carved out of the mountain.

  4. DOption (d)

    Sudden cyclone.

View answer & explanation

Correct answer

Answer: (a)A.

A violent or forceful flow of fluid.

Why this answer is correct

The correct answer is "Torrent"

Key Points

  • A torrent refers to a strong, fast-moving flow of water or other fluid, such as a stream or river that is flowing rapidly and forcefully, often as a result of heavy rain or melting snow. The term is commonly used to describe a powerful flow of water that can cause damage or destruction if it overflows its banks or breaks through barriers(धार, प्रवाह).
  • For example, During a heavy rainstorm, a small stream might turn into a raging torrent, overflowing its banks and flooding nearby homes and businesses. Or during the melt of heavy snow, a mountain stream may turn into a massive torrent that causes damage to the infrastructure and communities downstream.
  • Thus, the most appropriate meaning of the word "Torrent" is "A violent or forceful flow of fluid."

Correct answer: A violent or forceful flow of fluid.

Question 66

The question below consist of a set of labelled sentences. Out of four options given, select the most logical order of the sentences which form a paragraph.

The Grammys had their origin in the Hollywood Walk of Fame project in the 1950 s.

P. They determined to rectify this by creating awards given by their industry similar to the Oscars and the Emmys.

Q. After deciding to go forward with such awards, a question remained what to call them.

R. One working title was the 'Eddie', to honour Thomas Edison, the invent or of the phonograph.

S. As recording executives on the Walk of Fame committee compiled a list of significant recording industry people who might qualify for a Walk of Fame star, they realized that many leading people in their business would not earn a star on Hollywood Boulevard.

Eventually, the name was chosen after a mail-in contest whereby approximately 300 contestants submitted the name 'Grammy', with the earliest post mark from contest winner Jay Danna of New Orleans, Louisiana, as an abbreviated reference to the gramophone.

  1. AOption (a)

    SPQR

  2. BOption (b)

    QSRP

  3. COption (c)

    PQRS

  4. DOption (d)

    RSQP

View answer & explanation

Correct answer

Answer: (a)A.

SPQR

Why this answer is correct

The correct answer is "​SPQR".

Key Points

  • The correct sequence is "SPQR"
  • The first part should be that best connects the opening of the paragraph**.**
  • S will be the next part after starting as it best connects the opening, this sentence provides the background for the creation of the Grammys, explaining that the committee realized that many important people in the recording industry would not be honored with a star on Hollywood Boulevard.
  • P follows S, because this sentence explains that the recording executives decided to create an award to honor people in their industry, similar to the Oscars and the Emmys.
  • Q follows P, because this sentence states that after deciding to create the awards, the question of what to name them remained.
  • R follows Q because because it best connects the previous sentence and explain what question is remained, and, also it best connects the last given sentence.

Correct arrangement:   The Grammys had their origin in the Hollywood Walk of Fame project in the 1950s. As recording executives on the Walk of Fame committee compiled a list of significant recording industry people who might qualify for a Walk of Fame star, they realized that many leading people in their business would not earn a star on Hollywood Boulevard.They determined to rectify this by creating awards given by their industry similar to the Oscars and the Emmys. After deciding to go forward with such awards, a question remained what to call them. One working title was the 'Eddie', to honour Thomas Edison, the invent or of the phonograph. Eventually, the name was chosen after a mail-in contest whereby approximately 300 contestants submitted the name 'Grammy', with the earliest post mark from contest winner Jay Danna of New Orleans, Louisiana, as an abbreviated reference to the gramophone.

Question 67

Select the most appropriate meaning of the given proverb.

Shards bring luck

  1. AOption (a)

    If something breaks, you will be lucky.

  2. BOption (b)

    The younger ones let the older ones go first.

  3. COption (c)

    A favour is followed by a consideration.

  4. DOption (d)

    A single positive event does not mean that the turnaround has been made.

View answer & explanation

Correct answer

Answer: (a)A.

If something breaks, you will be lucky.

Why this answer is correct

The correct answer is "If something breaks, you will be lucky."

Key Points

  • The proverb "shards bring luck" is a saying that suggests that something that is broken or shattered can bring good luck or fortune(अगर कुछ टूट जाता है, तो आप भाग्यशाली होंगे").
  • This phrase is often used to express that even though something may seem bad or negative at first, it may ultimately lead to something positive. It's meaning is that even in the worst of situations, there is always a chance for good things to happen.
  • For example, If someone breaks a valuable vase, they might say "Well, shards bring luck," to express that even though the vase is broken, it might bring good luck or fortune in some way.
  • Thus, the most appropriate meaning of the given proverb is "If something breaks, you will be lucky".

Correct answer: If something breaks, you will be lucky.

Question 68

The sentences below are not complete. There are four options for each question. Choose the best word to complete the sentence

The best ______ other workers can give is to fight together for better pay ourselves.

  1. AOption (a)

    adoption

  2. BOption (b)

    discord

  3. COption (c)

    dissemination

  4. DOption (d)

    solidarity

View answer & explanation

Correct answer

Answer: (d)D.

solidarity

Why this answer is correct

The correct answer is "Solidarity"

Key Points

  • The best option to complete the sentence is "solidarity."
  • The sentence is talking about working together for better pay, so "solidarity" is the best word choice as it means unity and cooperation among individuals or groups, working towards a common goal.
  • Solidarity: Unity or cooperation among individuals or groups, working towards a common goal(एकजुटता).
  • For example, a labor union uses solidarity among its members to negotiate better wages and working conditions for all employees.
  • Thus, the most appropriate option is "Solidarity"

Correct Sentence: The best solidarity other workers can give is to fight together for better pay ourselves.

Additional Information 

  • Discord: Lack of agreement or harmony, especially among those expected to cooperate(असामंजस्य)..
  • For example, if there is discord among a group of friends, it can cause tension and make it difficult for them to do things together
  • Dissemination: The act of spreading something widely(प्रचार).
  • For example, a company might disseminate information about a new product through a press release, advertising, and social media to reach as many potential customers as possible.
  • Adoption: The act of taking something as one's own(अभिग्रहण).
  • For example, a family might adopt a child, legally making them their own and responsible for their upbringing.

Question 69

Select the most appropriate meaning of the given idiom.

Loose talk

  1. AOption (a)

    Treat someone badly.

  2. BOption (b)

    Uninhibited, indiscreet talk or conversation.

  3. COption (c)

    Keeping secrets.

  4. DOption (d)

    Behaving freely.

View answer & explanation

Correct answer

Answer: (b)B.

Uninhibited, indiscreet talk or conversation.

Why this answer is correct

The correct answer is "Uninhibited, indiscreet talk or conversation."

Key Points

  • "Loose talk" is an idiom that refers to casual or careless speech, particularly when it comes to discussing sensitive or confidential information. It can also refer to gossip or idle chatter that is not productive or appropriate in a given situation(बेहिचक, अविवेकपूर्ण बात या बातचीत).
  • Example: "I wouldn't trust him with company secrets. He has a tendency to engage in loose talk."
  • Thus, the most appropriate meaning of the idiom is "Uninhibited, indiscreet talk or conversation."

Correct answer: Uninhibited, indiscreet talk or conversation."

Question 70

Select the word that is opposite in meaning (ANTONYM) to the word given below.

Liberty

  1. AOption (a)

    Refusal

  2. BOption (b)

    Dependence

  3. COption (c)

    Anarchy

  4. DOption (d)

    Incarceration

View answer & explanation

Correct answer

Answer: (b)B.

Dependence

Why this answer is correct

The correct answer is "Dependence".

Key Points

  • In the above question, we have to choose the correct antonym for the word "LIBERTY".
  • Liberty refers to the state of being free within society from oppressive restrictions imposed by authority on one's way of life, behavior, or political views(स्वतंत्रता).
  • Example: "The citizens of this country enjoy the liberty to speak their minds freely without fear of government retaliation."
  • Dependence refers to the state of relying on someone or something else for support or supplies.(निर्भरता)
  • ​Example: "The country's dependence on foreign oil has become a major concern."
  • Thus, as per the above explanation, the most appropriate antonym of the word is "Dependence".

Therefore the Correct answer is 'option 2'.

Additional Information 

  • Let's understand the other words:
  • Refusal refers to the act of declining or rejecting a request or offer(इनकार).
  • Example: "The company's refusal to comply with the new regulations resulted in a fine."
  • Anarchy refers to a situation or society that lacks governing authority or regulatory systems.
  • Example: "In the dystopian novel, after a global catastrophe that wiped out all existing government structures, the world descended into anarchy.".
  • Incarceration refers to the state of being imprisoned or confined, typically as a punishment for a crime(क़ैद).
  • Example: "The criminal's incarceration lasted for 10 years."

Question 71

The sentence below has been divided into three parts. Select the part of the sentence that has an error. If the sentence has no error, select the option 'No Error'.

The first preliminary criteria to be fulfilled before / the state can claim moral right to fight against all these things is that the young / generation should be afforded facilities for contracting an early marriage.

  1. AOption (a)

    generation should be afforded facilities for contracting an early marriage.

  2. BOption (b)

    The first preliminary criteria to be fulfilled before

  3. COption (c)

    the state can claim moral right to fight against all these things is that the young

  4. DOption (d)

    No Error

View answer & explanation

Correct answer

Answer: (b)B.

The first preliminary criteria to be fulfilled before

Why this answer is correct

The correct answer is "​The first preliminary criteria to be fulfilled before".

Key Points

  • The error lies in the part "The first preliminary criteria to be fulfilled before".
  • The sentence is using "criteria" which is a plural noun, but it is being used with "the first" which is singular. So it should be "The first preliminary criterion" instead of "The first preliminary criteria".
  • Thus, it should be criterion instead of criteria.

Correct Sentence: The first preliminary criterion to be fulfilled before the state can claim the moral right to fight against all these things is that the young generation should be afforded facilities for contracting an early marriage.

Question 72

Select the word segment that substitutes (replaces) the bracketed word segment correctly and completes the sentence meaningfully. Select the option 'no correction required' if the sentence is correct as given.

Our values determine (those metrics under which we measure) ourselves and everyone else.

  1. AOption (a)

    the metrics by which we measure

  2. BOption (b)

    the metrics from which we measure

  3. COption (c)

    No correction required.

  4. DOption (d)

    those metrics from which we measure

View answer & explanation

Correct answer

Answer: (a)A.

the metrics by which we measure

Why this answer is correct

THe correct answer is "the metrics by which we measure".

Key Points

  • The word segment that substitutes the bracketed word segment correctly and completes the sentence meaningfully is 'the metrics by which we measure'
  • Because "by which" indicates the means by which something is done, or the standard by which something is judged or measured, and it makes sense in the context of the sentence.
  • Thus, the most appropriate substitution is "the metrics by which we measure".

Correct Sentence: Our values determine the metrics by which we measure ourselves and everyone else."

Question 73

Select the word that is opposite in meaning (ANTONYM) to the word given below.

HYPOCRITE

  1. AOption (a)

    CHARLATAN

  2. BOption (b)

    INGENUOUS

  3. COption (c)

    PHONY

  4. DOption (d)

    IMPOSTER

View answer & explanation

Correct answer

Answer: (b)B.

INGENUOUS

Why this answer is correct

The correct answer is "​INGENUOUS".

Key Points

  • In the given question, we have to choose the word that is opposite to the meaning of the word "HYPOCRITE".
  • Hypocrite refers to a person who pretends to have virtues, moral or religious beliefs, principles, etc., that he or she does not actually possess, especially a person whose actions belie stated beliefs(

पाखंडी).

  • Example: "He always claimed to care about the environment, but his actions proved him to be a hypocrite."
  • Ingenuous refers to a person who is innocent and unsuspecting; simple; credulous(निष्कपट).
  • Example: "The ingenuous child believed everything the stranger told them."
  • Thus, as per the above explanation, the most appropriate antonym of the word is "INGENUOUS"

Correct answer: INGENUOUS

Additional Information

  • Let's understand the other given words.
  • Charlatan refers to a person who makes elaborate, fraudulent, and often voluble claims to skill or knowledge, typically in a field like medicine, spirituality, or the occult, but who essentially lacks the ability or knowledge that he or she claims to possess(पाखंडी).
  • Example: "The self-proclaimed psychic was exposed as a charlatan when it was revealed that they had no real abilities."
  • Phony refers to a person or thing that is not genuine; fake(जाली).
  • Example: "The watch was a phony, made of cheap materials and not worth the price."
  • Impostor refers to a person who pretends to be someone else in order to deceive others, often for fraudulent gain.
  • Example: "The impostor successfully posed as a wealthy businessman for several years before being discovered(ढोंग)."

Question 74

Some parts of a sentence have been jumbled up, and labelled P, Q, R and S. Select the option that gives the correct sequence in which these parts can be rearranged to form a meaningful and grammatically correct sentence.

The EE British Academy Film Awards

 

P. British Academy of Film and

Q. or BAFTA Film Awards are presented in

R. Television Arts (BAFTA) to honour the

S. an annual award show hosted by the

 

best British and international contributions to film.

  1. AOption (a)

    PQRS

  2. BOption (b)

    QSPR

  3. COption (c)

    SRQP

  4. DOption (d)

    RQSP

View answer & explanation

Correct answer

Answer: (b)B.

QSPR

Why this answer is correct

The correct answer is "​QSPR".

Key Points

  • The correct sequence is "QSPR"
  • ​The fist sentence should be that best connects the starting of the paragraph "The EE British Academy Film Awards"
  • Q will be the first sentence after starting because it best connects the starting and tells about the other name of the awards.
  • S follows Q, because this part tells us about the event, an annual award show, and who is hosting it.
  • P follows S, because this part further tells us the name of the organization that is hosting the award show.
  • R follows P, because, this part tells us the purpose of the award show and tells that the award show is hosted by the British Academy of Film and Television Arts (BAFTA) and also it best connects the last sentence.

Correct arrangement:  The EE British Academy Film Awards or BAFTA Film Awards are presented in an annual award show hosted by the Television Arts (BAFTA) to honor the best British and international contributions to film.

Question 75

Select the word that is similar in meaning (SYNONYM) to the word given below.

PRIVILEGE

  1. AOption (a)

    PREROGATIVE

  2. BOption (b)

    RESPONSIBILITY

  3. COption (c)

    DISPENSATION

  4. DOption (d)

    DETRIMENT

View answer & explanation

Correct answer

Answer: (a)A.

PREROGATIVE

Why this answer is correct

The correct answer is "PREROGATIVE"

Key Points

  • In the give question, we have to choose the most appropriate synonym of the word "PRIVILEGE"
  • Privilege refers to a special right, advantage, or immunity granted or available only to a particular person or group of people(विशेषाधिकार,).
  • Example: "Being able to afford private schooling is a privilege that not everyone has."
  • Prerogative refers to a right or privilege exclusive to a particular individual or class(विशेषाधिकार,).
  • Example: "It is the prerogative of the President to veto any bill passed by Congress."
  • Thus, the most appropriate synonym of the word is "Prerogative"

Correct answer: Prerogative

Additional Information

  • Let's understand the other words.
  • Responsibility refers to the state or fact of having a duty to deal with something or of having control over someone(उत्तरदायित्व).
  • Example: "As a parent, it is my responsibility to ensure that my child is safe and well-cared for."
  • Dispensation refers to an exemption or release from a rule or obligation(व्यवस्था).
  • Example: "The employee was given a dispensation from the dress code policy for medical reasons."
  • Detriment refers to a harmful effect or damage(निरोध).
  • Example: "The company's disregard for safety regulations resulted in a detriment to both employees and the environment."

A passage is given with 5 questions following it. Read the passage carefully and choose the best answer to each question out of the four alternatives.

Most people use social media in one form or another. While there is nothing inherently wrong with that, and while social media can sometimes be beneficial, it's important to be aware that social media is associated with a number of issues and potential dangers, including stress, anxiety, loneliness, and depression. Understanding the dangers of social media is important, both so you can deal with them yourself, and so you can help others deal with them. As such, in the following article you will learn about the issues that are associated with social media, see who is most vulnerable to them, and find out what you can do to deal with them effectively. There are many reasons why people keep using social media even when it's bad for them, and these reasons vary across individuals and across circumstances. One notable reason why people continue to use social media even though it affects them negatively is that they're simply unaware of its harmful influence. Furthermore, in some cases, people are aware of the harmful influence that social media has on them, but they don't care enough about these dangers to want to change their behaviour. However, many people continue to use social media even though they know it's bad for them and even though they want to stop, because they're psychologically predisposed to keep using it. For example, one study found that people keep using Facebook despite the fact that it makes them feel bad, because they keep expecting it to make them feel better.

Question 76

Based on Passage: View directions & passage

What is the meaning of 'psychologically predisposed' from the above passage?

  1. AOption (a)

    Emotional addiction to the used of social media.

  2. BOption (b)

    Mentally preoccupied with the contents of social media.

  3. COption (c)

    Emotionally or mentally unavailable to other people around them.

  4. DOption (d)

    Mentally or emotionally inclined to the use of social media.

View answer & explanation

Correct answer

Answer: (d)D.

Mentally or emotionally inclined to the use of social media.

Why this answer is correct

The correct answer is "Mentally or emotionally inclined to the use of social media.".

Key Points

  • Psychologically predisposed refers to a person's tendency or inclination to think, feel, or behave in a certain way as a result of their psychological makeup or past experiences. It suggests that a person may be more likely to exhibit certain behaviors or reactions due to their psychological characteristics or history.
  • For example, someone who is psychologically predisposed to anxiety may be more likely to experience anxiety in certain situations. This is different from a person who may experience anxiety due to a specific trigger or event.
  • This passage suggests, many people continue to use social media even though they know it's bad for them and even though they want to stop, because they're psychologically predisposed to keep using it.
  • Thus, the correct answer is "Mentally or emotionally inclined to the use of social media".

Correct answer: Mentally or emotionally inclined to the use of social media.

Question 77

Based on Passage: View directions & passage

Three assertions are given below (A, B & C) on the basis of your understanding of the passage, which one of the options is correct?

A. There are a number of potential dangers and issues associated with social media.

B. People continue using social media despite knowing its ill effects.

C. People use Facebook with the expectation that it will make them feel better.

  1. AOption (a)

    A & B are correct

  2. BOption (b)

    All are correct.

  3. COption (c)

    B & C are correct.

  4. DOption (d)

    Only B is correct.

View answer & explanation

Correct answer

Answer: (b)B.

All are correct.

Why this answer is correct

The correct answer is "All are correct."

Key Points

  • Let's consider the following lines from the passage.
  • it's important to be aware that social media is associated with a number of issues and potential dangers, including stress, anxiety, loneliness, and depression.
  • However, many people continue to use social media even though they know it's bad for them and even though they want to stop, because they're psychologically predisposed to keep using it.
  • one study found that people keep using Facebook despite the fact that it makes them feel bad, because they keep expecting it to make them feel better..
  • These points clearly mentioned that there are a number of potential dangers and issues associated with social media, People continue using social media despite knowing its ill effects. People use Facebook with the expectation that it will make them feel better.
  • Thus, all the statements are correct.

Correct answer: All are correct

Question 78

Based on Passage: View directions & passage

On the basis of your reading of the above passage, select the most appropriate heading for the same.

  1. AOption (a)

    Ban Facebook

  2. BOption (b)

    Dangers of using Social Media

  3. COption (c)

    Subconscious Mind

  4. DOption (d)

    Social media- a boon to society

View answer & explanation

Correct answer

Answer: (b)B.

Dangers of using Social Media

Why this answer is correct

The correct answer is "Dangers of using social media"

Key Points

  • On the basis of the reading of the above passage the most appropriate heading for the passage is ""Dangers of using Social Media".
  • The title "Dangers of using Social Media" is most appropriate title because the passage is highlighting the potential negative impact of social media usage on individuals' mental health and well-being, and the importance of being aware of these dangers in order to mitigate them. The passage goes on to provide examples of the specific issues associated with social media, and discuss the reasons why people continue to use it despite its negative effects.

Question 79

Based on Passage: View directions & passage

Which of the following is not associated with health issues and potential dangers of using social media?

  1. AOption (a)

    Stress

  2. BOption (b)

    Anxiety

  3. COption (c)

    Sleeplessness

  4. DOption (d)

    Depression

View answer & explanation

Correct answer

Answer: (c)C.

Sleeplessness

Why this answer is correct

The correct answer is "Sleeplessness".

Key Points

  • Refer to the line:  While there is nothing inherently wrong with that, and while social media can sometimes be beneficial, it's important to be aware that social media is associated with a number of issues and potential dangers, including stress, anxiety, loneliness, and depression.
  • There is no mention of the "Sleeplessness" as a health issue and potential danger of using social media.
  • Thus, the correct answer is "Sleeplessness"

Correct answer: Sleeplessness

Question 80

Based on Passage: View directions & passage

Three assertions are given below (A, B & C) on the basis of your understanding of the passage, which one of the options is correct?

A. Understanding the dangers of social media is important so that one can help themselves and others around them.

B. People do not care enough about the dangers of social media so as to change their own behaviour.

C. Social media is always unhealthy and dangerous.

  1. AOption (a)

    Only A is correct.

  2. BOption (b)

    B & C are correct.

  3. COption (c)

    All are correct.

  4. DOption (d)

    A & B are correct

View answer & explanation

Correct answer

Answer: (d)D.

A & B are correct

Why this answer is correct

The correct answer is "A & B are correct".

Key Points

  • Let's consider the following lines.
  • Understanding the dangers of social media is important, both so you can deal with them yourself, and so you can help others deal with them.
  • In some cases, people are aware of the harmful influence that social media has on them, but they don't care enough about these dangers to want to change their behaviour.
  • The passage suggts that understanding the dangers of social media is important so that one can help themselves and others around them and People do not care enough about the dangers of social media so as to change their own behaviour.
  • The third assertions is incorrect because social media is not always unhealthy and dangerous, in fact it is very useful for some good purposes.
  • Hence, A and B are correct.

Correct answer: A and B are correct

General Hindi (20 questions)

Question 81

निम्नलिखित प्रश्न में ,चार विकल्पों में से उस विकल्प का चयन करें जो दिये गए संधि - विच्छेद शब्द का सही संधि का विकल्प हो।

इति + आदि -

  1. AOption (a)

    इत्याद

  2. BOption (b)

    इत्यादि

  3. COption (c)

    इत्यादी

  4. DOption (d)

    इत्यदि

View answer & explanation

Correct answer

Answer: (b)B.

इत्यादि

Why this answer is correct

दिए गए सभी विकल्पों में सही उत्तर ' इत्यादि है। 

Key Points

  • इति + आदि में "यण स्वर संधि" संधि है। इति + आदि की संधि "इत्यादि" होता है। तथा इसमें “यण स्वर संधि” लागू होती है।
  • ' इति + आदि ' = इत्यादि यण संधि का उदाहरण है। ह्रस्व या दीर्घ स्वर - इ, ई , उ , ऊ या ऋ के बाद यदि कोई भी असमान स्वर आता है तो इ / ई का ' य ' उ / ऊ का ' व ' तथा ऋ का ' र् ' हो जाता है इसे यण संधि कहते हैं।
  • जैसेः
  • इ + अ = य : यदि + अपि = यद्यपि
  • इ + आ = या : इति + आदि = इत्यादि
  • इ + ऊ = यू : उपरि + उक्त = उपर्युक्त
  • इ + ए = ये : प्रति + एक = प्रत्येक
  • उ + अ = व : सु + अच्छ = स्वच्छ

Additional Information

  • संधि के भेद
  • हिंदी व्याकरण में संधि मुख्य रूप से तीन प्रकार की होती हैं- स्वर संधि, व्यंजन संधि, और विसर्ग संधि।
  • स्वर संधि
  • व्यंजन संधि
  • विसर्ग संधि
  • स्वर संधि
  • जब स्वर के साथ स्वर का मेल होता है तब जो परिवर्तन होता है उसे स्वर संधि कहते हैं। हिंदी में स्वरों की संख्या ग्यारह होती है। बाकी के अक्षर व्यंजन होते हैं। जब दो स्वर मिलते हैं जब उससे जो तीसरा स्वर बनता है उसे स्वर संधि कहते हैं।
  • स्वर संधि के प्रकार
  • दीर्घ संधि
  • गुण संधि
  • वृद्धि संधि
  • यण संधि
  • अयादि संधि
  • व्यंजन संधि
  • जब व्यंजन को व्यंजन या स्वर के साथ मिलाने से जो परिवर्तन होता है , उसे व्यंजन संधि कहते हैं।
  • विसर्ग संधि
  • विसर्ग के बाद जब स्वर या व्यंजन आ जाये तब जो परिवर्तन होता है उसे विसर्ग संधि कहते हैं।

Question 82

निम्नलिखित प्रश्न में, चार विकल्पों में से उस विकल्प का चयन करें जो दिए गए मुहावरे का सही विकल्प है।

अढाई दिन की हुकूमत -

  1. AOption (a)

    कुछ दिनों की शानो-शौकत

  2. BOption (b)

    बात से फिर जाना

  3. COption (c)

    मौके की तलाश में होना

  4. DOption (d)

    मौत के समीप होना

View answer & explanation

Correct answer

Answer: (a)A.

कुछ दिनों की शानो-शौकत

Why this answer is correct

दिए गए सभी विकल्पों में सही उत्तर 'कुछ दिनों की शानो-शौकत' है। 

 Key Points

  • 'अढाई दिन की हुकूमत' मुहावरे का सही अर्थ है 
  • कुछ दिनों की शानो-शौकत
  • वाक्य प्रयोग –
  • जनाब, जरा होशियारी से काम लें। यह अढाई दिन की हुकूमत जाती रहेगी।
  • अन्य विकल्पों का विश्लेषण:-
  • बात से फिर जाना - के लिए मुहावरा है - थूक कर चाटना

वाक्य प्रयोग – महेश तो थूक कर चाटने वालो में से है, वह हमेशा शराब की लत को छोड़ने के वादे करता है और अगले ही दिन शराब पी लेता है।

  • मौके की तलाश में होना - के लिए मुहावरा है - ताक में बैठना​                                                                                                                                                       

 वाक्य प्रयोग - पुलिस ताक में बैठी है कि अपराधी दोबारा गलती करे और उसे रंगे हाथ पकड़ा जाए।

  • ​ मौत के समीप होना** -** के लिए मुहावरा है **- अंतिम घड़ी आना'**​​                                                                                                                                               

वाक्य प्रयोग - डॉक्टरों ने भी अब कह दिया है कि इनकी अंतिम घड़ी आ गयी है।

Additional Information

  • मुहावरा” अरबी भाषा का शब्द है जिसका अर्थ होता है “आदि होना” या “अभ्यास होना” l
  • हम अपने दैनिक जीवन में कहीं बार अपने मन के भाव या विचारों को मुहावरों के शब्दों का प्रयोग करके प्रकट करते हैं। मुहावरे पूर्ण रूप से स्वतंत्र नहीं होते इनका प्रयोग वाक्य सौंदर्य को बढ़ाने में किया जाता है।

Question 83

निम्नलिखित प्रश्न में ,चार विकल्पों में से उस विकल्प का चयन करें जो दिए गए शब्द का समानार्थी शब्द नहीं है।

दिल्लगी -

  1. AOption (a)

    परिहास

  2. BOption (b)

    तरफ

  3. COption (c)

    मज़ाक

  4. DOption (d)

    ठिठोली

View answer & explanation

Correct answer

Answer: (b)B.

तरफ

Why this answer is correct

दिए गए सभी विकल्पों में सही उत्तर 'तरफ' है। 

Key Points

  • 'दिल्लगी' का समानार्थी शब्द -  मज़ाक, मनोविनोद, हवाख़ोरी, ताज़ापन, ताज़गी, प्रसन्नता, ठट्ठा, मसख़री, मसख़रापन, रखैल, ठठोली, ठोली, हँसी, परिहास, प्रहसन, हास-परिहास, मनोरंजन, विनोद, क्रीड़ा, हँसी-ठट्ठा, हँसी-मज़ाक, मनबहलाव, हास्योक्ति ।
  • इन शब्दों में अर्थ की समानता होते हुए भी इनके प्रयोग एक तरह के नहीं हैं।
  • ये शब्द अपने में इतने पूर्ण हैं कि एक ही शब्द का प्रयोग सभी स्थितियों में और सभी स्थलों पर अच्छा नहीं लगता- कहीं कोई शब्द ठीक बैठता है और कहीं कोई।
  • प्रत्येक शब्द की महत्ता विषय और स्थान के अनुसार होती है ।

Additional Information

  • पर्यायवाची शब्द की परिभाषा - एक ही अर्थ में प्रयुक्त होने वाले शब्द जो बनावट में भले ही अलग हों, पर्यायवाची या समानार्थी शब्द कहलाते हैं।
  • उदाहरण - नृप- राजा, भूप, भूपति काली-भगवती, चंडी, अम्बा ।

Question 84

निम्नलिखित प्रश्न में, चार विकल्पों में से उस विकल्प का चयन करें जो उपसर्ग शब्द का सही विकल्प हो l

  1. AOption (a)

    दयावंत

  2. BOption (b)

    सहर्ष

  3. COption (c)

    तैराक

  4. DOption (d)

    भारी

View answer & explanation

Correct answer

Answer: (b)B.

सहर्ष

Why this answer is correct

दिए गए सभी विकल्पों में सही उत्तर ' सहर्ष' है। 

Key Points

  • सहर्ष : स + हर्ष
  • स : उपसर्ग
  • हर्ष : मूल शब्द
  • व्याख्या : उपसर्ग वे शब्दांश होते है, जो किसी शब्द के शुरु में लगते हैं। उपसर्ग उस शब्द के आगे लगकर एक तरह से विशेषण का कार्य करते हैं। उपसर्ग लगते ही उस शब्द या तो अर्थ बदल जाता है, या उस शब्द का अर्थ और अधिक विस्तृत हो जाता है।

उपसर्ग को अंग्रेजी में प्रिफिक्स (Prefix) कहते हैं

  • आ, वि, ना, स उपसर्ग से दो-दो शब्द बनाइए।
  • उ���्तर: आ-आजन्म, आकाश वि-विनय, विजयना-नाराज, नादानस–सहर्ष, सहित।
  • तैराक - तैरना + अक = तैराक
  • भारी - भार + ई = भारी
  • दयावंत - दया + वंत = दयावंत

Additional Information

  • उपसर्ग
  • जो शब्दांश किसी मूल शब्द के पहले जुड़कर उसके अर्थ मेँ परिवर्तन या विशेषता उत्पन्न कर देते हैँ, उन शब्दांशोँ को उपसर्ग कहते हैँ। जैसे –
  • ‘हार’ एक शब्द है। इस शब्द के पहले यदि 'सम्, वि, प्र', उपसर्ग जोड़ दये जायेँ, तो– सम्+हार = संहार, वि+हार = विहार, प्र+हार = प्रहार तथा उप+हार = उपहार शब्द बनेँगे। यहाँ उपसर्ग जुड़कर बने सभी शब्दोँ का अर्थ ‘हार’ शब्द से भिन्न है।
  • उपसर्गों का अपना स्वतंत्र अर्थ नही होता, मूल शब्द के साथ जुड़कर ये नया अर्थ देते हैँ। इनका स्वतंत्र प्रयोग नहीँ होता। जब किसी मूल शब्द के साथ कोई उपसर्ग जुड़ता है तो उनमेँ संधि के नियम भी लागू होते हैँ। संस्कृत उपसर्गोँ का अर्थ कहीँ–कहीँ नहीँ भी निकलता है। जैसे – ‘आरम्भ’ का अर्थ है– शुरुआत। इसमेँ ‘प्र’ उपसर्ग जोड़ने पर नया शब्द ‘प्रारम्भ’ बनता है जिसका अर्थ भी ‘शुरुआत’ ही निकलता है।

Question 85

निम्नलिखित प्रश्न में, चार विकल्पों में से उस विकल्प का चयन करें जो दिए गए वाक्य के काल के भेद का सही विकल्प है।

लड़कियाँ गाना गा रही हैं।

  1. AOption (a)

    पूर्ण भूत काल

  2. BOption (b)

    सामान्य वर्तमान काल

  3. COption (c)

    अपूर्ण वर्तमान काल

  4. DOption (d)

    संभाव्य भविष्यत् काल

View answer & explanation

Correct answer

Answer: (c)C.

अपूर्ण वर्तमान काल

Why this answer is correct

दिए गए विकल्पों में से सही 'अपूर्ण वर्तमान काल' है।

Key Points

  • 'जो क्रिया अभी हो रही है' उसे अपूर्ण वर्तमान कहते हैं।
  • अपूर्ण का अर्थ होता है – अधूरा।
  • क्रिया के जिस रूप से कार्य के लगातार होने का पता चलता है, उसे अपूर्ण वर्तमान काल कहते है।
  • जिन वाक्यों के अंत में रहा है, रहे है, रही है, रहा हूँ आदि आते है, उसे अपूर्ण वर्तमान काल कहते हैं।
  • लड़कियां गाना गा रही हैं । - इस वाक्य में अपूर्ण वर्तमान काल इसलिए हैं क्योकि इसमें कार्य के अधूरा होने का बोध हो रह��� हैं और कार्य पूरा नहीं हुआ हैं ।

अन्य विकल्प:

  • पूर्ण भूतकाल:- क्रिया के जिस रूप से यह ज्ञात हो कि कार्य पहले ही पूरा हो चुका है, उसे पूर्ण भूतकाल कहते हैं।
  • ​जैसे- उसने श्याम को मारा था।​
  • सामान्य वर्तमान काल:- जिस क्रिया से क्रिया के सामान्य रूप का वर्तमान में होने का पता चलता है, उसे सामान्य वर्तमान काल कहते हैं।​
  • ​जैसे- गरिमा कविता लिखती है।​
  • ​संभाव्य भविष्यत् काल:- क्रिया के जिस रूप से उसके भविष्य में होने की संभावना का पता चलता है, वहाँ संभाव्य भविष्यतकाल होता है।
  • ​जैसे- संभव है कि कल हम सिनेमा देखने जायेगे ।

Additional Information

काल - क्रिया के जिस रुप से कार्य व्यापार के समय तथा उसकी पूर्णता पूर्णता का बोध होता है, उसे काम कहते हैं। काल के तीन भेद माने गए हैं।
वर्तमान कालवर्तमान काल में क्रिया व्यापार की निरंतरता रहती है। इसके चार भेद माने गए हैं - सर्वमान्य वर्तमान, तात्कालिक, वर्तमान संदिग्ध वर्तमान और संभाव्य वर्तमान।उदाहरण सीता पढ़ती है।
भूत कालभूतकाल में क्रिया व्यापार की समाप्ति का बोध होता है। इसके पांच भेद हैं - सामान्य भूत, आसन भूत, अपूर्ण भूत, संदिग्ध भूत हेतुहेतुमद् भूत।उदाहरण सीता ने पढ़ाई की।
भविष्य कालभविष्य में होने वाली क्रिया का बोध भविष्य काल से होता है। इसके तीन भेद हैं-सामान्य भविष्य, संभाव्य भविष्य और हेतुहेतु मद् भविष्य।उदाहरण सीता पढ़ेगी।

Question 86

निम्नलिखित प्रश्न में ,चार विकल्पों में से उस विकल्प का चयन करें जो वाक्यांश के लिए प्रयुक्त एक शब्द का सही विकल्प हो।

दुसरों के काम में हाथ डालना-

  1. AOption (a)

    हस्तक्षेप

  2. BOption (b)

    पाथेय

  3. COption (c)

    चालबाज

  4. DOption (d)

    अडंगा

View answer & explanation

Correct answer

Answer: (a)A.

हस्तक्षेप

Why this answer is correct

सही उत्तर है हस्तक्षेप। 

Key Points

  • हस्तक्षेप का अर्थ है दूसरों के काम में हाथ डालना।
  • वाक्य में प्रयोग: खेत को लेकर दोनों भाइयों में विवाद इतना बढ़ गया कि पड़ोसियों को हस्तक्षेप करना पड़ा।
  • पर्यायवाची शब्द: दखलंदाज़ी।

​ 

Additional Informationअन्य विकल्प: 

  • पाथेय: वह भोजन जो पथिक रास्ते में खाने के लिए बांधकर ले जाता है।
  • चालबाज़: कपट भरी चालाकियां करने वाला।
  • अड़ंगा: रुकावट।

Important Points

  • जब किसी वाक्य में प्रयुक्त या स्वतन्त्र किसी वाक्यांश के लिए किसी एक शब्द का प्रयोग किया जाता है,
  • जो उस वाक्यांश के अर्थ को पूरी तरह सिद्ध करता हो तो उसे ;वाक्यांश के लिए एक शब्द कहते हैं।
  • वाक्यांश के लिए एक शब्द के उदाहरण- 
  • देखने योग्य- दर्शनीय
  • दूसरे के मन की बात जानने वाला- अन्तर्यामी
  • दस वर्षो का समय- दशक
  • दूर की बात सोचनेवाला- दूरदर्शी

Question 87

निम्नलिखित प्रश्न में, चार विकल्पों में से उस विकल्प का चयन करें जो सही विराम चिह्न युक्त वाक्य का सही विकल्प है।

  1. AOption (a)

    पानी पीकर राधा की माँ बोली -"महाराज, इसमे तो दवाई की गंध आ रही है।"

  2. BOption (b)

    ��ानी पीकर राधा की माँ बोली -महाराज, इसमे तो दवाई की गंध आ रही है।"

  3. COption (c)

    पानी पीकर राधा की माँ बोली -"महाराज, इसमे तो दवाई की गंध आ रही है ;

  4. DOption (d)

    पानी पीकर राधा की माँ बोली -"महाराज, इसमे तो दवाई की गंध आ रही है।

View answer & explanation

Correct answer

Answer: (a)A.

पानी पीकर राधा की माँ बोली -"महाराज, इसमे तो दवाई की गंध आ रही है।"

Why this answer is correct

सही उत्तर है- पानी पीकर राधा की माँ बोली -"महाराज, इसमे तो दवाई की गंध आ रही है।"

 Key Points

  • प्रस्तुत वाक्य में निर्देशक चिन्ह(-), अवतरण चिन्ह("...")तथा पूर्णविराम(। ) का प्रयोग किया गया है।
  • निर्देशक चिह्न (–) - किसी व्यक्ति के द्वारा कहे गए कथन  को अधिकृत करने से पहले निर्देशक चिह्न का प्रयोग किया जाता है।
  • अवतरण चिन्ह-( “…” )- किसी वाक्य के खास शब्द पर जोर देने के लिए अवतरण चिह्न (”…”) का प्रयोग किया जाता है। इसे उद्धरण चिन्ह भी कहा जाता है।
  • पूर्ण विराम-(।)-  लिखते वक़्त वाक्य ख़त्म होने के बाद पूर्ण विराम लगाते है।

Additional Information

  • विराम चिन्ह – भाषा के लिखित रूप में विशेष स्थानों पर रुकने का संकेत करने वाले चिन्हों को विराम चिन्ह कहते हैं।​
  • हिंदी के कुछ महत्वपूर्ण विराम चिन्ह: 
  • अर्द्ध विराम चिन्ह (;)- जहाँ अल्प विराम से कुछ अधिक ठहरते हैं तथा पूर्ण विराम से कम ठहरते हैं, वहाँ अर्द्ध विराम का चिन्ह ( ; ) लगाया जाता है।
  • मुझे लगा वह मेरा हितैषी है; वह तो आस्तीन का सांप निकला।
  • अल्प विराम चिन्ह (,)- वाक्य में जहाँ थोड़ा रुकना हो या अधिक वस्तुओं, व्यक्तियों आदि को अलग करना हो वहाँ अल्प विराम (,) चिन्ह का प्रयोग किया जाता है।
  • रमेश, राघव और राज कल जयपुर जायेंगे।
  • प्रश्नवाचक चिन्ह (?)- बातचीत के दौरान जब किसी से कोई बात पूछी जाती है अथवा कोई प्रश्न पूछा जाता है, तब वाक्य के अंत में प्रश्नसूचक-चिन्ह का प्रयोग किया जाता है।
  • तुम्हारी माताजी का नाम क्या है ?
  • विस्मयादिबोधक चिन्ह (!)- इसका प्रयोग हर्ष, विवाद, विस्मय, घृणा, आश्रर्य, करुणा, भय, संबोधन इत्यादि का बोध कराने के लिए इस चिन्ह का प्रयोग किया जाता है।
  • वाह! कितना सुंदर दृश्य है।
  • लाघव चिन्ह (०)- किसी बड़े तथा प्रसिद्ध शब्द को संक्षेप में लिखने के लिए उस शब्द का पहला अक्षर लिखकर उसके आगे शून्य (०) लगा देते हैं। यह शून्य ही लाघव-चिन्ह कहलाता है।
  • डॉक्टर – डॉ०, प्रोफेसर – प्रो०

Question 88

निम्नलिखित में से सही तद्भव-तत्सम युग्म की पहचान कीजिये।

  1. AOption (a)

    कई-कति

  2. BOption (b)

    कूदना-कुक्षि

  3. COption (c)

    कंकती-कंघी

  4. DOption (d)

    इलायची-अम्लिका

View answer & explanation

Correct answer

Answer: (a)A.

कई-कति

Why this answer is correct

सही उत्तर है - कई-कति। 

Key Points

  • कति का तद्भव रूप है कई।
  • अन्य विकल्प:
  • | तद्भव | तत्सम | | --- | --- | | कोख | कुक्षि | | कंघी | कंकति | | इलाइची | एला | | इमली | अम्लिका |

Additional Information

  • तत्सम शब्द- ऐसे शब्द जो संस्कृत से ज्यों-के-त्यों हिंदी में ले लिए गए हैं।
  • जैसे- आम्र, अग्नि, ज्येष्ठ, नव, अक्षि, हस्त, रात्रि, वानर, इत्यादि।
  • तद्भव शब्द- ऐसे शब्द जिनका संस्कृत से हिंदी में आने पर उनका रूप बदल गया।
  • जैसे- आम, आग, जेठ, नौ, आँख, हाथ, रात, बन्दर, इत्यादि।

Question 89

निम्नलिखित प्रश्न में, चार विकल्पों में से उस विकल्प का चयन करें जो शुद्ध शब्द का सही विकल्प हो।

  1. AOption (a)

    अनुदित

  2. BOption (b)

    मोलिक

  3. COption (c)

    आधुनीक

  4. DOption (d)

    सार्वजनिक

View answer & explanation

Correct answer

Answer: (d)D.

सार्वजनिक

Why this answer is correct

सही उत्तर है - सार्वजनिक। 

Key Pointsअन्य विकल्प:

अशुद्धशुद्ध
अनुदितअनूदित
मोलिकमौलिक
आधुनीकआधुनिक

 

Additional Informationवर्तनी- 

  • लिखने की रीति को वर्तनी कहते हैं।
  • ‘वर्तनी’ शब्द का अर्थ ‘पीछे चलना’ है।
  • उच्चारित होने वाले शब्द के लेखन में प्रयोग होने वाले लिपि चिह्नों के व्यवस्थित रूप को वर्तनी कहा जाता है।

Question 90

निम्नलिखित प्रश्न में ,चार विकल्पों में से उस विकल्प का चयन करें जो दिये गए शब्द का सही तत्सम शब्द है।

उलाहना

  1. AOption (a)

    शिकवा

  2. BOption (b)

    शिकायत

  3. COption (c)

    उपालंभ

  4. DOption (d)

    अवक्षेपण

View answer & explanation

Correct answer

Answer: (c)C.

उपालंभ

Why this answer is correct

सही उत्तर है उपालंभ।  

Key Points

  • उपालंभ का अर्थ है उलाहना।
  • पर्यायवाची शब्द- शिकायत, ताना, गिला, शिकवा।

Additional Information

  • अन्य विकल्प: 
  • अवक्षेपण का अर्थ है नीच फेंकना या गिराना।

Important Points

  • शिकायत और शिकवा उलाहना के पर्यायवाची हैं किन्तु ये संस्कृत के शब्द नहीं हैं।
  • तत्सम शब्द वे शब्द होते हैं जो संस्कृत भाषा से ज्यों के त्यों हिंदी में लिए जाते हैं।

Question 91

निम्नलिखित प्रश्न में, चार विकल्पों में से उस विकल्प का चयन करें जो दिए गए शब्द -युग्म का सही अर्थ वाला विकल्प हो।

ताप- तप

  1. AOption (a)

    सधवा, सनातन

  2. BOption (b)

    अज्ञ, शोभा

  3. COption (c)

    आंच, तपस्या

  4. DOption (d)

    कंधा, आग

View answer & explanation

Correct answer

Answer: (c)C.

आंच, तपस्या

Why this answer is correct

सही उत्तर है ताप- तप। 

Key Points

  • ताप का अर्थ है आंच।
  • तप का अर्थ है तपस्या।
  • अन्य विकल्प:
  • अज्ञ: मूर्ख, यज्ञ: हवन
  • शोभा: बहार, आँगन में - बाहर

Additional Information

  • हिंदी व्याकरण में अनेक प्रकार के ऐसे शब्द होते हैं जिन का उच्चारण एक समान होता है।
  • परंतु उनके अर्थ अलग-अलग होते हैं। तो इसे शब्द-युग्म या समोच्चरित भिन्नार्थक शब्द कहा जाता है।

कुछ शब्द युग्म:-

  • अनल - आग, अनिल - हवा
  • निर्जर - देवता, निर्झर - झरना
  • सुचि - सुई, सूची - तालिका
  • तरणि - सूर्य, तरणी - नाव
  • अलि- भौंरा, आली- सखी

Question 92

निम्नलिखित प्रश्न में ,चार विकल्पों में से उस विकल्प का चयन करें जो दिए गए वाक्य के प्रकार के भेद का सही विकल्प है।

अब आप जा सकते हैं

  1. AOption (a)

    आज्ञावाचक वाक्य

  2. BOption (b)

    इच्छावाचक

  3. COption (c)

    विस्मयादि वाचक

  4. DOption (d)

    संकेत वाचक

View answer & explanation

Correct answer

Answer: (a)A.

आज्ञावाचक वाक्य

Why this answer is correct

सही उत्तर है आज्ञावाचक वाक्य। 

Key Points

  • आज्ञावाचक वाक्य: जिन वाक्यों में आज्ञा या प्रार्थना करने का बोध होता है वे आज्ञावाचक वाक्य कहलाते हैं।
  • उदहारण: अभी यहाँ आ जाओ।

Important Pointsअन्य विकल्प:

  • इच्छावाचक वाक्य - जिन वाक्यों में किसी इच्छा अथवा शुभकामना आदि का बोध होता है वे वाक्य इच्छवाचक कहलाते हैं।
  • उदहारण: आपकी यात्रा मंगलमय हो।
  • विस्मयादिवाचक वाक्य - जिन वाक्यों में आश्चर्य, शोक, घृणा, ख़ुशी, स्तब्धता होती है वे विस्मयादिबोधक वाक्य होते हैं।
  • उदहारण: वाह! कितना सुन्दर चित्र है।
  • संकेतवाचक वाक्य- जिन वाक्यों में क्रिया एक दूसरे पर निर्भर हो या किसी संकेत का बोध होता है वे वाक्य संकेत वाचक होते हैं।
  • उदहारण: यदि बारिश हुई, तो मैं नहीं आ पाऊँगी।

Additional Informationअर्थ के आधार पर वाक्य के अन्य भेद:

  • विधानवाचक वाक्य - ऐसे वाक्य जिनसे किसी काम के होने या किसी के अस्तित्व का बोध हो। इन्हें विधिवाचक वाक्य भी कहते हैं।
  • उदहारण: वह कल आ जायेगा।
  • निषेधवाचक वाक्य - जिन वाक्यों में किसी काम के न होने का बोध होता है वे निषेधवाचक वाक्य कहलाते हैं।
  • उदहारण: मैं खाना नहीं खाऊंगा।
  • प्रश्नवाचक वाक्य- जिन वाक्यों में किसी से प्रश्न करने का बोध हो वे प्रश्न वाचक वाक्य कहलाते हैं।
  • उदहारण: क्या तुम दिल्ली जाओगे?
  • संदेहवाचक वाक्य- जिन वाक्यों में किसी प्रकार का संदेह उत्पन्न हो वे संदेहवाचक वाक्य कहलाते हैं।
  • उदहारण: विनय शायद बाज़ार जाये।

Question 93

निम्नलिखित प्रश्न में ,चार विकल्पों में से उस विकल्प का चयन करें जो दिए गए वाक्य के भेद का सही विकल्प है।

रीना अस्वस्थ होने के कारण कल विद्यालय नहीं गई थी।

  1. AOption (a)

    विधान वाचक वाक्य

  2. BOption (b)

    संयुक्त वाक्य

  3. COption (c)

    मिश्र वाक्य

  4. DOption (d)

    सरल वाक्य

View answer & explanation

Correct answer

Answer: (d)D.

सरल वाक्य

Why this answer is correct

सही उत्तर है-  सरल वाक्य।

Key Points

  • इस वाक्य में एक ही कर्ता व क्रिया है इसलिए यह सरल वाक्य है।
  • यहाँ कोई उपवाक्य नहीं है।
  • सरल वाक्य- ऐसे वाक्य जिनमें एक ही क्रिया एवं एक ही कर्ता होता है या जिस वाक्य में एक ही उद्देश्य एवं एक ही विधेय होता है, वे वाक्य सरल वाक्य कहलाते हैं। इसमें कर्ता एक से अधिक हो सकते हैं लेकिन मुख्य क्रिया एक ही होगी।
  • उदहारण:
  • ​रमा और सीता बाहर खेल रही हैं।
  • रजत बीमार होने के कारण नहीं आ पायेगा।

Additional Information संरचना के आधार पर वाक्य के अन्य भेद: 

  • संयुक्त वाक्य- जिस वाक्य में दो या दो से अधिक उपवाक्य मिले हों, परन्तु सभी वाक्य प्रधान हो तो ऐसे वाक्य को संयुक्त वाक्य कहते है। ये ‘और, एवं, तथा, या, अथवा, इसलिए, अतः, फिर भी, तो, नहीं तो, किन्तु, परन्तु, लेकिन, पर’ आदि से जुड़े होते हैं।
  • उदहारण -  रीना अस्वस्थ थी इसीलिए वह विद्यालय नहीं गयी।
  • मिश्रित वाक्य- जिस वाक्य में एक से अधिक वाक्य मिले हों, किन्तु एक प्रधान उपवाक्य तथा शेष आश्रित उपवाक्य हों, मिश्रित वाक्य कहलाता है।
  • उदहारण - वह औरत जो पार्क में बैठी हैं मेरी मौसी हैं।

Question 94

निम्नलिखित प्रश्न में ,चार विकल्पों में से उस विकल्प का चयन करें जो दिए गए लोकोक्ति का सही अर्थ हो।

टके की मुर्गी नौ टके महसूल=

  1. AOption (a)

    परिणाम कुछ नहीं निकलना, बात वहीं की वहीं रहना

  2. BOption (b)

    धन-दौलत से ही सब कार्य सिद्ध होते हैं।

  3. COption (c)

    शांत प्रकृति वाला मनुष्य क्रोधी मनुष्य को हरा देता है।

  4. DOption (d)

    कम कीमती वस्तु अधिक मूल्य पर देना

View answer & explanation

Correct answer

Answer: (d)D.

कम कीमती वस्तु अधिक मूल्य पर देना

Why this answer is correct

सही उत्तर है- कम कीमती वस्तु अधिक मूल्य पर देना। 

Key Points

  • टके की मुर्गी नौ टके महसूल - कम कीमती वस्तु अधिक मूल्य पर देना।
  • अन्य विकल्प:
  • ढाक के वही तीन पात - परिणाम कुछ नहीं निकलना, बात वहीं की वहीं रहना
  • टके का सब खेल - धन-दौलत से ही सब कार्य सिद्ध होते हैं।
  • ठंडा लोहा गरम लोहे को काटता है- शांत प्रकृति वाला मनुष्य क्रोधी मनुष्य को हरा देता है।

Important Points 

  • लोकोक्ति -
  • जब कोई पूरा कथन किसी प्रसंग विशेष में उद्धत किया जाता है तो लोकोक्ति कहलाता है।
  • लोकोक्ति वाक्यांश न होकर स्वतंत्र वाक्य होते हैं।

 Additional Informationहिंदी के कुछ प्रचलित मुहावरे:

  • इतनी-सी जान, गज भर की जुबान - बहुत बढ़-बढ़ कर बातें करना
  • गुड़ खाए गुलगुलों से परहेज - कोई बड़ी बुराई करना और छोटी से बचना
  • जिन ढूँढ़ा तिन पाइयाँ गहरे पानी पैठ - परिश्रम का फल अवश्य मिलता है
  • आ बैल मुझे मार- स्वयं मुसीबत मोल लेना
  • आँख का अन्धा नाम नयनसुख - गुण के विरुद्ध नाम होना।
  • ईश्वर की माया, कहीं धूप कहीं छाया - संसार में सभी एक जैसे नहीं हैं- कोई अमीर है, कोई गरीब
  • एक तो करेला दूजे नीम चढ़ा - कुटिल स्वभाव वाले मनुष्य बुरी संगत में पड़ कर और बिगड़ जाते है।

Question 95

निम्नलिखित प्रश्न में ,चार विकल्पों में से उस विकल्प का चयन करें जो दिये गए संधि शब्द का सही संधि विच्छेद का विकल्प हो।

वधूर्जा

  1. AOption (a)

    वधू + उर्जा

  2. BOption (b)

    वधू + ओर्जा

  3. COption (c)

    वधू + और्जा

  4. DOption (d)

    वधू + ऊर्जा

View answer & explanation

Correct answer

Answer: (d)D.

वधू + ऊर्जा

Why this answer is correct

सही उत्तर है- वधु+ऊर्जा।  

Key Points

  • वधू+ऊर्जा = वधूर्जा।
  • ऊ और ऊ स्वरों के मेल के कारण यहाँ दीर्घ स्वर संधि होगी।

**Additional Informationसन्धि:**​

  • संधि का शाब्दिक अर्थ है- मेल या समझौता।
  • जब दो वर्णों का मिलन अत्यन्त निकटता के कारण होता है तब उनमें कोई-न-कोई परिवर्तन होता है और वही परिवर्तन संधि के नाम से जाना जाता है।
  • संधि के तीन प्रकार हैं - 1. स्वर, 2. व्यंजन और 3. विसर्ग
  1. स्वर संधि: स्वर वर्ण के साथ स्वर वर्ण के मेल से जो विकार उत्पत्र होता है, उसे 'स्वर संधि' कहते हैं। जैसे- वधू + ऊर्जा=वधूर्जा।
  2. व्यंजन संधि: व्यंजन से स्वर अथवा व्यंजन के मेल से उत्पत्र विकार को व्यंजन संधि कहते है। जैसे- सत् + धर्म = सद्धर्म।
  3. विसर्ग संधि:  विसर्ग के साथ स्वर या व्यंजन के मेल से उत्पन्न विकार को विसर्ग संधि कहते हैं। जैसे- मनः + रथ = मनोरथ।

नीचे दिए गए गद्यांश को ध्यानपूर्वक पढ़ें और पूछे गए प्रश्नों का सर्वोत्तम उत्तर चुनें।

अहिंसा और कायरता कभी साथ नहीं चलती। मैं पूरी तरह शस्त्र-सज्जित मनुष्य के हृदय से कायर होने की कल्पना कर सकता हूँ। हथियार रखना कायरता नहीं तो डर का होना तो प्रकट करता ही है, परन्तु सच्ची अहिंसी शुद्ध निर्भयता के बिना असम्भव है।

क्या मुझमें बहादुरों की वह अहिंसा है? केवल मेरी मृत्यु ही इसे बताएगी। अगर कोई मेरी हत्या करे और मैं मुँह से हत्यारे के लिए प्रार्थना करते हुए तथा ईश्वर का नाम जपते हुए और हृदये मन्दिर में उसकी जीती-जागती उपस्थिति का भान रखते हुए मरूं तो ही कहा जाएगा कि मुझमें बहादुरों की अहिंसा थी। मेरी सारी शक्तियों के क्षीण हो जाने से अपंग बनकर मैं एक हारे हुए आदमी के रूप में नहीं मरना चाहता। किसी हत्यारे की गोली भले मेरे जीवन का अन्त कर दे, मैं उसका स्वागत करूंगा। लेकिन सबसे ज्यादा तो मैं अन्तिम श्वास तक अपना कर्तव्य पालन करते हुए ही मरना पसन्द करूंगा।

मुझे शहीद होने की तमन्ना नहीं है। लेकिन अगर धर्म की रक्षा का उच्चतम कर्तव्य पालन करते हुए मुझे शहादत मिल जाए तो मैं उसका पात्र माना जाऊँगा। भूतकाल में मेरे प्राण लेने के लिए मुझ पर अनेक बार आक्रमण किए गए हैं, परन्तु आज तक भगवान ने मेरी रक्षा की है और प्राण लेने का प्रयत्न करने वाले अपने किए पर पछताए हैं। लेकिन अगर कोई आदमी यह मानकर मुझ पर गोली चलाए कि वह एक दुष्ट का खात्मा कर रहा है, तो वह एक सच्चे गाँधी की हत्या नहीं करेगा, बल्कि उस गाँधी की करेगा जो उसे दुष्ट दिखाई दिया था।

Question 96

Based on Passage: View directions & passage

'उपस्थिति' का विलोम शब्द क्या है?

  1. AOption (a)

    कमी

  2. BOption (b)

    अनुपस्थिति

  3. COption (c)

    अभाव

  4. DOption (d)

    गैरहाजिरी

View answer & explanation

Correct answer

Answer: (b)B.

अनुपस्थिति

Why this answer is correct

सही उत्तर है - अनुपस्थिति। 

Key Points

  • उपस्थिति का अर्थ- हाज़िरी।
  • अनुपस्थिति का अर्थ- गैरहाज़िरी।

 Mistake Points

  • हालाँकि गैरहाज़िरी अनुपस्थिति का पर्यायवाची है और अर्थ की दृष्टि से उपस्थिति का विलोम हो सकता है।
  • किन्तु हमे प्रश्न में शब्द जिस रूप में दिया गया होता है उसी रूप में उसके विकल्प का चयन करना होता है।
  • यहाँ उपस्थिति का विलोम शब्द पूछा गया है इसलिए हम उससे मिलते सबसे उपयुक्त विकल्प का चयन करेंगे।
  • यहाँ अनुपस्थिति सबसे उचित शब्द है।
  • यदि प्रश्न में हाज़िरी शब्द दिया गया होता तब गैरहाज़िरी का चयन करना उचित होता।

Additional Information

  • विलोम शब्द: जो शब्द किसी दूसरे शब्द का उल्टा अर्थ बताते हैं, उन्हें विलोम शब्द या विपरीतार्थक शब्द कहते है।
  • जैसे-  
  • आय- व्यय
  • आजादी- गुलामी
  • दिन- रात
  • दुर्गम - सुगम

Question 97

Based on Passage: View directions & passage

अनुच्छेद के अनुसार सच्चे रूप में अहिंसावादी बनना किसके बिना संभव नहीं है?

  1. AOption (a)

    शुद्ध निर्भयता के बिना

  2. BOption (b)

    कायरता के बिना

  3. COption (c)

    असहायता के बिना

  4. DOption (d)

    निराश्रित हुए बिना

View answer & explanation

Correct answer

Answer: (a)A.

शुद्ध निर्भयता के बिना

Why this answer is correct

सही उत्तर है - शुद्ध निर्भयता के बिना। 

 Key Pointsगद्यांश की निम्न पंक्ति से यह स्पष्ट होता है -

  • हथियार रखना कायरता नहीं तो डर का होना तो प्रकट करता ही है, परन्तु सच्ची अहिंसी शुद्ध निर्भयता के बिना असम्भव है।

Additional Informationअहिंसावादी - 

  • अर्थ- अहिंसा में विश्वास करने वाला।
  • विलोम शब्द- हिंसावादी।
  • वाक्य में प्रयोग- गांधी जी अहिंसावादी तरीके से स्वतंत्रता प्राप्त करने के पक्ष में थे।

Question 98

Based on Passage: View directions & passage

लेखक को क्या होने की तमन्ना नहीं है?

  1. AOption (a)

    बड़ा क्रांतिकारी

  2. BOption (b)

    शहीद

  3. COption (c)

    जीने की

  4. DOption (d)

    जीवन में अहिंसावादी होने की

View answer & explanation

Correct answer

Answer: (b)B.

शहीद

Why this answer is correct

सही उत्तर है - शहीद। 

 Key Points

गद्यांश में लेखक कह रहा है कि- 

  • मुझे शहीद होने की तमन्ना नहीं है।
  • लेकिन अगर धर्म की रक्षा का उच्चतम कर्तव्य पालन करते हुए मुझे शहादत मिल जाए तो मैं उसका पात्र माना जाऊँगा।
  • लेखक कायरों कि तरह ना ही जीना चाहता है ना ही मरना चाहता है।
  • वह अंतिम श्वास तक अपने कर्तव्यों का पालन करते हुए मरना चाहता है।

Additional Informationशहीद का अर्थ- देश के प्रति कर्त्तव्य के लिए अपने प्राण देने वाला।

Question 99

Based on Passage: View directions & passage

अनुच्छेद के लिए उचित शीर्षक दीजिये।

  1. AOption (a)

    निर्भीकता और अहिंसा

  2. BOption (b)

    अहिंसावादी लेखक

  3. COption (c)

    लेखक का अहिंसा आचरण को अपनाना

  4. DOption (d)

    बहादुर ही अहिंसा धर्म को अपना सकता है।

View answer & explanation

Correct answer

Answer: (b)B.

अहिंसावादी लेखक

Why this answer is correct

सही उत्तर है- अहिंसावादी लेखक। 

Key Points

  • उपरोक्त गद्यांश में एक अहिंसावादी लेखक के विचार व्यक्त हुए हैं।
  • अतः इसका उचित शीर्षक अहिंसावादी लेखक होगा।
  • लेखक के अंतिम वाक्य से पता लगता है कि वह खुद एक अहिंसावादी लेखक है - लेकिन अगर कोई आदमी यह मानकर मुझ पर गोली चलाए कि वह एक दुष्ट का खात्मा कर रहा है, तो वह एक सच्चे गाँधी की हत्या नहीं करेगा, बल्कि उस गाँधी की करेगा जो उसे दुष्ट दिखाई दिया था।

 Additional Informationअन्य विकल्प:

  • निर्भीकता और अहिंसा-
  • इस गद्यांश में निर्भीकता का ज़िक्र सिर्फ एक वाक्य में किया गया है वह भी लेखक के अहिंसावाद के प्रति विचार दर्शाने।
  • इसके अतिरिक्त गद्यांश में निर्भीकता और अहिंसा का एक साथ कोई उल्लेख नहीं है।
  • अतः यह इसका शीर्षक नहीं रखा जा सकता।
  • लेखक का अहिंसा आचरण को अपनाना-
  • गद्यांश में लेखक के अहिंसा आचरण को अपनाने कि कहानी नहीं बताई गयी है।
  • इसमें लेखक के अहिंसावादी विचारों का उल्लेख है और यह स्पष्ट है कि अहिंसा उसके जीवन का अंग बन चूका है।
  • अतः यह शीर्षक अनुचित होगा।
  • बहादुर ही अहिंसा धर्म को अपना सकता है-
  • लेखक ने ऐसा कहीं स्पष्ट नहीं किया कि बहादुर ही अहिंसा धर्म को अपना सकता है।
  • ना ही इस गद्यांश का सार यह निकाला जा सकता है कि बहादुर ही अहिंसा धर्म को अपना सकता है।
  • अतः यह शीर्षक सार्थक नहीं होगा।

Important Pointsउचित शीर्षक का चुनाव: 

  • शीर्षक मूल विषय/भाव से सम्बंधित ही होना चाहिए।
  • शीर्षक संक्षित, आकर्षक तथा सार्थक होना चाहिए।
  • शीर्षक गद्यांश को समग्र रूप से अपने में समाहित करता होना चाहिए न कि केवल किसी एक अंश को।

Question 100

Based on Passage: View directions & passage

भूतकाल में लेखक के साथ क्या हुआ था?

  1. AOption (a)

    उनके ऊपर अनेक बार आक्रमण हुआ था

  2. BOption (b)

    सरकार द्वारा पुरस्कार से सम्मानित किया गया था।

  3. COption (c)

    अनहोनी हुआ था

  4. DOption (d)

    बहुत अत्याचार हुआ था

View answer & explanation

Correct answer

Answer: (a)A.

उनके ऊपर अनेक बार आक्रमण हुआ था

Why this answer is correct

सही उत्तर है - उनके ऊपर अनेक बार आक्रमण हुआ था

 Key Pointsगद्यांश में लेखक कहता है कि-

  • भूतकाल में मेरे प्राण लेने के लिए मुझ पर अनेक बार आक्रमण किए गए हैं, परन्तु आज तक भगवान ने मेरी रक्षा की है और प्राण लेने का प्रयत्न करने वाले अपने किए पर पछताए हैं।

Additional Information

  • भूतकाल - बीता हुआ समय, गत-काल।
  • आक्रमण - हमला।
  • प्रयत्न- कोशिश

Discipline 1 (20 questions)

Question 101

Compute the inverse of the matrix A=[423 222 101] A = \begin{bmatrix} 4 & 2 & 3\ 2 & 2 & 2 \ 1 & 0 & 1 \end{bmatrix}

  1. AOption (a)

    [1−1−1 01/2−1 112]\begin{bmatrix} 1 & -1 & -1\ 0 & 1/2 & -1 \ 1 & 1 & 2 \end{bmatrix}

  2. BOption (b)

    [1−1−1 01/2−1 −112]\begin{bmatrix} 1 & -1 & -1\ 0 & 1/2 & -1 \ -1 & 1 & 2 \end{bmatrix}

  3. COption (c)

    [1−11 01/2−1 −112]\begin{bmatrix} 1 & -1 & 1\ 0 & 1/2 & -1 \ -1 & 1 & 2 \end{bmatrix}

  4. DOption (d)

    [1−1−1 01/2−1 −113]\begin{bmatrix} 1 & -1 & -1\ 0 & 1/2 & -1 \ -1 & 1 & 3 \end{bmatrix}

View answer & explanation

Correct answer

Answer: (b)B.

[1−1−1 01/2−1 −112]\begin{bmatrix} 1 & -1 & -1\ 0 & 1/2 & -1 \ -1 & 1 & 2 \end{bmatrix}

Why this answer is correct

Concept:

The inverse of the matrix 

A-1 = adj(A) / det(A)

Let [Aij\rm _{ij}] be a square matrix.

Steps for obtaining adjoint of the matrix 

1. We need to calculate cofactors of all elements

⇒ Cij  or Aij = (-1)(i + j)  Mij

Here, Mij  = Minor of matrix A

2. Transpose of the cofactor is called adjoint of the matrix

⇒ Adj (A) = CT

Calculation:

A=[423 222 101] A = \begin{bmatrix} 4 & 2 & 3\ 2 & 2 & 2 \ 1 & 0 & 1 \end{bmatrix}

⇒ |A| = 4(2) - 2(0) + 3(-2)

⇒ |A| =  8 - 6 = 2

⇒ Cof A = [20−2−212−2−24]\begin{bmatrix} 2 & 0 & -2 \\ -2 & 1 & 2 \\ -2 & -2 & 4 \end{bmatrix}

∵ adj(A) = (cof A)T

 adj A=[2−2−201−2−224]adj\ A=\begin{bmatrix} 2 & -2 & -2 \\ 0 & 1 & -2 \\ -2 & 2 & 4 \end{bmatrix}

∵ A-1 = adj(A) / det(A)

= 12[2−2−201−2−224]\frac{1}{2}\begin{bmatrix} 2 & -2 & -2 \\ 0 & 1 & -2 \\ -2 & 2 & 4 \end{bmatrix}

⇒ A-1 = [1−1−101/2−1−112]\begin{bmatrix} 1 & -1 & -1 \\ 0 & 1/2 & -1 \\ -1 & 1 & 2 \end{bmatrix}

Hence, option (2) is correct.

Question 102

This polynomial x3 + 3x2 + 4x - 8 has

  1. AOption (a)

    three real zeros

  2. BOption (b)

    one positive and two imaginary zeros

  3. COption (c)

    no zeros

  4. DOption (d)

    one negative and two imaginary zeros

View answer & explanation

Correct answer

Answer: (b)B.

one positive and two imaginary zeros

Why this answer is correct

Concept:

Zeros of a polynomial: The values of the variable where polynomial becomes zero, called zeros of a polynomial. Mathematically,

Let a  P(x) is a polynomial in variable x, and  p(α) = 0, than 'x = α' is a zero of polynomial.

The number of the real root of a polynomial: 

  • The number of roots of a polynomial is the highest power of the variable in the polynomial.
  • The number of sign changes of polynomial P(x) is the maximum number of positive roots.
  • The number of sign changes of polynomial P(- x) is the maximum number of negative roots.
  • Before using the Rule of Signs the polynomial must have a constant term.
  • Imaginary roots always remain in pairs.

 

Calculation:

Let P(x) = x3 + 3x2 + 4x - 8

We can see that P(x) is the polynomial of degree 3. Hence, 

Maximum three roots of a polynomial are possible. Again, 

P(x) = x3 + 3x2 + 4x - 8

We can see that P(x) has only one sign change i.e. on positive real value.

Since, imaginary roots always remain in pairs, 

Hence, P(x) has one positive and two imaginary roots.

Alternate Method

Let P(x) =x3 + 3x2 + 4x - 8

at x = 1

P(x) = 1 + 3 + 4 - 8 = 0

Therefore, (x - 1) will be factor of P(x). 

By deviding P(x) with (x - 1), we will get quadratic 

Q(x) = x2 + 4x + 8 

We know that discriminent of quadratic euqation ax2 + bx + c is given by

D = b2 - 4ac

⇒ D = 16 - 32 = -16 

⇒ D < 0  which represent imaginary root of quadratic.

Hence, P(x) has one real root (x = 1) and two imaginary roots.

Question 103

Discover the number of possible relations in an anti-symmetric relation with 19 elements.

  1. AOption (a)

    219 × 3171

  2. BOption (b)

    19.34 × 791

  3. COption (c)

    3.5 × 932

  4. DOption (d)

    1.9 × 364

View answer & explanation

Correct answer

Answer: (a)A.

219 × 3171

Why this answer is correct

Concept:

Antisymmetric relations:

  • Antisymmetric relations have no restrictions on the diagonal elements (a, a) so there are 2n ways to set such elements.
  • As to the non-diagonal elements, there are options for each pair {(a, b), (b, a)} where a ≠ b.
  • An antisymmetric relation may contain either (a, b) or (b, a) or none of them.
  • Therefore, there are 3n2 − n23^{\frac{n^2\ -\ n}{2}}  ways to set the off-diagonal elements.

The total number of antisymmetric relations is given by the expression

N = 2n.3n2 − n2N\ =\ 2^n.3^{\frac{n^2\ -\ n}{2}}

Calculation:

Given that,

n = 19

We know that the total number of antisymmetric relations is given by the expression

N = 2n.3n2 − n2N\ =\ 2^n.3^{\frac{n^2\ -\ n}{2}}

⇒ N = 219.3192 − 192⇒\ N\ =\ 2^{19}.3^{\frac{{19}^2\ -\ 19}{2}}

⇒ N = 219 × 3171

Question 104

Find out which of the following statements is not true if A is a Hermitian matrix.

  1. AOption (a)

    The diagonal entries of A are all real.

  2. BOption (b)

    There exists a unitary U such that U × AU is a diagonal matrix.

  3. COption (c)

    If A3 = I, then A = I

  4. DOption (d)

    If A2 = I, then A = I

View answer & explanation

Correct answer

Answer: (d)D.

If A2 = I, then A = I

Why this answer is correct

Concept:

Conjugate transpose: 

A∗ = (aˉji)A^*\ = \ (\bar a_{ji}) is called the conjugate transpose of the matrix A, where

A∗ =Aˉt =AtˉA^*\ = \bar{A}^t\ = \bar{A^t}

Hermitian matrix: A square matrix  A with complex entries is called a Hermitian matrix if A* = A

Properties of Hermitian matrix: 

  • The elements of the principal diagonal of a hermitian matrix are all real numbers.
  • The non-diagonal elements of a hermitian matrix are complex numbers.
  • The sum and product of any two hermitian matrices are hermitian.
  • If inverse A id exist, then the inverse of a hermitian matrix is a hermitian.
  • The determinant of a hermitian matrix is real.

Unitary matrix: A is called unitary matrix if AA* = A*A = I

Spectral theorem: spectral theorem states that if A equals the transpose of A, then A is diagonalizable.

There exists a Unitary matrix U such that U-1MU is a diagonal matrix.

Diagonal matrix:  Any matrix for which all entries off the main diagonal are zero is called a diagonal matrix.

Cayley-Hamilton theorem: it that if p(λ) is the characteristic polynomial of a square matrix A, obtained from p(λ) = det (λI − A), then substituting A for λ in the polynomial gives the zero matrices.

Thus, by applying the theorem, matrix A satisfies its own characteristic polynomial, p(A) = 0 

Calculation:

Option 1: From the property of the hermitian matrix, this statement is correct. 

Option 2: According to the spectral theorem, this statement is also true. 

Option 3: Let us consider a hermitian matrix  

A = [01 10 ]A\ =\ \left[ \begin{matrix} 0 & 1 \ 1 & 0 \ \end{matrix} \right]

A2 = [01 10 ][01 10 ] = [10 01 ]A^2\ =\ \left[ \begin{matrix} 0 & 1 \ 1 & 0 \ \end{matrix} \right]\left[ \begin{matrix} 0 & 1 \ 1 & 0 \ \end{matrix} \right]\ =\ \left[ \begin{matrix} 1 & 0\ 0 & 1 \ \end{matrix} \right]           ---(1)

A3 = A2A = [01 10 ][10 01 ] = [01 10 ] = AA^3\ = \ A^2A\ =\ \left[ \begin{matrix} 0 & 1 \ 1 & 0 \ \end{matrix} \right]\left[ \begin{matrix} 1& 0 \ 0 & 1 \ \end{matrix} \right]\ =\ \left[ \begin{matrix} 0 & 1\ 1 & 0 \ \end{matrix} \right]\ =\ A

Option 4: From equation (1) can see that,

Although A2 = I, it is not necessary A = I 

Hence, this statement is not true.

Question 105

Calculate the volume of sphere by triple integration?

  1. AOption (a)

    4a3π3\frac{4a^3\pi}{3}

  2. BOption (b)

    8a3π3\frac{8a^3\pi}{3}

  3. COption (c)

    2a3π3\frac{2a^3\pi}{3}

  4. DOption (d)

    a3π3\frac{a^3\pi}{3}

View answer & explanation

Correct answer

Answer: (a)A.

4a3π3\frac{4a^3\pi}{3}

Why this answer is correct

Explanation -

The equation for a sphere of radius R centered at the origin is x2 + y2 + z2 = R2 .

In spherical coordinates ( ρ, θ, ϕ ):

ρ is the radial distance from the origin.

θ is the azimuthal angle in the xy-plane from the x-axis.

ϕ is the polar angle from the z-axis.

The relationships between Cartesian coordinates (x, y, z) and spherical coordinates (ρ, θ, ϕ) are:

x = ρ sin ϕ cos θ

y = ρ sin ϕ sin θ

z = ρ cos ϕ

The volume element in spherical coordinates is:

dV = ρ2 sin ϕ  dρ dϕ dθ

The volume V of the sphere is given by:

V=∫02π∫0π∫0Rρ2sin⁡ϕ,dρ,dϕ,dθV = \int_0^{2\pi} \int_0^\pi \int_0^R ρ^2 \sin ϕ , dρ , dϕ , dθ

Let's evaluate this integral step by step.

  1. Integrate with respect to ρ:

∫0Rρ2,dρ=[ρ33]0R=R33\int_0^R ρ^2 , dρ = \left[ \frac{ρ^3}{3} \right]_0^R = \frac{R^3}{3}

  1. Integrate with respect to ϕ:

∫0πsin⁡ϕ,dϕ=[−cos⁡ϕ]0π=(−cos⁡π)−(−cos⁡0)=1+1=2\int_0^\pi \sin ϕ , dϕ = \left[ -\cos ϕ \right]_0^\pi = (-\cos \pi) - (-\cos 0) = 1 + 1 = 2

  1. Integrate with respect to θ:

∫02πdθ=[θ]02π=2π\int_0^{2\pi} dθ = \left[ θ \right]_0^{2\pi} = 2\pi

Putting it all together:

V=(R33)×2×2π=2R33×2π=4πR33V = \left( \frac{R^3}{3} \right) \times 2 \times 2\pi = \frac{2R^3}{3} \times 2\pi = \frac{4\pi R^3}{3}

Thus, the volume of a sphere of radius R is:

V = 4πR33\frac{4\pi R^3}{3}

Hence Option (1) is correct.

Question 106

Inner product spaces over the field of complex numbers are sometimes referred to as

  1. AOption (a)

    real vector space

  2. BOption (b)

    abstract vector space

  3. COption (c)

    vector space

  4. DOption (d)

    unitary space

View answer & explanation

Correct answer

Answer: (d)D.

unitary space

Why this answer is correct

Concept:

  • Let v = (v1, . . . , vn) and w = (w1, . . . , wn) ∈ Rn .
  • We define the inner product (or dot product or scalar product) of v and w by the following formula: < v, w > = v1w1 + · · · + vnwn.
  • Define the length or norm of v by the formula || v || = √< v, v > =√ { v12 1 + · · · + vn2 } .

 

We have the following properties for the inner product:

  1. (Bilinearity) For all v, u, w ∈ R , < v + u, w > = < v + w > + < u + w > and < v ,u + w > = < v + u > + < v + w >. For all v, w ∈ Rn and t ∈ R, < tv, w > = < vt, w > = t< v, w >.
  2. (Symmetry) For all v, w ∈ Rn , < v, w > = < w, v >.
  3. (Positive definiteness) For all v ∈ Rn , < v, v > = || v ||2 ≥ 0, and  < v, v > = 0  if and only if v = 0.

Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces.

​Hence, the correct answer is option 4)

Question 107

What is the slope of the tangent to the curve y = 4x ÷ (x2 + 1) at (0, 0)?

  1. AOption (a)

    0

  2. BOption (b)

    4

  3. COption (c)

    8

  4. DOption (d)

    2

View answer & explanation

Correct answer

Answer: (b)B.

4

Why this answer is correct

Concept:

Slope of a tangent to a curve is given by differentiating equation of the curve with the independent variable, i.e.

Slope (m) = dydx\frac {dy}{dx}

ddx(p(x)g(x))=p′(x).g(x) − g′(x).p(x)(g(x))2\frac {d}{dx}\left( \frac {p(x)}{g(x)} \right ) = \frac {p'(x).g(x)\ -\ g'(x).p(x)}{(g(x))^2}

Where, p'(x) = ddxp(x)\frac {d}{dx}p(x)

Calculation:

Given:

y = 4x ÷ (x2 + 1)

Slope of the curve can be calculated by differentiating w.r.t x

dydx\frac {dy}{dx} = ddx(4xx2+1)\frac {d}{dx}\left( \frac {4x}{x^2 + 1} \right )

Using ddx(p(x)g(x))=p′(x).g(x) − g′(x).p(x)(g(x))2\frac {d}{dx}\left( \frac {p(x)}{g(x)} \right ) = \frac {p'(x).g(x)\ -\ g'(x).p(x)}{(g(x))^2}

= 4 . (x2+1) − 2x . 4x(x2+1)2 \frac {4\ .\ (x^2 +1)\ -\ 2x\ .\ 4x}{(x^2 +1)^2}

= 4x2+4 − 8x2(x2+1)2 \frac {4x^2 +4\ -\ 8x^2}{(x^2 +1)^2}

dydx\frac {dy}{dx} = 4 − 4x2(x2+1)2 \frac {4\ -\ 4x^2}{(x^2 +1)^2}

(dydx)(0,0)\left (\frac {dy}{dx} \right )_{(0,0)} = 4 − 4(0)2((0)2+1)2 \frac {4\ -\ 4(0)^2}{((0)^2 +1)^2}

(dydx)(0,0)\left (\frac {dy}{dx} \right )_{(0,0)} = 4

Question 108

A quadratic field is called a simple quadratic field if:

  1. AOption (a)

    it also follows that πi is associate of some yj and conversely.

  2. BOption (b)

    divisor, unity, associate and prime in Q(w) are same as those in k(i).

  3. COption (c)

    the representation is unique apart from the order of the primes, the presence of unities and ambiguities between associated primes.

  4. DOption (d)

    every algebraic integer can be expressed as a product of prime element uniquely up to change of order and multiplication by units.

View answer & explanation

Correct answer

Answer: (d)D.

every algebraic integer can be expressed as a product of prime element uniquely up to change of order and multiplication by units.

Why this answer is correct

Concept Used:

An algebraic integer of the form a+bD a+b \sqrt{D} where D is square free forms a quadratic field and is denoted Q(D) \mathbb{Q}(\sqrt{D}). If D>0, the field is called a real quadratic field, and if D<0, it is called an imaginary quadratic field. The integers in Q(1) \mathbb{Q}(\sqrt{1}) are simply called integers. The integers in Q(−1) \mathbb{Q}(\sqrt{-1}) are called Gaussian integers, and the integers in Q(−3) \mathbb{Q}(\sqrt{-3}) are called Eisenstein integers.

Simple Quadratic Field: If for all algebraic integers in the quadratic field can be expressed uniquely as the product of primes, then the quadratic field is called the Simple Quadratic Field.

Question 109

Zero of x2 - 100 will be

  1. AOption (a)

    10

  2. BOption (b)

    -100

  3. COption (c)

    0

  4. DOption (d)

    100

View answer & explanation

Correct answer

Answer: (a)A.

10

Why this answer is correct

Concept:

Zeros of a polynomial:

Zeros of polynomial p(x) are the value of 'x' for which p(x) becomes zero.

Geometrically, these are the value of x on which the graph cut the x-axis.

Formula used:

(a - b)(a + b) = a2 - b2 

Calculation:

Let p(x) = x2 - 100

For zeros, p(x) should be zero

⇒ x2 - 100 = 0

⇒ x2 - 102 = 0

∵ (a - b)(a + b) = a2 - b2 

⇒ (x - 10)(x + 10) = 0          

⇒ x = 10, -10

Hence,  x = 10 is the zero of the polynomial.

Question 110

Find the value of the limit, lim⁡x→0[25x]\mathop {\lim }\limits_{x \to 0} [\frac{{25}}{{x}}]?

  1. AOption (a)

    0

  2. BOption (b)

    3 ÷ 2

  3. COption (c)

    ∞ 

  4. DOption (d)

    1 ÷ 2

View answer & explanation

Correct answer

Answer: (c)C.

∞ 

Why this answer is correct

Concept:

Limit: Limit is the value that a function approaches as the input approaches some value.

There are seven indeterminate forms of limit that is ∞ - ∞, 0/0, ∞/∞, 0 × ∞, 00, ∞0, 1∞  

Calculation:

Let y =  lim⁡x→0[25x]\mathop {\lim }\limits_{x \to 0}[\frac{25}{x}]

On putting the value of x = 0, we will get

y  = 25/0

⇒ y = ∞ 

Hence, the value of the given expression is ∞

Question 111

Find out the number of equivalence classes that can be defined by the set {1, 2, 3}.

  1. AOption (a)

    5

  2. BOption (b)

    72

  3. COption (c)

    16

  4. DOption (d)

    125

View answer & explanation

Correct answer

Answer: (a)A.

5

Why this answer is correct

Concept:

  • An equivalence class is a name that we give to the subset of S which includes all elements that are equivalent to each other.
  • “Equivalent” is dependent on a specified relationship, called an equivalence relation.
  • If there's an equivalence relation between any two elements, they're called equivalent.

 

​Calculation:

The smallest equivalence relation is the identity relation R1 = {(a, a), (b, b), (c, c)}

The two ordered pairs of two distinct elements can be added to give three more equivalence relations.

R2 = {(a, a), (b, b), (c, c), (a, b), (b, a)}

Similarly, R3 and R4 can be made by taking (b, c), (c, b) and (a, c), (c, a) respectively

Finally the largest equivalence relation i.e., the universal relation R5 = {(a, a), (b, b), (c, c), (a, b), (b, a), (a, c), (c, a), (b, c), (c, b)}.

Hence totally 5 equivalence relations can be created

Hence, the correct answer is option 1)

Question 112

What will be the average rate of change of function Y = [25 - x2] at x = 2?

  1. AOption (a)

    2

  2. BOption (b)

    -4

  3. COption (c)

    21

  4. DOption (d)

    4

View answer & explanation

Correct answer

Answer: (b)B.

-4

Why this answer is correct

Concept Used:

The average rate of change over the interval [a, b] for a function f(x) is f(b)−f(a)b−a \frac{f(b)-f(a)}{b-a}

Now, if b = a + h for some h > 0

⇒ Average rate of change =f(a+h)−f(a)a+h−a=f(a+h)−f(a)h =\frac{f(a+h)-f(a)}{a+h-a}=\frac{f(a+h)-f(a)}{h}

The average rate of change at any point x = a is

lim⁡h→0f(a+h)−f(a)h\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} [derivative of f(x) at x = a]

Calculation:

Here, y=[25−x2]=f(x) y=\left[25-x^2\right]=f(x)     (say)

Average rate of change at x = 2 is lim⁡h→0[(25−(2+h)2−(25−22)h] \lim _{h \rightarrow 0} \left[\frac{\left(25-(2+h)^2-\left(25-2^2\right)\right.}{h}\right]

⇒lim⁡h→0[25−4−h2−4h−25+4]h=lim⁡h→0h(−h−4)h ⇒ \lim _{h \rightarrow 0} \frac{\left[25-4-h^2-4 h-25+4\right]}{h}=\lim _{h \rightarrow 0} \frac{h(-h-4)}{h}

⇒lim⁡h→0(−h−4)=−4 ⇒ \lim _{h \rightarrow 0}(-h-4)=-4

Question 113

If A be a n × n real matrix. Then, which of the following statements is satisfied conditions for the matrix?

  1. AOption (a)

    If rank (A) = r, then A has r non-zero eigenvalues.

  2. BOption (b)

    If A ^ k = 0 for some k > 0, then trace (A) = 0

  3. COption (c)

    if the eigenvalues of A are λ1,...,λn, then A is similar to a diagonal matrix with λ1,...,λn along the diagonal

  4. DOption (d)

    If A has a repeated eigenvalue, then A is not diagonalizable

View answer & explanation

Correct answer

Answer: (b)B.

If A ^ k = 0 for some k > 0, then trace (A) = 0

Why this answer is correct

The correct answer is option 2 i.e if A ^ k = 0 for some k > 0, then trace (A) = 0.

Concept :

Option 1 : 

Take an example (01 00) \begin{pmatrix} 0& 1 \ 0 & 0 \end{pmatrix}

here , we have rank of matrix is 1 but there is no non-zero eigen value .So, option 1 is not satisfying condition. 

Option 2: 

A is a Nilpotent matrix and all the eigen values of nilpotent matrix are zero and 

trace = sum of eigen values .

  ⟹  \impliestrace = 0

So, option 2 is satisfying condition .

Option 3: 

The matrix A is similar to a diagonal matrix with eigen values λ1,λ2,...,λn\lambda1,\lambda2,...,\lambda n if and only if A has n L.I. vectors  .

So, option 3 is not satisfying condition.

Option 4: 

Here , we can think of identity matrix. Identity matrix has eigen values 1,1 but it is Diagonalizable .

So, option 4 is not satisfying condition.

Question 114

What will be the value of x3 + y3 + z3, if x + y + z = 0?

  1. AOption (a)

    xyz

  2. BOption (b)

    2xyz

  3. COption (c)

    3xyz

  4. DOption (d)

    xyz(xy + yz + zx)

View answer & explanation

Correct answer

Answer: (c)C.

3xyz

Why this answer is correct

Formula used:

The value of the expression 

x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx).

Calculation:

Given,

x + y + z = 0

From the above discussion, we can conclude that, 

 x3 + y3 + z3 = 3xyz

Hence, 3xyz will be correct option.

Question 115

Find out the area enclosed by parabolas x2 = y and y2 = x.

  1. AOption (a)

    1 ÷ 12 

  2. BOption (b)

    1 ÷ 6

  3. COption (c)

    1 ÷ 9

  4. DOption (d)

    1 ÷ 3

View answer & explanation

Correct answer

Answer: (d)D.

1 ÷ 3

Why this answer is correct

Solution:

x2 = y . . .(i)

y2 = x . . .(ii)

Using (i) and (ii), we get

⇒ y4 = y ⇒ y(y3 -1) = 0

⇒ y = 0 or y = 1

⇒ x = 0 or x =1

Point of intersection of both curves are (0, 0) and (1, 1)

The area enclosed between curves=∫01xdx−∫01x2dx = \int_0^1{\sqrt{x}}dx - \int_0^1 x^2 dx

⇒x3/23/2∣01−x33∣01=2x3/23∣01−13+0 ⇒ \left.\frac{x^{3/2}}{3/2} \right|_0^{1} - \left.\frac{x^3}{3}\right|_0^1 = \left.\frac{2x^{3/2}}{3} \right|_0^{1} -\frac{1}{3}+0

⇒ 23−0−13=13 \frac{2}{3} - 0 -\frac{1}{3} = \frac{1}{3}

Hence, the area enclosed between both curves is 1÷3.

Question 116

Divergence and Curl of a vector field are _______

  1. AOption (a)

    Scalar & Scalar

  2. BOption (b)

    Scalar & Vector

  3. COption (c)

    Vector & Vector

  4. DOption (d)

    Vector & Scalar

View answer & explanation

Correct answer

Answer: (b)B.

Scalar & Vector

Why this answer is correct

Divergence - Scalar operator

Curl - Vector operator

Explanation

The curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional space.

The curl of a scalar field is undefined. It is defined only for 3D vector fields.

Important Point:

For a vector F = F1i + F2j + F3k

Div=∇.F=∂F1∂x+∂F2∂y+∂F3∂zDiv = \nabla .F = \frac{{\partial {F_1}}}{{\partial x}} + \frac{{\partial {F_2}}}{{\partial y}} + \frac{{\partial {F_3}}}{{\partial z}}

Curl = \nabla \times F = \left| {\begin{array}{*{20}{c}} i&j&k\ {\frac{\partial }{{\partial x}}}&{\frac{\partial }{{\partial y}}}&{\frac{\partial }{{\partial z}}}\ {{F_1}}&{{F_2}}&{{F_3}} \end{array}} \right|

Question 117

If A is a 9 × 10 matrix of rank 3. Which of the following is equal to the dimension of the null space of A?

  1. AOption (a)

    9

  2. BOption (b)

    3

  3. COption (c)

    7

  4. DOption (d)

    10

View answer & explanation

Correct answer

Answer: (c)C.

7

Why this answer is correct

Concept Used:

for any m × n matrix A, the Rank-Nullity theorem states that rank⁡(A)+dim⁡N(A)=n \operatorname{rank}(A)+\operatorname{dim} N(A)=n

where rank (A) represents the rank of a matrix, A and dim N(A) represent a dimension of the null space of A.

Calculation:

Here, Matrix A is of order 9 × 10, and rank(A) = 3

rank⁡(A)+dim⁡N(A)=n \operatorname{rank}(A)+\operatorname{dim} N(A)=n

⇒3+dim⁡N(A)=10\Rightarrow 3+\operatorname{dim} N(A)=10

⇒dim⁡N(A)=10−3=7\Rightarrow \operatorname{dim} N(A)=10-3=7

Hence, the dimension of the null pace of A is 7.

Question 118

What will be the differential function of log(x2 + 2)?

  1. AOption (a)

    2x ÷ (x2 - 2) dx

  2. BOption (b)

    -2x ÷ (x2 + 2) dx

  3. COption (c)

    2x ÷ (x2 + 2) dx

  4. DOption (d)

    -2x ÷ (x2 - 2) dx

View answer & explanation

Correct answer

Answer: (c)C.

2x ÷ (x2 + 2) dx

Why this answer is correct

Formula used:

dydx log⁡x= 1x\frac{dy}{dx}\ \log x= \ \frac{1}{x}  and 

dydxxn = nxn−1\frac{dy}{dx}x^n\ = \ nx^{n - 1}

Calculation:

Let,

y = log(x2 + 2)

⇒ dydx = ddxlog⁡(x2+2)\frac{dy}{dx}\ = \ \frac{d}{dx}\log(x^2 + 2)

⇒ dydx =1(x2+2)ddx(x2)\frac{dy}{dx}\ = \frac{1}{(x^2 + 2)} \frac{d}{dx}(x^2)

⇒ dydx =2x(x2+2)\frac{dy}{dx}\ = \frac{2x}{(x^2 + 2)}

Therefore differential function,

dy = 2x ÷ (x2 + 2) dx

Hence, option '3' will be correct option.

Question 119

Which of the following characteristic is included in the study of general procedure for tracing the algebraic curve?

  1. Symmetry
  2. Orthogonality
  3. Tangents to the Curve at the origin
  1. AOption (a)

    1 & 2 are correct

  2. BOption (b)

    only 1

  3. COption (c)

    1 & 3 are correct

  4. DOption (d)

    only 2

View answer & explanation

Correct answer

Answer: (c)C.

1 & 3 are correct

Why this answer is correct

Concept:

The general procedure for tracing the algebraic curve consists of the study of the following characteristics of the curve:

Symmetry, Region/Extent, Asymptotes, Origin, Tangents to the curve at the origin, Maxima and Minima, Sign of the first derivative, Sign of the second derivative, and Inflection Point.

So, 1 & 3 are correct

Hence, the correct answer is option 3)

Question 120

Find out the slope of the tangent line for the curve xy3 - yx3 = 2 at the point (1, -1).

  1. AOption (a)

    -2

  2. BOption (b)

    2

  3. COption (c)

    0

  4. DOption (d)

    -1

View answer & explanation

Correct answer

Answer: (d)D.

-1

Why this answer is correct

Concept:

The slope of function: The slope of a curve y = f(x) at the point P means the slope of the tangent at the point P.

The value of dy/dx = f'(x) at point P gives the slope of function.

Formula used:

If m & n are two functions of x, then

ddx(mn) = mddxn+nddxm\frac{d}{dx}(mn)\ = \ m\frac{d}{dx}n + n\frac{d}{dx}m   

ddxxn =nxn−1\frac{d}{dx} x^n\ = nx^{n-1}

Calculation:

Given that,

xy3 - yx3 = 2

Differentiating with respect to 'x'

ddxxy3 − ddxyx3 = ddx2\frac{d}{dx}xy^3\ -\ \frac{d}{dx}yx^3\ = \ \frac{d}{dx}2

Using chain rule of differentiation

x.3y2dydx + y3 − y.3x2 − x3dydx = 0x.3y^2\frac{dy}{dx} \ + \ y^3\ - \ y.3x^2 \ - \ x^3\frac{dy}{dx} \ = \ 0

⇒  dydx(3xy2−x3)=3x2y−y3\Rightarrow \ \ \frac{dy}{dx}(3xy^2 - x^3) = 3x^2y - y^3

⇒dydx = 3x2y − y33xy2 − x3\Rightarrow \frac{dy}{dx}\ =\ \frac{3x^2y\ -\ y^3}{3xy^2\ -\ x^3}

Therefore, the slope of a curve at the point (1, -1)

dydx =3.(1)2(−1) − (−1)33.(1)(−1)2 − (1)3 = −1\frac{dy}{dx}\ = \frac{3.(1)^2(-1)\ -\ (-1)^3}{3.(1)(-1)^2\ -\ (1)^3}\ = \ -1

Hence, -1 is the slope of a curve at the point (1, -1)

Discipline 2 (20 questions)

Question 121

What is FEM?

  1. AOption (a)

    Fixed element mesh

  2. BOption (b)

    Finite element mesh

  3. COption (c)

    Forward element method

  4. DOption (d)

    Finite element method

View answer & explanation

Correct answer

Answer: (d)D.

Finite element method

Why this answer is correct

Concept Used:

The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Some problems which can be solved using this method are traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.

Question 122

Solve: lim⁡x→0sin[1x]\mathop {\lim }\limits_{x \to 0} sin[\frac{1}{x}]

  1. AOption (a)

    -1

  2. BOption (b)

    0

  3. COption (c)

    does not exist

  4. DOption (d)

    1

View answer & explanation

Correct answer

Answer: (c)C.

does not exist

Why this answer is correct

Concept:

Left-hand limit: A left-hand limit means the limit of a function as it approaches from the left-hand side.

If 'a' approaches the function from the left side, the function is said to have a left-hand limit let (P). 

lim⁡x→a−f(x) = P\mathop {\lim }\limits_{x \to a^-}f(x) \ =\ P

Right-hand limit: A right-hand limit means the limit of a function as it approaches from the right-hand side.

If 'a' approaches the function from the right side, the function is said to have a right-hand limit let (Q). 

lim⁡x→a+f(x) = Q\mathop {\lim }\limits_{x \to a^+}f(x) \ =\ Q

For the limit to exist, both side limits should be the same, i.e.

P = Q

Calculation:

Let,

y =lim⁡x→0sin[1x]y \ = \mathop {\lim }\limits_{x \to 0} sin[\frac{1}{x}]

Hence left-hand limit (LHL)

lim⁡x→a−sin(1x) =  lim⁡x→0sin(1x)\mathop {\lim }\limits_{x \to a^-}sin(\frac{1}{x})\ = \ \ \mathop {\lim }\limits_{x \to 0}sin(\frac{1}{x})

On replacing x from '-a'

⇒ lim⁡x→a−(1x) =  lim⁡a→0sin(−1a)\Rightarrow \ \mathop {\lim }\limits_{x \to a^-}(\frac{1}{x})\ = \ \ \mathop {\lim }\limits_{a \to 0}sin(\frac{-1}{a})

∵ sin(-θ) = sinθ  

⇒lim⁡x→a−sin(1x)=−lim⁡a→0sin(1a)\Rightarrow \mathop {\lim }\limits_{x \to a^-}sin(\frac{1}{x}) = - \mathop {\lim }\limits_{a \to 0}sin(\frac{1}{a})   ----(1)

Similarly 

⇒lim⁡x→a+sin(1x)=lim⁡a→0sin(1a)\Rightarrow \mathop {\lim }\limits_{x \to a^+}sin(\frac{1}{x}) = \mathop {\lim }\limits_{a \to 0}sin(\frac{1}{a})        -----(2)

From the equation (1) and (2), we can see that

LHL ≠  RHL

Hence, the limit of a given function does not exist.

Question 123

Calculate lim⁡n→∞7n + 35n +n\mathop {\lim }\limits_{n \to \infty} \frac{7n\ +\ 3}{5n\ +\sqrt n }

  1. AOption (a)

    1/5 

  2. BOption (b)

    1/3

  3. COption (c)

    7 /5

  4. DOption (d)

    7/3

View answer & explanation

Correct answer

Answer: (c)C.

7 /5

Why this answer is correct

Concept:

L’Hospital’s Rule: If we have an indeterminate form 0/0 or ∞/∞, we need to differentiate the numerator and denominator and then take the limit.

Formula used:

ddxxn = nxn−1\frac{d}{dx}x^n\ = \ nx^{n- 1}

ddxx = 12x\frac{d}{dx}{\sqrt x}\ = \ \frac{1}{2\sqrt x}

Calculation:

Let y = lim⁡n→∞7n + 35n +n\mathop {\lim }\limits_{n \to \infty} \frac{7n\ +\ 3}{5n\ +\sqrt n }

Since this is a ∞/∞ indeterminate form, therefore we can apply l-Hopital's rule to solve this.

Therefore, differentiating numerator and denominator separately with respect to n 

y = lim⁡n→∞[75 + 12n]\mathop {\lim }\limits_{n \to \infty}[ \frac{7 }{5\ +\ \frac{1}{2 \sqrt n}}]

Taking the limit on both sides, we will get 

y = 7/5             (∵ 1/∞ = 0)

Hence, 7/5 will be the correct option.

AlternateMethodLet, 

y = lim⁡n→∞7n + 35n + n\mathop {\lim }\limits_{n \to \infty} \frac{7n \ + \ 3 }{5n\ +\ \sqrt n}

⇒ y = lim⁡n→∞7 + 3n5 + 1n\mathop {\lim }\limits_{n \to \infty} \frac{7\ +\ \frac{3}{n} }{5\ +\ \frac{1}{ \sqrt n}}

Taking the limit of 'n on both sides, we will get

⇒ y = 7/5        (∵ 1/∞ = 0)

Question 124

Derivate of  δδxln(2x+3)\frac{\delta}{\delta x}ln(2x + 3) is equals to

  1. AOption (a)

    −1(2x+3)\frac{-1}{(2x + 3)}

  2. BOption (b)

    2(2−3x)\frac{2}{(2 - 3x)}

  3. COption (c)

    −2(2x+3)\frac{-2}{(2x + 3)}

  4. DOption (d)

    2(2x+3)\frac{2}{(2x + 3)}

View answer & explanation

Correct answer

Answer: (d)D.

2(2x+3)\frac{2}{(2x + 3)}

Why this answer is correct

Formula used:

ddxln⁡x = 1x\frac{d}{dx} \ln x\ = \ \frac{1}{x}

ddxxn = nxn−1\frac{d}{dx}x^n\ = \ nx^{n - 1}

Calculation:

Let,

y = δδxln⁡(2x+3)y\ = \ \frac{\delta}{\delta x}\ln(2x + 3)

From the above discussion

 y =12x+3ddx(2x)y\ = \frac{1}{2x + 3}\frac{d}{dx}(2x)

⇒y = 22x + 3\Rightarrow y\ = \ \frac{2}{2x\ +\ 3}

Hence, 22x + 3 \frac{2}{2x\ +\ 3}  will be the correct answer of the derivative.

Question 125

What is the angle which the tangent to the curve at P makes with the positive direction of x axis? The value of f'(x) is -1 at the point P on a continuous curve y = f(x).

  1. AOption (a)

    π ÷ 4

  2. BOption (b)

    3π ÷ 4

  3. COption (c)

    3π ÷ 2

  4. DOption (d)

    π ÷ 2

View answer & explanation

Correct answer

Answer: (b)B.

3π ÷ 4

Why this answer is correct

Formula used:

tan(π/4) = 1

tan(π - θ) = - tanθ

Calculation:

Given that, f'(x)P = -1

Let, θ  be the required angle which the tangent to the curve y = f(x) at P makes with the positive direction of the x-axis.

Therefore,

tanθ = f’(x) = -1

⇒ tanθ = - tan(π/4)              [∵ tan(π/4) = 1] 

This can be written as,

tanθ = tan(π – π/4)              [∵ tan(π - θ) = - tanθ]

⇒ tanθ = tan(3π/4)

⇒ θ = 3π/4

Hence, option 2 is correct.

Question 126

If [x] is the greatest integer in x, then what is: lim⁡x→1[x + 1]\mathop {\lim }\limits_{x \to 1} [x\ +\ 1] ?

  1. AOption (a)

    0

  2. BOption (b)

    -1

  3. COption (c)

    1

  4. DOption (d)

    limit does not exist

View answer & explanation

Correct answer

Answer: (d)D.

limit does not exist

Why this answer is correct

Concept:

Greatest Integer Function: (Floor function)

The function f (x) = [x] is called the greatest integer function and means greatest integer less than or equal to x i.e [x] ≤ x.

The domain of [x] is R and range is I, where R is the set of real numbers and I is the set of integers.

Condition of limit to exist:

Let y = f(x) be a function. Then for a function we say, lim⁡x→af(x)\mathop {\lim }\limits_{x \to a} f(x) exists i.e ;lim⁡x→af(x)=l;\mathop {\lim }\limits_{x \to a} f(x) = l,

where l is a finite value if 

 lim⁡x→a−f(x)=lim⁡x→a+f(x)\rm\mathop {\lim }\limits_{x → a^{-}} f(x)=\mathop {\lim }\limits_{x → a^{+}} f(x)

Calculation:

Let, y = lim⁡x→1[x + 1]\mathop {\lim }\limits_{x \to 1} [x\ +\ 1]

Taking the left-hand limit (LHL)

y = lim⁡x→1−[x + 1]y\ =\ \mathop {\lim }\limits_{x \to 1^-} [x\ +\ 1]

⇒ y = lim⁡h→0[1 − h + 1]⇒ \ y\ =\ \mathop {\lim }\limits_{h \to 0} [1\ -\ h\ +\ 1]

⇒ y = lim⁡h→0[− h]⇒ \ y\ =\ \mathop {\lim }\limits_{h \to 0} [-\ h]

⇒ y = -1       (first integer less than zero)

Now,taking the right-hand limit (RHL)

y = lim⁡x→1+[x + 1]y\ =\ \mathop {\lim }\limits_{x \to 1^+} [x\ +\ 1]

⇒ y = lim⁡h→0[1 + h + 1]⇒ \ y\ =\ \mathop {\lim }\limits_{h \to 0} [1\ +\ h\ +\ 1]

⇒ y = lim⁡h→0[2 + h]⇒ \ y\ =\ \mathop {\lim }\limits_{h \to 0} [2\ +\ h]

⇒ y = 2       (first integer ≥ 2) 

Clearly, we can see that, LHL ≠ RHL, therefore the limit of this function does not exist

Hence, option 4 is correct.

Shortcut Trick Graph of y = [x + 1] is shown below

We can see, when we approach from the left-hand side (LHL) of 1, we get the value 1 and when approached from the right-hand side (RHL), we get the value 2.

It means LHL ≠ RHL

Hence, the limit of a given function does not exist.

Question 127

What is the order of the differential equation d2ydx2 + 2x = 0\frac{d^2y}{dx^2}\ +\ 2x\ =\ 0

  1. AOption (a)

    2

  2. BOption (b)

    1

  3. COption (c)

    3

  4. DOption (d)

    0

View answer & explanation

Correct answer

Answer: (a)A.

2

Why this answer is correct

Concept:

Order of a differential equation: It is the order of the highest derivative (also known as differential coefficient) present in the equation.

The degree of the differential equation: It is represented by the power of the highest order derivative in the given differential equation.

If the given differential equation is not a polynomial equation in derivatives, then the degree of this equation is not defined.

Calculation:

Given differential equation is

d2ydx2 + 2x = 0\frac{d^2y}{dx^2}\ +\ 2x\ =\ 0

We can see that the highest order of differential equation is 2.

Hence, the order of the differential equation is 2.

Question 128

Find out the common ratio of the GP if the first term of a GP is 1 and the sum of the third term and fifth term is 90

  1. AOption (a)

    3

  2. BOption (b)

    9

  3. COption (c)

    2

  4. DOption (d)

    4

View answer & explanation

Correct answer

Answer: (a)A.

3

Why this answer is correct

Given:

First-term of GP = 1 and

The Sum of the third and fifth terms = 90

Concept:

Geometric progression: In a geometric progression, the ratio between 2 consecutive terms remains constant.

If 'a' is the first term of GP and 'r' is the common ration, then GP is written as

a, ar2, ar3, ar4, ar5 ...........

Calculation:

Let, 'a' is the first term and 'r' is a common ratio,

According to question

a = 1        ---(1)

ar2 + ar4 = 90   

From equation (1)

r4 + r2 = 90

⇒ r4 + 10r2 - 9r2 - 90 = 0

⇒ r2(r2 + 10) - 9(r2 + 10) = 0

⇒ (r2 + 10)(r2 - 9) = 0

Since r2 + 10 can not be zero, this is a positive number.

⇒ r2 - 9 = 0

⇒ r = ± 3

Hence, from the given answer, 3 is the common ratio of given geometric progression.

Question 129

What is the sum of the next three numbers in the sequence?

1, 3, 6, 10, 15, 21,...

  1. AOption (a)

    109

  2. BOption (b)

    76

  3. COption (c)

    108

  4. DOption (d)

    74

View answer & explanation

Correct answer

Answer: (a)A.

109

Why this answer is correct

Calculation:

Given series is 

1, 3, 6, 10, 15, 21 .......

This can be written as 

1, 1 + 2, 3 + 3, 6 + 4, 10 + 5, 15 + 6, 21 + 7, 28 + 8, 36 + 9, ........

= 1, 3, 6, 10, 15, 21, 28, 36, 45, ......

According to question, sum of next three numbers in sequence

= 28 + 36 + 45 

= 109

Hence, the sum will be 109

Question 130

What will be the order and degree of the differential equation:

3x(d3ydx3) + 2x2(dydx)3 − 5xy = 03x(\frac{d^3y}{dx^3}) \ + \ 2x^2(\frac{dy}{dx})^3\ -\ 5xy\ =\ 0

  1. AOption (a)

    First order, fourth degree

  2. BOption (b)

    Third order, first degree

  3. COption (c)

    Third order, Third degree

  4. DOption (d)

    First order, third degree

View answer & explanation

Correct answer

Answer: (b)B.

Third order, first degree

Why this answer is correct

Concept:

Order: The order of a differential equation is the order of the highest derivative (also known as differential coefficient) present in the equation.

The degree: The degree of the differential equation is represented by the power of the highest order derivative in the given differential equation.

If the given differential equation is not a polynomial equation in derivatives, then the degree of this equation is not defined.

​Calculation:

Given differential equation is

3x(d3ydx3) + 2x2(dydx)3 − 5xy = 03x(\frac{d^3y}{dx^3}) \ + \ 2x^2(\frac{dy}{dx})^3\ -\ 5xy\ =\ 0

Here d3ydx3\frac{d^3y}{dx^3} term represents the order of the equation which is 3.

Since the power of term d3ydx3\frac{d^3y}{dx^3} is one, therefore the degree of the equation is 1.

Hence, the differential equation has third order and first degree.

So option (ii) is correct.

Question 131

If n is any positive integer then 1 + 3 + 5 +...+(2n - 1) = ?

  1. AOption (a)

    n - 1

  2. BOption (b)

    n

  3. COption (c)

    n2

  4. DOption (d)

    2n - 1

View answer & explanation

Correct answer

Answer: (c)C.

n2

Why this answer is correct

Concept:

Arithmetic Progression (AP): The series of numbers where the difference of any two consecutive terms is the same, is called an Arithmetic Progression.

If three numbers a, b and c are in AP, then

b - a = c - b ⇒ 2b = a + c.

Formula used: 

Nth term of AP: 

Tn = a + (n - 1)d

Sum of n terms of AP:

Sn = n2(a + l)S_n\ =\ \frac{n}{2}(a\ + \ l)   where,

a = first term, d = common difference

Tn = nth term , l = last term and

n = number of terms in AP.

Calculation:

Let,

S = 1 + 3 + 5 +...+(2n - 1)    ----(1)

First term a = 1, common difference d =  3 - 1 = 2 

Let, given AP has 'm' terms and from equation (1)

Last term or mth term = 2n - 1

∵ Tn = a + (n - 1)d

⇒ 2n - 1 =  1 + (m - 1)2

⇒ 2n - 2 = 2m - 2

⇒  m = n

Hence, the sum of n (or m) terms of AP, by using the formula

 Sn = n2(a + l)S_n\ =\ \frac{n}{2}(a\ + \ l)

⇒ Sn = n2(1 + 2n − 1)S_{n}\ =\ \frac{n}{2}(1 \ + \ 2n\ -\ 1)

⇒ Sn = n2

Hence, the sum of the series will be n2.

Mistake PointsNote that, last term of the given AP is '2n - 1', not 'n'.

Hence, you have to let a total number of terms other than 'n'. 

Additional InformationGeometric Progression (GP): The series of numbers where the ratio of any two consecutive terms is the same, is called a Geometric Progression.

If three numbers a, b and c are in GP, thenba=cb\rm \dfrac ba=\dfrac cb ⇒ b2 = ac.

Harmonic Progression (HP): The series of numbers where the reciprocals of the terms are in Arithmetic Progression, is called a Harmonic Progression.

If three numbers a, b and c are in HP, then 1a+1c=2b\rm \dfrac{1}{a}+\dfrac{1}{c}=\dfrac{2}{b}.

Question 132

Which one is not the characteristics of Transcritical bifurcation:

  1. AOption (a)

    Two steady-state branches collide

  2. BOption (b)

    Dashed line denotes unstable steady-state

  3. COption (c)

    Non-linear implicit scalar equation

  4. DOption (d)

    Solid line denotes stable steady-state

View answer & explanation

Correct answer

Answer: (c)C.

Non-linear implicit scalar equation

Why this answer is correct

Explanation:

A bifurcation occurs in a nonlinear differential equation when a small change in a parameter results in a qualitative change in the long-time solution.

The corresponding differential equation will be written as:

x˙=fr(x)\dot x=f_r(x)

Transcritical Bifurcation:

A transcritical bifurcation occurs when there is an exchange of stabilities between two fixed points. The normal form for a transcritical bifurcation is given by,

x˙=rx−x2\dot x=rx - x^2

The fixed points are x∗ = 0  and x∗ = r .

The derivative of the right-hand-side is  f′(ẋ) = r − 2x,

so that  f′(0) = r  and  f′(r) = −r .

Therefore, for r < 0, x∗ = 0 is stable and x∗ = r is unstable, while for r > 0, x∗ = r  is stable (solid line) and x∗ = 0 is unstable.

The two fixed points thus exchange stability as r passes through zero.

Question 133

If dy = 2x5dx; then what is the equation of the y in terms of x if curves passes through (1, 2)

  1. AOption (a)

    x6 + 3y - 5 = 0

  2. BOption (b)

    x6 - 3y + 5 = 0

  3. COption (c)

    x5 + y - 2 = 0

  4. DOption (d)

    x5 - y + 2 = 0

View answer & explanation

Correct answer

Answer: (b)B.

x6 - 3y + 5 = 0

Why this answer is correct

Concept: 

If the function y = f(x)  passes through point (a, b), then the point will setisfy f(x).

Formula used:

∫xndx = xn + 1n + 1 + C\int x^ndx\ =\ \frac{x^{n\ +\ 1}}{n\ +\ 1}\ +\ C

Calculation:

Given that,

dy = 2x5dx         

Integrating both sides

⇒ ∫dy=∫2x5dx⇒ \ \int dy = \int 2x^5dx

⇒ y = 2.x66 + C⇒ \ y\ =\ 2.\frac{x^6}{6}\ +\ C              .......(1)

According to the question, equation (1) passes through (1, 2), therefore it will satisfy the equation.

⇒ 2 = 2.166 + C⇒ \ 2\ =\ 2.\frac{1^6}{6}\ +\ C

⇒ C = 5/3

So that, equation (1) becomes 

⇒ y = x63 + 53⇒ \ y\ =\ \frac{x^6}{3}\ +\ \frac{5}{3}

⇒ x6 - 3y + 5 = 0

Hence, option 2 will be correct.

Question 134

At what value or values of x this function is discontinuous:   {\rm{f}}\left( {\rm{x}} \right) = \left{ {\begin{array}{*{20}{c}} { x\ +\ 4}&{{\rm}}&{{\rm{x}} \le -1}\ {{\rm{x^2}} {\rm{}}}&{{\rm{}}}&{-1 < {\rm{x}} < 1}\ 2\ -\ x&{{\rm}}&{{\rm{x}} \ge 1} \end{array}} \right.

  1. AOption (a)

    1

  2. BOption (b)

    2

  3. COption (c)

    -1

  4. DOption (d)

    0

View answer & explanation

Correct answer

Answer: (c)C.

-1

Why this answer is correct

Concept:

  1. A function f(x) is said to be continuous at a point x = a, in its domain if lim⁡x→af(x)=f(a)\mathop {\lim }\limits_{x \to {\rm{a}}} f\left( x \right) = f\left( a \right) exists or its graph is a single unbroken curve.
  2. f(x) is Continuous at x = a ⇔ lim⁡x→a+f(x)=lim⁡x→a−f(x)=lim⁡x→af(x)\mathop {\lim }\limits_{x \to {{\rm{a}}^ + }} f\left( x \right){\rm{}} = {\rm{}}\mathop {\lim }\limits_{x \to {{\rm{a}}^ - }} f\left( x \right){\rm{}} = {\rm{}}\mathop {\lim }\limits_{x \to {\rm{a}}} f\left( x \right)

Calculation:

Given, {\rm{f}}\left( {\rm{x}} \right) = \left{ {\begin{array}{*{20}{c}} { x\ +\ 4}&{{\rm}}&{{\rm{x}} \le -1}\ {{\rm{x^2}} {\rm{}}}&{{\rm{}}}&{-1 < {\rm{x}} < 1}\ 2\ -\ x&{{\rm}}&{{\rm{x}} \ge 1} \end{array}} \right.

Using the relation 

lim⁡x→a+f(x)=lim⁡x→a−f(x)=lim⁡x→af(x)\mathop {\lim }\limits_{x \to {{\rm{a}}^ + }} f\left( x \right){\rm{}} = {\rm{}}\mathop {\lim }\limits_{x \to {{\rm{a}}^ - }} f\left( x \right){\rm{}} = {\rm{}}\mathop {\lim }\limits_{x \to {\rm{a}}} f\left( x \right)

Hence, left-hand limit (LHL) at x = -1

lim⁡x→−1−f(x)=lim⁡x→−1−(x + 4)\mathop {\lim }\limits_{x \to {{\rm{-1}}^ - }} f( x ) = \mathop {\lim }\limits_{x \to {{\rm{-1}}^ - }} (x\ +\ 4)

⇒ lim⁡x→−1−f(x)=−1+ 4 = 3\Rightarrow \ \mathop {\lim }\limits_{x \to {{\rm{-1}}^ - }} f( x ) = -1 +\ 4\ =\ 3        .......(1)

and right-hand limit (RHL) at x = -1

lim⁡x→−1+f(x)=lim⁡x→−1+(x2)\mathop {\lim }\limits_{x \to {{\rm{-1}}^ + }} f( x ) = \mathop {\lim }\limits_{x \to {{\rm{-1}}^ + }} (x^2)

⇒ lim⁡x→−1+f(x)=lim⁡x→−1+(12) = 1\Rightarrow \ \mathop {\lim }\limits_{x \to {{\rm{-1}}^ + }} f( x ) = \mathop {\lim }\limits_{x \to {{\rm{-1}}^ + }} (1^2)\ =\ 1       .......(2)

We can see that, LHL ≠ RHL, therefore function is discontinuous at x = -1

Hence, option 3 is correct.

Question 135

The difference equation Pn + 1 = aPn + b. Then calculate P2 if P0 = 2, a = 3, and b = -1.

  1. AOption (a)

    -7

  2. BOption (b)

    1

  3. COption (c)

    14

  4. DOption (d)

    5

View answer & explanation

Correct answer

Answer: (c)C.

14

Why this answer is correct

Solution:

Pn + 1 = aPn + b

given, a = 3 and b = -1

Pn+1 = 3Pn - 1 . . .(i) 

Put n=0 

P1 = 3 P0 - 1 = 6-1 = 5 

Put  n = 1 in equation (i) 

P2 = 3P2 - 1 = 15-1 = 14

Hence, P2 is 14.

Question 136

In which option we can evaluate the limit by using L'Hop̂ital's rule? 

  1. AOption (a)

    lim⁡x→ππcos⁡x(x − π)\mathop {\lim }\limits_{x \to \pi}\pi \cos x(x \ -\ \pi)

  2. BOption (b)

    lim⁡x→∞e−x\mathop {\lim }\limits_{x \to \infty} e^{-x}

  3. COption (c)

    lim⁡x→0x tan⁡(x)\mathop {\lim }\limits_{x \to 0}x\ \tan (x)

  4. DOption (d)

    lim⁡x→1ln⁡(x)x − 1\mathop {\lim }\limits_{x \to 1}\frac{\ln(x)}{x\ -\ 1}

View answer & explanation

Correct answer

Answer: (d)D.

lim⁡x→1ln⁡(x)x − 1\mathop {\lim }\limits_{x \to 1}\frac{\ln(x)}{x\ -\ 1}

Why this answer is correct

Concept:

L’Hopital’s Rule: If we have an indeterminate form 0/0 or ∞/∞ we can apply L'Hopital's rule.

We need to differentiate the numerator and denominator and then take the limit.

Calculation:

We can simply check options one by one.

1) lim⁡x→ππcos⁡x(x − π)\mathop {\lim }\limits_{x \to π}π \cos x(x \ -\ π)

If we put x = π, we will get 

π cos0 = π       (∵ cos 0 = 1) 

2) lim⁡x→∞e−x\mathop {\lim }\limits_{x \to ∞} e^{-x}

This can also be written as 

lim⁡x→∞1ex\mathop {\lim }\limits_{x \to ∞} \frac{1}{e^{x}}

If we will put x = ∞, we will get 

  1∞ = 0\frac{1}{\infty}\ = \ 0

3) lim⁡x→0x tan⁡(x)\mathop {\lim }\limits_{x \to 0}x\ \tan(x)

If we put x = 0, we will get

0 × tan0 = 0

4) lim⁡x→1ln⁡(x)x − 1\mathop {\lim }\limits_{x \to 1}\frac{\ln(x)}{x\ -\ 1}

If we put x = 1, we will get

ln(1)/0 = 0/0        (∵ ln1 = 0)

Hence, from the above discussion, we can see that we can solve

lim⁡x→1ln⁡(x)x − 1\mathop {\lim }\limits_{x \to 1}\frac{\ln(x)}{x\ -\ 1} by L'Hopital's rule.

Question 137

Calculate lim⁡n→∞(1+1n)n\mathop {\lim }\limits_{n \to \infty}{\left(1+\frac{1}{n}\right)^n}

  1. AOption (a)

    n

  2. BOption (b)

    e

  3. COption (c)

    1

  4. DOption (d)

    0

View answer & explanation

Correct answer

Answer: (b)B.

e

Why this answer is correct

Concept:

L’Hospital’s Rule: If we have an indeterminate form 0/0 or ∞/∞, we need to differentiate the numerator and denominator and then take the limit.

Formula used: 

eln⁡x = x e^{\ln x}\ = \ x

ddx(1x) = −1x2\frac{d}{dx}(\frac{1}{x})\ = \ \frac{-1}{x^2}  and  ddx(ln⁡x) = 1x\frac{d}{dx}( \ln x)\ = \ \frac{1}{x}

lim⁡n→∞(1+an)n=ea\mathop {\lim }\limits_{n \to \infty}\left ( 1 + \frac{a}{n} \right )^n = e^a

Calculation:

Let, y = lim⁡n→∞(1+1n)n\mathop {\lim }\limits_{n \to \infty}{\left(1+\frac{1}{n}\right)^n}

by using the above logarithmic property,

y=eln⁡lim⁡n→∞(1+1n)ny = e^{{\ln } \mathop {\lim }\limits_{n \to \infty}{\left(1+\frac{1}{n}\right)^n}}

y=elim⁡n→∞ln⁡(1+1n)ny = e^{ \mathop {\lim }\limits_{n \to \infty}\ln {\left(1+\frac{1}{n}\right)^n}}

 y=elim⁡n→∞nln⁡(1 +1n)y =e^{{\mathop {\lim }\limits_{n \to \infty}}n\ln(1\ + \frac{1}{n})}

⇒ y = elim⁡n→∞ln(1+1n)1n⇒ \ y\ = \ e^{\mathop {\lim }\limits_{n \to \infty }\frac{ln\left ( 1 + \frac{1}{n} \right )}{\frac{1}{n}}}

now, the power of 'e' has an indeterminate form of 0/0, we can apply L-hospital's rule and differentiate the numerator and denominator.

⇒ y = elim⁡n→∞−1n2(1+1n)(−1n2)⇒ \ y\ =\ e^{\mathop {\lim }\limits_{n \to \infty }\frac{\frac{-1}{n^2}}{\left ( 1 + \frac{1}{n} \right )\left ( \frac{-1}{n^2} \right )}}

⇒ y= elim⁡n→∞11+1n⇒ \ y= \ e^{\mathop {\lim }\limits_{n \to \infty }\frac{1}{1+\frac{1}{n}}}

⇒ y  = e

Hence, the value of a given expression is e.

Shortcut Trick

Let, y = lim⁡n→∞(1+1n)n\mathop {\lim }\limits_{n \to \infty}{\left(1+\frac{1}{n}\right)^n}

This is the form of 1∞ which can be solved using the above formula.

On comparing it from the above formula, we can see that,

a = 1

Therefore,

y = e

Hence, the value of the given expression will be e.

Question 138

Let us say a, b, c are in AP then which of the following is not true?

  1. AOption (a)

    b + b = a + c

  2. BOption (b)

    b + a = c + b

  3. COption (c)

    2b = a + c

  4. DOption (d)

    b - a = c - b

View answer & explanation

Correct answer

Answer: (b)B.

b + a = c + b

Why this answer is correct

Concept:

Arithmetic progression (AP) is a mathematical sequence in which the difference between two consecutive terms is always a constant.

Common difference (d) in AP:

Let’s suppose, \({{\rm{a}}{\rm{1}}},{{\rm{a}}{{\rm{2;}}}},{{\rm{a}}{{\rm{3;}}}}, \ldots .,{{\rm{a}}{\rm{n}}}\) is an AP, the common difference ‘ d ’can be obtained as:

d = \({{\rm{a}}{\rm{2}}} - {{\rm{a}}{{\rm{1;}}}} = {{\rm{a}}{\rm{3}}} - {{\rm{a}}{{\rm{2;}}}} = ; \ldots \ldots = {{\rm{a}}{\rm{n}}} - {{\rm{a}}{{\rm{n - 1}}}}\)

Calculation:

Given that, 

a, b, c are in AP

we know that, for an AP, the difference between two consecutive terms is always a constant.

⇒ b - a = c - b   

⇒ b + b = a + c

⇒ 2b = a + c

Therefore, we can see that, 

b + a = c + b does not satisfied.

Hence, option 2 is correct.

Additional Information To find nth term of an AP:

Tn = a + (n − 1) × d

Tn = Sn – Sn-1

To find sum of ‘n’ terms of an AP:

Sn=n2[2a;+;(n;−;1);×;d]{{\rm{S}}_{\rm{n}}} = {\rm{}}\frac{{\rm{n}}}{2}\left[ {2{\rm{a;}} + {\rm{;}}\left( {{\rm{n;}} - {\rm{;}}1} \right){\rm{;}} \times {\rm{;d}}} \right] also,

Sn=n2[first;term+last;term]{{\rm{S}}_{\rm{n}}} = \frac{{\rm{n}}}{2}\left[ {{\rm{first;term}} + {\rm{last;term}}} \right]

Where,

a = First term, d = Common difference

n = number of terms, Tn = nth term

Sn = sum of ‘n’ terms

Question 139

The half-life of radium is 1600 years, i.e., it takes 1600 years for half of any quantity to delay. If a sample initially contains 50 g, how long will it be unit it contains 45 g?

  1. AOption (a)

    212.3 years

  2. BOption (b)

    344.3 years

  3. COption (c)

    243.2 years

  4. DOption (d)

    282.4 years

View answer & explanation

Correct answer

Answer: (c)C.

243.2 years

Why this answer is correct

Formula Used:

A=A0⋅(1/2)t/hA=A_0 \cdot(1 / 2)^{t/ h}

Where A is the final amount, A0 is the initial amount, t is the time taken to reach the final amount and h is the half-life of the substance.

Calculation:

Here, A=45 gm, A0 = 50 gm,  h=1600 years and we have to calculate t.

⇒  45=50⋅(1/2)t/1600 45=50\cdot(1 / 2)^{t / 1600}

⇒ 910=(1/2)t/1600 \frac{9}{10}=(1 / 2)^{t / 1600}

take log10 both sides

 ⇒log⁡10910=t1600log⁡10(1/2) \log _{10} \frac{9}{10}=\frac{t}{1600} \log _{10}(1 / 2)

 ⇒ log⁡1032−log⁡1010=−t1600log⁡102 \log _{10} 3^2-\log _{10} 10=\frac{-t}{1600} \log _{10} 2

 log⁡103≈0.4771 and log⁡102≈0.3010 \log _{10} 3 \approx 0.4771 \text { and } \log _{10} 2 \approx 0.3010

 ⇒ 2×0.4771−1=−t1600×0.3010 2 \times 0.4771-1=\frac{-t}{1600} \times 0.3010

 ⇒ 0.9542−1=−t1600×0.301 0.9542-1=-\frac{t}{1600} \times 0.301

 ⇒ −0.0458×16000.301=−t \frac{-0.0458 \times 1600}{0.301}=-t

 ⇒ t=73.280.301≈243.46 t=\frac{73.28}{0.301} \approx 243.46

Note: For a more exact answer we can take the more accurate values of log10 2 and log10 3.

Hence, 243.2 years is the closest option here.

Question 140

If Y is in terms of variable x as Y = f(x) then y is known as

  1. AOption (a)

    implicit function

  2. BOption (b)

    linear function

  3. COption (c)

    identity function

  4. DOption (d)

    explicit function

View answer & explanation

Correct answer

Answer: (d)D.

explicit function

Why this answer is correct

Concept: 

Implicit function: A function or a relation in which the dependent variable is not isolated on one side of the equation.

Linear function: A mathematical function in which the variables appear only in the first degree, are multiplied by constants and are combined only by addition and subtraction called a linear function.

Identity function: If f is a function defined on R i.e f: R → R such that f(x) = x ∀ x ∈ R then f is an identity function.

Explicit function:

  1. A mathematical function containing only the independent variable or variables as opposed to an implicit function called explicit function.
  2. A function in which dependent variables can be written as in terms of the independent variable called explicit function.

Calculation:

Given that,

Y = f(x)

This means y can be written in terms of x. 

Hence, it will be an explicit function.

Discipline 3 (20 questions)

Question 141

What is the Green's value for the functions F = y2 and G = x2 for the region x = 1 and y = 2 from origin.

  1. AOption (a)

    2

  2. BOption (b)

    -1

  3. COption (c)

    -2

  4. DOption (d)

    1

View answer & explanation

Correct answer

Answer: (c)C.

-2

Why this answer is correct

Concept:

Green Theorem: A line integral around a simple closed curve C to a double integral over the plane region D bounded by C, i.e.

∫∫D(dMdx−dNdy)dx dy=∫(N dx+M dy)\displaystyle \int \int_{D} \left ( \frac {dM}{dx} - \frac {dN}{dy} \right) dx\ dy = \int (N\ dx + M\ dy)

Calculation:

Given:

F = y2 and G = x2 for the region x = 1 and y = 2

Here M = G & N = F

Applying Green's theorem

∫∫D(dGdx−dFdy)dx dy=∫∫D(dx2dx−dy2dy)dx dy\displaystyle \int \int_{D} \left ( \frac {dG}{dx} - \frac {dF}{dy} \right) dx\ dy = \displaystyle \int \int_{D} \left ( \frac {dx^2}{dx} - \frac {dy^2}{dy} \right) dx\ dy

=∫01∫02(2x − 2y)dx dy= \displaystyle \int_{0}^{1} \int_{0}^{2} (2x\ -\ 2y) dx\ dy

=∫02∣x2−2xy∣01dy= \displaystyle \int_{0}^{2} |x^2 - 2xy|_{0}^{1} dy

=∫02(1−2y)dy= \displaystyle \int_{0}^{2} (1-2y) dy

=∣y − y2∣02= |y\ -\ y^2|_{0}^{2}

= 2 - 4

= -2

Question 142

A function f on [0, 1] by  \rm f(x) = \left { \begin{matrix} \rm 1 & \rm x \ \epsilon\ \frac{1}{n}\ for\ some\ n\ \epsilon\ N \ \rm 0 & \rm \ otherwise \end{matrix}\right.  find ∫f with limits 0 to 1.

  1. AOption (a)

    n

  2. BOption (b)

    0

  3. COption (c)

    -1

  4. DOption (d)

    1

View answer & explanation

Correct answer

Answer: (b)B.

0

Why this answer is correct

Concept:

The given function is defined on the interval [0,1] [0,1] as

f(x)={1,if x=1n for some n∈N, 0,otherwise. f(x)= \begin{cases} 1, & \text{if } x=\frac{1}{n} \text{ for some } n\in\mathbb{N},\ 0, & \text{otherwise}. \end{cases}

The set \left{\frac{1}{n} : n\in\mathbb{N}\right} is a countable subset of [0,1] [0,1] .

A countable set has Lebesgue measure zero.

The function f(x) f(x) differs from the zero function only on this measure-zero set.

Changing the values of a function on a set of measure zero does not affect the value of its integral.

Hence,

∫01f(x),dx=0 \int_0^1 f(x),dx = 0

The correct answer is option (2).

Question 143

Find the maximum value from the extreme values of this function:

u = x2y2 - 5x2 - 8xy - 5y2

  1. AOption (a)

    (0, 0)

  2. BOption (b)

    (-1, 1)

  3. COption (c)

    (1, -1)

  4. DOption (d)

    (3, 3)

View answer & explanation

Correct answer

Answer: (a)A.

(0, 0)

Why this answer is correct

Concept:

Extreme values: If y = f(x) is any function, than the value of x where f'(x) becomes zero called extreme values.

On these points, slope of function becomes zero i.e.

dy/dx = f'(x) = 0

Calculation:

Given :

u=x2y2−5x2−8xy−5y2 u = x^2y^2 - 5x^2 - 8xy - 5y^2

Step 1: First partial derivatives

∂u∂x=2xy2−10x−8y \frac{\partial u}{\partial x} = 2xy^2 - 10x - 8y

∂u∂y=2yx2−8x−10y \frac{\partial u}{\partial y} = 2yx^2 - 8x - 10y

For extreme values, set both partial derivatives equal to zero.

Step 2: Critical points

Solving the equations

2xy2−10x−8y=0 2xy^2 - 10x - 8y = 0

2yx2−8x−10y=0 2yx^2 - 8x - 10y = 0

We obtain the critical points:

(0,0), (1,−1), (−1,1), (3,3), (−3,−3) (0,0),\ (1,-1),\ (-1,1),\ (3,3),\ (-3,-3)

Step 3: Second partial derivatives

uxx=2y2−10 u_{xx} = 2y^2 - 10

uyy=2x2−10 u_{yy} = 2x^2 - 10

uxy=4xy−8 u_{xy} = 4xy - 8

The discriminant is

D=uxxuyy−(uxy)2 D = u_{xx}u_{yy} - (u_{xy})^2

Step 4: Nature of critical points

At (0,0):

uxx=−10, uyy=−10, uxy=−8 u_{xx} = -10,\ u_{yy} = -10,\ u_{xy} = -8

D=100−64=36>0 D = 100 - 64 = 36 > 0

Since uxx<0 u_{xx} < 0 , this point gives a maximum.

u(0,0)=0 u(0,0) = 0

At (1,-1) and (-1,1):

The discriminant is negative, so these are saddle points.

At (3,3) and (-3,-3):

The discriminant is also negative, so these are saddle points.

Step 5: Maximum value

The only maximum occurs at (0,0) (0,0)

The correct answer is (0,0)

Question 144

Which of these functions is not uniformly continuous on (0, 1)?

  1. AOption (a)

    x2

  2. BOption (b)

    sin(x) ÷ x

  3. COption (c)

    1 ÷ x2

  4. DOption (d)

    sin(x)

View answer & explanation

Correct answer

Answer: (c)C.

1 ÷ x2

Why this answer is correct

Concept:

Uniform continuity on an Interval: The function f is uniformly continuous on I if for every ε > 0, there exists a δ > 0 such that

|x − y| < δ  ⇒  |f(x) − f(y)| < ε.

Here, the δ may depend on ε but NOT on the points.

  1. Uniform continuity is stronger than continuity,
  2. If 'f' is uniformly continuous on an interval (a, b), then it is continuous on (a, b).

Calculation:

Option 1: x2 is uniformly continuous on (0,1) because continuous functions are always uniformly continuous on a compact set.

Option 2: f(x) = sinxxf(x) \ =\ \frac{sinx}{x}

We know that this is the standard form which is uniformly continuous on (0, 1)

Option 3: 1x2\frac{1}{x^2}

We can see that δ is not dependent only on ε but it is dependent on point.

Hence 1x2\frac{1}{x^2} is not uniformly continuous on (0, 1).

Option 4: sinx is uniformly continuous on (0,1) because continuous functions are always uniformly continuous on a compact set.

Additional Information

Continuity on an Interval: The real-valued function 'f' defined on an open interval I is said to be continuous at a ϵ I,

if for any given number ε > 0 however small, we can find a δ > 0 such that

|x - a| < δ  ⇒   |f(x) - f(a)| < ε

Or we can also write

x ϵ (a - δ , a + δ)   ⇒  f(x)  ϵ (f(a) - ε, f(a) + ε)

Here, the δ may depend BOTH on a and ϵ.

Question 145

Find the value of ∫cot3(x) cosec4 (x)dx

  1. AOption (a)

    − (cot4x4 + cosec6x6)-\ (\frac{cot^4x}{4}\ +\ \frac{cosec^6x}{6})

  2. BOption (b)

    − (cosec4x4 + cosec6x6)-\ (\frac{cosec^4x}{4}\ +\ \frac{cosec^6x}{6})

  3. COption (c)

    − (cosec4x4 + cot6x6)-\ (\frac{cosec^4x}{4}\ +\ \frac{cot^6x}{6})

  4. DOption (d)

    − (cot4x4 + cot6x6)-\ (\frac{cot^4x}{4}\ +\ \frac{cot^6x}{6})

View answer & explanation

Correct answer

Answer: (d)D.

− (cot4x4 + cot6x6)-\ (\frac{cot^4x}{4}\ +\ \frac{cot^6x}{6})

Why this answer is correct

Formula used:

1 + cot2(x) = cosec2(x)

ddxcosec2x = − cot2x\frac{d}{dx}cosec^2x\ =\ -\ cot^2x

∫xndx = xn + 1n + 1 + C\int {x^n}dx\ =\ \frac{x^{n\ +\ 1}}{n\ +\ 1}\ +\ C

Calculation:

Let I = ∫cot3(x) cosec4(x)dx

⇒ I = ∫cot3(x) cosec2(x) cosec2(x)dx

⇒ I = ∫cot3(x)[1 + cot2(x)] cosec2(x)dx        ........(1)

⇒ cot(x)  = t

we know that, 

ddxcosec2x = − cot2x\frac{d}{dx}cosec^2x\ =\ -\ cot^2x

Differentiating both side, we will get

  • cosec2xdx = dt               

Hence, from equation (1) we will get

I = - ∫t3(1 + t2}dt 

⇒ I = - ∫ (t3 + t5)dt 

⇒ I = − (t44 + t66)-\ (\frac{t^4}{4}\ +\ \frac{t^6}{6})

But, t = cot(x)

⇒ I = − (cot4x4 + cot6x6)-\ (\frac{cot^4x}{4}\ +\ \frac{cot^6x}{6})

Hence, option 4 is the correct answer.

Question 146

Which conditions are necessary to explain that the upper and lower integrals of a bounded function exist?

  1. AOption (a)

    a, b ϵ R, a < b, and f ∶ [a, b] → R

  2. BOption (b)

    a, b ϵ R, a < b, and f ∶ [a, b] → R be bounded

  3. COption (c)

    a, b ϵ R, a < b, and f ∶ [a, b] → Z be bounded

  4. DOption (d)

    a, b ϵ R, a > b, and f ∶ [a, b] → R

View answer & explanation

Correct answer

Answer: (b)B.

a, b ϵ R, a < b, and f ∶ [a, b] → R be bounded

Why this answer is correct

Concept:

Partition: By partition of [a, b], we mean a finite set P of points x1, x2, ....., xn Where,

a = x0 ≤ x1 ≤ ....≤ xn-1 ≤ xn = b

Riemann integral: A bounded function f: [a, b] → R is said to be Riemann integral over [a, b] if 

∫abˉf(x)dx = ∫aˉbf(x)dx\int_{a}^{\bar{b}}f(x)dx\ =\ \int_{\bar{a}}^{b}f(x)dx

and f is denoted by ∫abf(x)dx \int_{a}^{b}f(x)dx

1. The upper integral of f on [a, b]:

U∫abf(x)dx = inf [U(f,p)]U\int_{a}^{b}f(x)dx\ =\ inf\ [U(f, p)]

2. The upper integral of f on [a, b]:

L∫abf(x)dx = sup [L(f,p)]L\int_{a}^{b}f(x)dx\ =\ sup\ [L(f, p)]

Where P is a partition of [a, b].

Conditions of integrability:

  • A bounded function is said to be integrable when upper and lower integrals are equal.
  • Riemann’s criterion for integrability: f is integrable on [a, b] ⇔ for every ϵ > 0 there exists a partition P such that U(P, f) − L(P, f) < ϵ
  • If f is continuous on [a, b] then f is integrable.
  • If f is a monotone function then f is integrable.

Calculation:

From the above discussion, we can say that if upper and lower integrals are the same then integral is said to be Riemann integrable i.e.,

∫abf(x)dx = U∫abf(x)dx = L∫abf(x)dx \int_{a}^{b}f(x)dx\ =\ U\int_{a}^{b}f(x)dx\ =\ L\int_{a}^{b}f(x)dx

A necessary condition for providing the upper and lower integrals of a bounded function exists if and only if

a, b ϵ R, a < b, and f : [a, b] → R to be bounded.

Hence, the correct answer is option 2**).**

Question 147

Find it (x, y) → (0, 0) (sin h(x) × sin h(y)) ÷ (xy)

  1. AOption (a)

    0

  2. BOption (b)

    ∞ 

  3. COption (c)

    990

  4. DOption (d)

    1

View answer & explanation

Correct answer

Answer: (d)D.

1

Why this answer is correct

Concept Used:

sinh⁡(x)=ex−e−x2 and cos⁡h(x)=ex+e−x2 \sinh (x)=\frac{e^x-e^{-x}}{2} \text { and } \cos h(x)=\frac{e^x+e^{-x}}{2}

L'horpital's rule,

  if lim⁡x→af(x)g(x)→00 \text { if } \lim _{x \rightarrow a} \frac{f(x)}{g(x)} \rightarrow \frac{0}{0} (indeterminart form)

 ⇒lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x) \Rightarrow \lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)}

and lim⁡x→a[f(x)g(x)]=lim⁡x→af(x)⋅lim⁡x→ag(x) \lim _{x \rightarrow a}[f(x) g(x)]=\lim _{x \rightarrow a} f(x) \cdot \lim _{x \rightarrow a} g(x)

Calculation:

lim⁡x→0sinh⁡(x)x=00 \lim _{x \rightarrow 0} \frac{\sinh (x)}{x}=\frac{0}{0}

 ⇒lim⁡x→0cosh⁡(x)1=lim⁡x→0ex+e−x2=e0+e02=22=1 \Rightarrow \lim _{x \rightarrow 0} \frac{\cosh (x)}{1}=\lim _{x \rightarrow 0} \frac{e^x+e^{-x}}{2}=\frac{e^0+e^0}{2}=\frac{2}{2}=1

lim⁡(x,y)→0sinh⁡(x)⋅sinh⁡(y)xy=lim⁡x→0sinh⁡(x)xlim⁡sinh⁡(y)y→0=1 \lim _{(x, y) \rightarrow 0} \frac{\sinh (x) \cdot \sinh (y)}{x y}=\lim _{x \rightarrow 0} \frac{\sinh (x)}{x} \lim \frac{\sinh (y)}{y \rightarrow 0}=1

Hence, the required limit is 1.

Question 148

Calculate the value of the integral ∮C\oint_C(xydy - y2dx) where C is the square, cut from the first quadrant by the lines x = 1 and y = 1, will be? (Using green's theorem)

  1. AOption (a)

    3 ÷ 2

  2. BOption (b)

    7 ÷ 2

  3. COption (c)

    5 ÷ 2

  4. DOption (d)

    1 ÷ 2

View answer & explanation

Correct answer

Answer: (a)A.

3 ÷ 2

Why this answer is correct

Concept Used:

Let C be a positively oriented, piecewise smooth, simple closed curve in a plane, and let D be the region bounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there, then

∮C(Ldx+Mdy)=∬D(∂M∂x−∂L∂y)dxdy \oint_C(L d x+M d y)=\iint_D\left(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right) d x d y

where the path of integration along C is anticlockwise.

Calculation:

Here,  ∮C\oint_C(xydy - y2dx)

L = -y2 and M = xy, using green's theorem

⇒  ∮C\oint_C(xydy - y2dx) = ∬R(∂(xy)∂x−∂(−y2)∂y)dxdy \iint_R\left(\frac{\partial (xy)}{\partial x}-\frac{\partial (-y^2)}{\partial y}\right) d x d y

⇒ ∫x=01∫y=01(y−(−2y))dxdy=∫x=01∫y=01(y+2y)dydx \int_{x=0}^1 \int_{y=0}^1\left({y}- (-2y)\right) d x d y = \int_{x=0}^1 \int_{y=0}^1\left({y}+ 2y\right) dydx

⇒ ∫x=01∫y=01(3y)dydx=∫x=013y22∣01dx \int_{x=0}^1 \int_{y=0}^1\left(3y\right) dydx =\int_{x=0}^1 \left.\frac{3y^2}{2}\right|_0^1dx

⇒∫x=01(32−0)dx=3x2∣01=32 \int_{x=0}^1 \left(\frac{3}{2} - 0\right)dx = \left.\frac{3x}{2}\right|_0^1= \frac{3}{2}

Hence, the required integral is 3÷2.

Question 149

A vector field with a vanishing curl is known as _______

  1. AOption (a)

    Cycloidal

  2. BOption (b)

    Irrotational

  3. COption (c)

    Rotational

  4. DOption (d)

    Solenoidal

View answer & explanation

Correct answer

Answer: (b)B.

Irrotational

Why this answer is correct

A vector field with a vanishing curl is called an irrotational ​vector.

Explanation:

Irrotational Vector: ​A vector point function F is said to be a rotational vector if curl F = 0

curl F = ∇ × F = \left[ {\begin{array}{*{20}{c}} i&j&k\ {\frac{\delta }{{\delta x}}}&{\frac{\delta }{{\delta y}}}&{\frac{\delta }{{\delta z}}}\ {{F_1}}&{{F_2}}&{{F_3}} \end{array}} \right]

Additional Information

If a vector point function F(x, y, z) = F1i  + F2j + F3k is defined and differentiable at each point in some region of space then the divergence of F is denoted by div F or ∇.F and define as,

div F = ∇.F = δF1δx+δF2δy+δF3δz\frac{{\delta {F_1}}}{{\delta x}} + \frac{{\delta {F_2}}}{{\delta y}} + \frac{{\delta {F_3}}}{{\delta z}} 

Solenoidal Vector: A vector function F is said to be a solenoidal vector if ∇.F = 0.

Question 150

Strokes theorem is used to convert the _______ integrals to _______ integrals

  1. AOption (a)

    line to surface

  2. BOption (b)

    line to area

  3. COption (c)

    surface to line

  4. DOption (d)

    volume to area

View answer & explanation

Correct answer

Answer: (c)C.

surface to line

Why this answer is correct

Stokes theorem:

  • Stoke's theorem enables us to transform the surface integral of the curl of the vector field A into the line integral of that vector field A over the boundary C of that surface and vice-versa. The theorem states.
  • The flux of the curl of a vector function A over any surface S of any shape is equal to the line integral of the vector field A over the boundary C of that surface i.e.

 

∬S,curl,,A.ds=∮C,A.ds\underset{S}{\mathop \iint },curl,,A.ds=\underset{C}{\mathop \oint },A.ds

Stokes Theorem is given as:

∮CF⃗⋅dr⃗=∫!!!∫Scurl;F⃗⋅n^ds\mathop \oint \nolimits_C \vec F \cdot d\vec r = \mathop \int!!!\int \nolimits_S^{} curl;\vec F \cdot \hat nds

It converts a line integral to a surface integral and uses the curl operation.

Curl;F⃗=∇⃗×F⃗{\rm{Curl;}}\vec F = \vec \nabla \times \vec F

∇⃗=∂∂xi^+∂∂yj^+∂∂zk^\vec \nabla = \frac{\partial }{{\partial x}}\hat i + \frac{\partial }{{\partial y}}\hat j + \frac{\partial }{{\partial z}}\hat k

Question 151

What is the divergence of the vector field  f⃗ = 6x2i^ + 3xy2j^ + xyz3k^\vec{f}\ =\ 6x^2\hat{i}\ +\ 3xy^2\hat{j}\ +\ xyz^3\hat{k}  at the point (2, 3, 4).

  1. AOption (a)

    106

  2. BOption (b)

    124

  3. COption (c)

    348

  4. DOption (d)

    100

View answer & explanation

Correct answer

Answer: (c)C.

348

Why this answer is correct

Concept:

Divergence of the vector function:

The net outward flux from a volume element around a point is a measure of the divergence of the vector field at that point.

Divergence of a function F⃗=F1i^+F2j^+F3k^\vec F = {F_1}\hat i + {F_2}\hat j + {F_3}\hat k

∇⋅F⃗=∂F1∂x+∂F2∂y+∂F3∂z \nabla \cdot \vec F = \frac{{\partial {F_1}}}{{\partial x}} + \frac{{\partial {F_2}}}{{\partial y}} + \frac{{\partial {F_3}}}{{\partial z}}

Calculation:

Given that,

f⃗ = 6x2i^ + 3xy2j^ + xyz3k^\vec{f}\ =\ 6x^2\hat{i}\ +\ 3xy^2\hat{j}\ +\ xyz^3\hat{k}

Therefore, the divergence of a given vector is

∇⋅f⃗=∂f1∂x+∂f2∂y+∂f3∂z \nabla \cdot \vec f = \frac{{\partial {f_1}}}{{\partial x}} + \frac{{\partial {f_2}}}{{\partial y}} + \frac{{\partial {f_3}}}{{\partial z}}        -----(1)

Here,

f1 = 6x2, f2 = 3xy2, f3 = xyz3      

Therefore, from equation (1)

∇⋅f⃗=∂∂x6x2+∂∂y3xy2+∂∂zxyz3 \nabla \cdot \vec f = \frac{{\partial {}}}{{\partial x}}6x^2 + \frac{{\partial {}}}{{\partial y}}3xy^2 + \frac{{\partial {}}}{{\partial z}}xyz^3

⇒ ∇⋅f⃗ \nabla \cdot \vec f  = 12x + 6xy + 3xyz2

Therefore, divergence at point (2, 3, 4)

∇⋅f⃗ \nabla \cdot \vec f  = 12(2) + 6(2)(3) + 3(2)(3)(4)2

⇒ ∇⋅f⃗ \nabla \cdot \vec f   = 24 + 36 + 288 = 348

Hence, option 3 is correct.

Additional Information

Gauss divergence theorem: 

It states that the surface integral of the normal component of a vector function F⃗\vec F taken over a closed surface ‘S’ is equal to the volume integral of the divergence of that vector function F⃗\vec F taken over a volume enclosed by the closed surface ‘S’.

∬S,F⃗.n^ds=∭V∇.F⃗dv\underset{S}{\mathop \iint },\vec{F}.\hat{n}ds=\iiint\limits_{V}{\nabla .\vec{F}dv}

Question 152

Find out the differential function of  x2 + 2\sqrt {x^2\ +\ 2} ?

  1. AOption (a)

    xx2 − 2 dxx\sqrt {x^2\ -\ 2}\ dx

  2. BOption (b)

    xx2 + 2 dxx\sqrt {x^2\ +\ 2}\ dx

  3. COption (c)

    xx2 − 2dx \frac{x}{\sqrt {x^2\ -\ 2}}{dx}

  4. DOption (d)

    xx2 + 2dx \frac{x}{\sqrt {x^2\ +\ 2}}{dx}

View answer & explanation

Correct answer

Answer: (d)D.

xx2 + 2dx \frac{x}{\sqrt {x^2\ +\ 2}}{dx}

Why this answer is correct

Formula used:

ddxx = 12x\frac{d}{dx}\sqrt x\ =\ \frac{1}{2\sqrt x}

ddx(xn) = nxn − 1\frac{d}{dx}(x^n)\ =\ nx^{n\ -\ 1}

Calculation:

Let y = x2 + 2\sqrt {x^2\ +\ 2}

Differentiating both sides with respect to x, we will get

dydx = x2 + 2\frac{dy}{dx}\ =\ \sqrt {x^2\ +\ 2}

We know that 

ddxx = 12x\frac{d}{dx}\sqrt x\ =\ \frac{1}{2\sqrt x}

⇒ dydx = 12x2 + 2× d(2x)dx\Rightarrow \ \frac{dy}{dx}\ =\ \frac{1}{2\sqrt {x^2\ +\ 2}}\times \ \frac{d(2x)}{dx}

⇒ dydx = xx2 + 2\Rightarrow \ \frac{dy}{dx}\ =\ \frac{x}{\sqrt {x^2\ +\ 2}}

⇒ dy = xx2 + 2dx\Rightarrow \ dy\ =\ \frac{x}{\sqrt {x^2\ +\ 2}}{dx}

Hence, option 4 is correct.

Question 153

Determine the series of sin θ using Taylor's theorem.

  1. AOption (a)

    1 − θ22! + θ44! − … 1\ -\ \frac{\theta^2 }{2!}\ +\ \frac{\theta ^4}{4!}\ -\ \dots

  2. BOption (b)

    1+ θ22! + θ44! − …1+\ \frac{\theta^2 }{2!}\ +\ \frac{\theta ^4}{4!}\ -\ \dots

  3. COption (c)

    θ − θ33! + θ55! − …\theta\ -\ \frac{\theta^3 }{3!}\ +\ \frac{\theta ^5}{5!}\ -\ \dots

  4. DOption (d)

    θ + θ33! + θ55! − …\theta\ +\ \frac{\theta^3 }{3!}\ +\ \frac{\theta ^5}{5!}\ -\ \dots

View answer & explanation

Correct answer

Answer: (c)C.

θ − θ33! + θ55! − …\theta\ -\ \frac{\theta^3 }{3!}\ +\ \frac{\theta ^5}{5!}\ -\ \dots

Why this answer is correct

Concept:

Taylor Series:

If f(z) is analytic inside a circle 'C', centre at z = a, and radius 'r', then for all z inside 'C'; the Taylor series is given by-

f(z)=f(a);+;(z−a)f′(a);+;(z−a)22!f′′(a);+;........(z−a)nn!fn(a)f(z)=f(a);+;(z-a)f'(a);+;\frac{(z-a)^2}{2!}f''(a);+;........\frac{(z-a)^n}{n!}f^n(a)

∴f(z)=∑n=0∞(z−a)nn!fn(a)\therefore f(z)=\sum_{n=0}^{\infty}\frac{(z-a)^n}{n!}f^n(a)

∴∑n=0∞an(z−a)n\therefore \sum_{n=0}^{\infty}a_n(z-a)^n where an=fn(a)n!a_n=\frac{f^n(a)}{n!}

Calculation:

Let f(θ)  = sinθ  and f(0) = sin0 = 0

Differentiating  with respect to θ 

f'(θ) = cosθ  and  f'(0) = cos0 = 1 

f''(θ) = - sinθ and f''(0) = - sin0 = 0

f'''(θ) = - cosθ  and f'''(θ) = - cos0 = -1

f''''(θ) =  sinθ and f''''(θ) = sin0 = 0

Therefore putting these values on Taylor series expansion at θ = 0, we will get 

sinθ = 0 + (θ − 0)(1) + (θ − 0)22!(0) + (θ − 0)33!(− 1) + (θ − 0)44!(0) …sinθ \ = \ 0\ +\ (θ\ -\ 0)(1)\ +\ \frac{(θ\ -\ 0)^2}{2!}(0)\ +\ \frac{(θ\ -\ 0)^3}{3!}(-\ 1) \ +\ \frac{(θ\ -\ 0)^4}{4!}(0)\ \dots

⇒ sinθ =θ + θ33! − θ55!…\Rightarrow \ sinθ \ = θ\ +\ \frac{θ^3 }{3!}\ -\ \frac{θ ^5}{5!}\dots

Important Points

Maclaurin's Series:

When a = 0 in Taylor's series i.e. about origin or powers of z, the series formed is Maclaurin's series.

f(z)=f(0);+;(z)f′(0);+;(z)22!f′′(0);+;........(z)nn!fn(0)f(z)=f(0);+;(z)f'(0);+;\frac{(z)^2}{2!}f''(0);+;........\frac{(z)^n}{n!}f^n(0)

Question 154

Which one is not the properties of pointwise convergence?

  1. AOption (a)

    There must be uniform spaces

  2. BOption (b)

    functions taking values in metric spaces

  3. COption (c)

    The values of the functions fn need not be real numbers

  4. DOption (d)

    Every sequence of N has a least element

View answer & explanation

Correct answer

Answer: (d)D.

Every sequence of N has a least element

Why this answer is correct

Concept: 

Pointwise convergence: In mathematics, pointwise convergence is one of the various senses in which a sequence of functions can converge to a particular function.   

Consider real functions defined on a closed interval [a,b],  a sequence of functions (fn) defined on [a,b] converges pointwise to the function f if and only if

for all x ∈ [a,b]  Pointwise convergence is weaker than uniform convergence.

lim⁡x→∞fn(x) = f(x)\mathop {\lim }\limits_{x \to \infty} f_n(x)\ =\ f(x)

Properties of pointwise convergence: 

  • Pointwise convergence does not, in general, preserve continuity.
  • A pointwise convergent sequence of functions need not be bounded.

Question 155

Integration of [sin (x) + cos (x)]ex is _______

  1. AOption (a)

    ex sin (x)

  2. BOption (b)

    ex cos (x)

  3. COption (c)

    ex (sin (x) + cos (x))

  4. DOption (d)

    ex tan (x)

View answer & explanation

Correct answer

Answer: (a)A.

ex sin (x)

Why this answer is correct

Concept: 

Integration by parts: Integration by parts is a method to find integrals of products.

The formula for integrating by parts is given by:

⇒ ∫uvdx=u∫vdx−∫(dudx×∫vdx)dx\rm ∫ u vdx=u ∫ vdx- ∫ \left({du\over dx}\times ∫ vdx\right)dx  + C

where u is the function u(x) and v is the function v(x) 

ILATE rule is Usually, the preference order of this rule is based on some functions such as Inverse, Logarithm, Algebraic, Trigonometric and Exponent.

Formula used:

\()1. ∫ sinx dx = - cosx + C

2. ∫ cosx dx =  sinx + C

3. ∫xndx = xn + 1n + 1 + C3.\ ∫ x^ndx\ = \ \frac{x^{n\ +\ 1}}{n\ +\ 1}\ +\ C

4. ∫exdx = ex + C 

 5.  ddxsinx = cosx5. \ \ \frac{d}{dx} sinx \ =\ cosx

6.  ddxcosx = − sinx6. \ \ \frac{d}{dx} cosx\ =\ -\ sinx

Calculation:

Let I =  ∫[sin (x) + cos (x)]exdx              

⇒ I = ∫sinx.exdx +  ∫cosx.exdx  + C          .........(1) 

Let  I1 = ∫sinx.exdx

Using integration by part method 

First part = sinx, second part = ex

I1 =  (sinx)∫exdx - ∫[ddx(sinx)\frac{d}{dx}( sinx )ex]dx 

⇒ ∫sinx.exdx = (sinx)ex - ∫(cosx )exdx

Hence, put this value in the equation (1)

⇒ I = (sinx)ex - ∫(cosx )exdx +  ∫cosx.exdx + C

⇒ I =  (sinx)ex + C

Hence, option 1 will be correct.

Shortcut TrickLet I =  ∫[sin (x) + cos (x)]ex dx     

We know that

∫[f(x) + f'(x)]exdx = exf(x) + C 

Here, f(x) = sinx and f'(x) = cosx

⇒ I =  (sinx)ex + C

Question 156

Find the volume V under the plane z = 8x + 6y over the rectangle R = [0, 1] × [0, 2].

  1. AOption (a)

    40 units

  2. BOption (b)

    10 units

  3. COption (c)

    30 units

  4. DOption (d)

    20 units

View answer & explanation

Correct answer

Answer: (d)D.

20 units

Why this answer is correct

Formula used:

∫xndx = xn + 1n + 1 + C\int x^ndx\ =\ \frac{x^{n\ +\ 1}}{n\ +\ 1}\ +\ C

Calculation:

Given that,

z = 8x + 6y and R = [0, 1] × [0, 2].

⇒ V = ∫01∫02(8x + 6y)dydx⇒\ V\ =\ \int_{0}^{1}\int_{0}^{2}(8x\ +\ 6y)dydx

⇒ V = ∫01[8xy + 6y22]01dx⇒\ V\ =\ \int_{0}^{1}[8xy\ +\ \frac{6y^2}{2}]_0^1dx

⇒ V = ∫01[16x + 12]dx⇒\ V\ =\ \int_{0}^{1}[16x\ +\ 12]dx

⇒ V = ∫01[16x + 12]dx⇒\ V\ =\ \int_{0}^{1}[16x\ +\ 12]dx

⇒ V = [16x2 + 12x]01⇒\ V\ =\ [\frac{16x}{2}\ +\ 12x]_0^1

⇒ V = 8 × 1 + 12 × 1

⇒ V = 20

Question 157

A function g ∶ [0, π2\frac{\pi}{2}] → R by  \rm g(x) = \left { \begin{matrix} \rm cos^2\ x & \rm x \ \epsilon\ Q \ \rm 0 & \rm \ otherwise \end{matrix}\right. Find the upper Riemann integrals of g over [0, π2\frac{\pi}{2}].

  1. AOption (a)

    π2\frac{\pi}{2}

  2. BOption (b)

    ​π3\frac{\pi}{3}

  3. COption (c)

    π4\frac{\pi}{4}

  4. DOption (d)

    ​π 

View answer & explanation

Correct answer

Answer: (c)C.

π4\frac{\pi}{4}

Why this answer is correct

Concept:

The upper Riemann integral of a bounded function on an interval is defined using the supremum of the function on each subinterval.

Given the function

g(x)={cos⁡2x,x∈Q, 0,otherwise, g(x)= \begin{cases} \cos^2 x, & x\in \mathbb{Q},\ 0, & \text{otherwise}, \end{cases}

defined on the interval [0,π2] \left[0,\frac{π}{2}\right] .

Since rational numbers are dense in every interval, each subinterval of [0,π2] \left[0,\frac{π}{2}\right] contains infinitely many rational points.

Therefore, on any subinterval [a,b] [a,b] ,

sup⁡g(x)=sup⁡cos⁡2x:x∈[a,b] \sup g(x) = \sup { \cos^2 x : x \in [a,b] }

The function cos2(x) is decreasing on [0,π2] \left[0,\frac{π}{2}\right] , hence the supremum on [a,b] [a,b] occurs at x=a.

Thus, the upper sum corresponds to the function cos2(x) itself.

Hence, the upper Riemann integral of g(x) g(x) is

∫0π/2‾g(x),dx=∫0π/2cos⁡2x,dx \displaystyle \overline{\int_0^{π/2}} g(x),dx = \int_0^{π/2} \cos^2 x , dx

Calculation:

cos⁡2x=1+cos⁡2x2 \cos^2 x = \frac{1+\cos 2x}{2}

∫0π/2cos⁡2x,dx=∫0π/21+cos⁡2x2,dx \displaystyle \int_0^{π/2} \cos^2 x , dx = \int_0^{π/2} \frac{1+\cos 2x}{2} , dx

=12∫0π/2dx+12∫0π/2cos⁡2x,dx = \frac{1}{2} \int_0^{π/2} dx + \frac{1}{2} \int_0^{π/2} \cos 2x , dx

=12[x]0π/2+12[sin⁡2x2]0π/2 = \frac{1}{2}\left[ x \right]_0^{π/2} + \frac{1}{2}\left[ \frac{\sin 2x}{2} \right]_0^{π/2}

=12(π2)+14(sin⁡π−sin⁡0) = \frac{1}{2}\left( \frac{π}{2} \right) + \frac{1}{4} \left( \sin π - \sin 0 \right)

=π4+0=π4 = \frac{π}{4} + 0 = \frac{π}{4}

The correct answer is π/4.

Question 158

Let f be the uniform limit of a sequence of continuous functions {fn}. Then f is

  1. AOption (a)

    Discontinuous

  2. BOption (b)

    Infinite

  3. COption (c)

    Continuous

  4. DOption (d)

    Limited

View answer & explanation

Correct answer

Answer: (c)C.

Continuous

Why this answer is correct

Concept:

If fn is a sequence of continuous functions that converges uniformly to a function f , then f is continuous. 

Uniform convergence means that for every ϵ > 0, there exists an N such that for all n ≥ N and for all x in the domain, |fn(x) - f(x)| < ϵ.

Explanation:

  1. Uniform Convergence of Continuous Functions:

When a sequence of functions {fn} converges uniformly to a function f , it means the rate of convergence of fn  to f does not depend on x .

This uniform convergence preserves the continuity of the limit function f .

  1. Continuity of the Limit Function:

Since each fn  is continuous and the convergence to f is uniform, f will also be continuous. This result is a well-known theorem in real analysis.

Given that f is the uniform limit of a sequence of continuous functions {fn} , f is continuous.

Therefore, the correct answer is Continuous.

Question 159

What is the area inside function 2x3 + 5x2 − 4x2\frac{2x^3\ +\ 5x^2\ -\ 4}{x^2} form x = 1 to a.

  1. AOption (a)

    a22 + 5a − 112\frac{a^2}{2}\ +\ \frac{5a\ -\ 11}{2}

  2. BOption (b)

    a2 + 5a + 4a − 10a^2\ +\ 5a\ +\ \frac{4}{a} \ -\ 10

  3. COption (c)

    a22 + 5a − 4ln(a)\frac{a^2}{2}\ +\ 5a\ -\ 4ln(a)

  4. DOption (d)

    a22 + 4ln(a) − 112\frac{a^2}{2}\ +\ 4ln(a)\ -\ \frac{11}{2}

View answer & explanation

Correct answer

Answer: (b)B.

a2 + 5a + 4a − 10a^2\ +\ 5a\ +\ \frac{4}{a} \ -\ 10

Why this answer is correct

Concept:

The area under the curve: The area under a curve between two points can be found by doing a definite integral between the two points.

Therefore, area of function y = f(x) b/w x = a and a = b is given by

A = ∫abf(x)dxA\ =\ \int_{a}^{b}f(x)dx

Formula used:

∫xndx = xn + 1(n + 1) + C\int x^ndx\ =\ \frac{x^{n\ +\ 1}}{(n\ +\ 1)} \ +\ C

Calculation:

Let the area of given function 2x3 + 5x2 − 4x2\frac{2x^3\ +\ 5x^2\ -\ 4}{x^2} is A. Therefore

 A = ∫1a2x3 + 5x2 − 4x2dxA\ =\ \int_{1}^{a}\frac{2x^3\ +\ 5x^2\ -\ 4}{x^2}dx

⇒A = ∫1a(2x + 5 − 4x2)dx\Rightarrow A\ =\ \int_{1}^{a}(2x\ +\ 5\ -\ \frac{4}{x^2})dx

Integrating each part,

 ⇒ A = ( x2 + 5x + 4x)1a\Rightarrow \ A\ =\ (\ x^2\ +\ 5x\ +\ \frac{4}{x})_{1}^{a}

Taking the limit of x, we will get

A = a2 + 5a + 4a − 1 − 5 − 4A\ =\ a^2\ +\ 5a\ +\ \frac{4}{a} \ -\ 1\ -\ 5\ -\ 4

⇒  A = a2 + 5a + 4a − 10\Rightarrow\ \ A\ =\ a^2\ +\ 5a\ +\ \frac{4}{a} \ -\ 10

Hence, option 2 is correct.

Question 160

What is the expansion of y = sin-1 x?

  1. AOption (a)

     x36 − 340x5 + 5112x7 − ......\ \frac{x^3}{6}\ -\ \frac{3}{40} x^5\ +\ \frac{5}{112}x^7\ -\ ......

  2. BOption (b)

     x36 + 340x5 + 5112x7 + ......\ \frac{x^3}{6}\ +\ \frac{3}{40} x^5\ +\ \frac{5}{112}x^7\ +\ ......

  3. COption (c)

    x − x36 + 340x5 − 5112x3 + ......x\ -\ \frac{x^3}{6}\ +\ \frac{3}{40} x^5\ -\ \frac{5}{112}x^3\ +\ ......

  4. DOption (d)

    x + x36 + 340x5 + 5112x7 + ......x\ +\ \frac{x^3}{6}\ +\ \frac{3}{40} x^5\ +\ \frac{5}{112}x^7\ +\ ......

View answer & explanation

Correct answer

Answer: (d)D.

x + x36 + 340x5 + 5112x7 + ......x\ +\ \frac{x^3}{6}\ +\ \frac{3}{40} x^5\ +\ \frac{5}{112}x^7\ +\ ......

Why this answer is correct

Concept:

Binomial expansion of (1 - x)-n is given by

(1 - x)-n = 1 + nx + n(n + 1)2.1x2 +  n(n + 1)(n + 2)3.2.1x3 ............1\ +\ nx\ +\frac{\ n(n\ +\ 1)}{2.1}x^2\ +\ \frac{\ n(n\ +\ 1)(n\ +\ 2)}{3.2.1}x^3 \ ............

Formula used:

ddxsin−1x = 11 − x2\frac{d}{dx}sin^{-1}x\ =\ \frac{1}{\sqrt{1\ -\ x^2}}

∫xndx = xn + 1n + 1 + C\int x^ndx \ =\ \frac{x^{n\ +\ 1}}{n\ +\ 1}\ + \ C

Calculation:

Given,

y = sin-1 x

dydx = 11 − x2 =(1 − x2)−12\frac{dy}{dx}\ =\ \frac{1}{\sqrt{1\ -\ x^2}}\ =(1\ -\ x^2)^{\frac{-1}{2}}              ........(1)

Using the above binomial expansion formula

⇒ dydx = 1 + 12x2 + 12.322.1x4 + 12.32523.2.1x6 + ......\Rightarrow \ \frac{dy}{dx}\ =\ 1\ + \ \frac{1}{2}x^2\ +\ \frac{\frac{1}{2}.\frac{3}{2}}{2.1}x^4\ +\ \frac{\frac{1}{2}.\frac{3}{2}\frac{5}{2}}{3.2.1}x^6\ +\ ......

Integrating both sides with respect to x, 

⇒ ∫d (sin−1x)dxdx = ∫(1 + 12x2 + 12.322.1x4 + 12.32523.2.1x6 + ......)dx\Rightarrow \ \int\frac{d\ (sin^{-1}x)}{dx}dx\ =\ \int (1\ + \ \frac{1}{2}x^2\ +\ \frac{\frac{1}{2}.\frac{3}{2}}{2.1}x^4\ +\ \frac{\frac{1}{2}.\frac{3}{2}\frac{5}{2}}{3.2.1}x^6\ +\ ......)dx

⇒ sin−1x  = x + 12x33 + 123212x55 + 12325216x77 + .......\Rightarrow \ sin^{-1}x\ \ =\ x \ + \ \frac{1}{2}\frac{x^3}{3}\ +\ \frac{1}{2}\frac{3}{2}\frac{1}{2}\frac{x^5}{5}\ +\ \frac{1}{2}\frac{3}{2}\frac{5}{2}\frac{1}{6}\frac{x^7}{7}\ +\ .......

⇒ sin−1x  = x + x36 + 340x5+ 5112x7+ .......\Rightarrow \ sin^{-1}x\ \ =\ x \ + \ \frac{x^3}{6}\ +\ \frac{3}{40}x^5 +\ \frac{5}{112}x^7 +\ .......

Hence, option 4 is the correct answer.

Discipline 4 (20 questions)

Question 161

If d1, d2 are two metrics on the same set M, Which statement is not equivalent from the following statements:

  1. AOption (a)

    the union of any finite collection of closed sets is closed 

  2. BOption (b)

    For every metric space (N, d), every g ∶ N → M is d1 continuous if it is d2 continuous.

  3. COption (c)

    d1 open and d2 open sets coincide

  4. DOption (d)

    For every metric space (N, d), every f ∶ M → N is d1 continuous if it is d2 continuous

View answer & explanation

Correct answer

Answer: (a)A.

the union of any finite collection of closed sets is closed 

Why this answer is correct

Concept:

If d1 and d2 are two metrics on M, then the following are equivalent:

  • The open sets with respect to d1 and d2 coincide.

  • Every sequence converges in d1 if and only if it converges in d2.

  • Every function continuous in one metric is continuous in the other.

Now, check each statement:

  1. "the union of any finite collection of closed sets is closed"

This is always true for any topology (by definition of closed sets), regardless of whether the topologies coincide or not.

  1. "For every metric space (N, d), every g : N → M is d1 continuous if it is d2 continuous."

This is equivalent to all d1-open subsets being d2-open, i.e., the d1-topology is coarser than the d2-topology.

  1. "d1 open and d2 open sets coincide"

This is the definition of the topologies being equivalent.

  1. "For every metric space (N, d), every f : M → N is d1 continuous if it is d2 continuous"

This is equivalent to all d2-open sets being d1-open, i.e., the d2-topology is coarser than the d1-topology.

Which is not equivalent?

Option 1 is not equivalent because it is always true and does not require the topologies to coincide.

Options 2, 3, and 4 express the equivalence of the two metrics' topologies.

∴The union of any finite collection of closed sets is closed is not equivalent.

Question 162

Elaborate cos (x2) as a power series.

  1. AOption (a)

    1 + x42! + x84! −  x126! + …1\ +\ \frac{x^4}{2!} \ +\ \frac{x^8}{4!}\ -\ \ \frac{x^{12}}{6!} \ +\ \dots

  2. BOption (b)

    1 − x42! − x84! −  x126! + …1\ -\ \frac{x^4}{2!} \ -\ \frac{x^8}{4!}\ -\ \ \frac{x^{12}}{6!} \ +\ \dots

  3. COption (c)

    1 − x42! + x84! +  x126! + …1\ -\ \frac{x^4}{2!} \ +\ \frac{x^8}{4!}\ +\ \ \frac{x^{12}}{6!} \ +\ \dots

  4. DOption (d)

    1 − x42! + x84! −  x126! + …1\ -\ \frac{x^4}{2!} \ +\ \frac{x^8}{4!}\ -\ \ \frac{x^{12}}{6!} \ +\ \dots

View answer & explanation

Correct answer

Answer: (d)D.

1 − x42! + x84! −  x126! + …1\ -\ \frac{x^4}{2!} \ +\ \frac{x^8}{4!}\ -\ \ \frac{x^{12}}{6!} \ +\ \dots

Why this answer is correct

Concept:

Maclaurin's Series:

When a = 0 in Taylor's series i.e. about origin or powers of z, the series formed is Maclaurin's series.

f(z)=f(0);+;(z)f′(0);+;(z)22!f′′(0);+;........(z)nn!fn(0)f(z)=f(0);+;(z)f'(0);+;\frac{(z)^2}{2!}f''(0);+;........\frac{(z)^n}{n!}f^n(0)

Calculation:

Let f(x) = cos(x)

⇒ f(0) = 1 

Differentating f(x) with respect to 'x'

f'(x) = - sinx     and           f'(0) = 0

f''(x) = - cosx        and          f''(0) = - 1

f'''(x) = sinx         and           f'''(0) = 0

f''''(x) = cosx       and         f''''(0) = 1

Using the above Maclaurin;s expansion, we will get

Cosx = 1 + x.0 + x22!.(−1)+ x33!(0) + x44!(1)…1\ +\ x. 0 \ + \ \frac{x^2}{2!}.(-1) +\ \frac{x^3}{3!}(0)\ +\ \frac{x^4}{4!}(1) \dots

⇒ Cosx = 1 − x22! + x44! +…1\ - \ \frac{x^2}{2!}\ +\ \frac{x^4}{4!}\ + \dots

Replace x with x2 on the above expansion, 

Cos x2 = 1 − x42! + x84! +…1\ - \ \frac{x^4}{2!}\ +\ \frac{x^8}{4!}\ + \dots

Shortcut Trick We know that

cos⁡θ=1−θ22!+θ44!−θ66!+…\cos \theta = 1 - \frac{{{\theta ^2}}}{{2!}} + \frac{{{\theta ^4}}}{{4!}} - \frac{{{\theta ^6}}}{{6!}} + \ldots

Therefore 

⇒ Cosx=1−x22!+x44!−x66!+…⇒ \ Cos x = 1 - \frac{{{x ^2}}}{{2!}} + \frac{{{x ^4}}}{{4!}} - \frac{{{x ^6}}}{{6!}} + \dots

Hence, to find the value of cosx2, we will replace x with x2 on the above expansion. Hence,

⇒ Cosx=1−x42!+x84!−x126!+…⇒ \ Cos x = 1 - \frac{{{x ^4}}}{{2!}} + \frac{{{x ^8}}}{{4!}} - \frac{{{x ^{12}}}}{{6!}} + \dots

Additional Information

Taylor Series:  If f(z) is analytic inside a circle 'C', centre at z = a, and radius 'r', then for all z inside 'C'; the Taylor series is given by-

f(z)=f(a);+;(z−a)f′(a);+;(z−a)22!f′′(a);+;........(z−a)nn!fn(a)f(z)=f(a);+;(z-a)f'(a);+;\frac{(z-a)^2}{2!}f''(a);+;........\frac{(z-a)^n}{n!}f^n(a)

∴f(z)=∑n=0∞(z−a)nn!fn(a)\therefore f(z)=\sum_{n=0}^{\infty}\frac{(z-a)^n}{n!}f^n(a)

∴∑n=0∞an(z−a)n\therefore \sum_{n=0}^{\infty}a_n(z-a)^n  where  an=fn(a)n!a_n=\frac{f^n(a)}{n!}

Question 163

Let * be a group of G on X. For x ϵ X, let Gx denote the subset {g * x ∶ g ∈  G} of X. Define an equivalence relation v on X by setting xvy if Gx = Gy The equivalence class of x is known as the _______.

  1. AOption (a)

    orbit and stabiliser of x

  2. BOption (b)

    space of orbits Ox

  3. COption (c)

    orbit Ox of x

  4. DOption (d)

    disjoint union of x 

View answer & explanation

Correct answer

Answer: (c)C.

orbit Ox of x

Why this answer is correct


**Concept -**

In group theory, an action of a group \\(G\\) on a set \\(X\\) is a formal way of interpreting the elements of \\(G\\) as "motions" of the elements in \\(X\\), in a way that the group operation is compatible with the action. For any element \\(x \in X\\), the subset \\(Gx = \{g * x : g \in G\}\\) of \\(X\\) is called the orbit of \\(x\\) under the action of \\(G\\). The notion of orbits partitions the set \\(X\\) into disjoint subsets where each subset consists of elements that are related to each other by the group action.

The equivalence relation \\(v\\) defined by \\(xvy\\) if \\(Gx = Gy\\) means that two elements are considered equivalent if they have the same orbit. The equivalence class of an element \\(x\\), under this relation, is the orbit of \\(x\\), which includes all elements that can be reached from \\(x\\) by the action of any element of \\(G\\).

The correct answer is, therefore, option 3: **orbit \\(Ox\\) of \\(x\\)**.

**Explanation -**

The concept of an orbit is fundamental in group action theory. Given a group action of \\(G\\) on \\(X\\), every element \\(x \in X\\) is associated with an orbit, which is the set of elements that can be "reached" by applying the group elements to \\(x\\). This orbit reflects the way the group \\(G\\) interacts with the elements of \\(X\\) through its action. Since \\(Gx = Gy\\) establishes an equivalence relation based on the equality of orbits, the equivalence class of \\(x\\) under this relation is precisely the orbit of \\(x\\), indicating all elements of \\(X\\) that are equivalent to \\(x\\) under the group action.

Therefore, the notion of orbits helps in understanding the structure of the set \\(X\\) under the action of the group \\(G\\) by partitioning \\(X\\) into disjoint orbits, each of which is an equivalence class under the relation defined by the equality of orbits.

Question 164

U(5) and U(12) are finite collection of groups; U(5) = {1, 2, 3, 4} and U(12) = {1, 5, 7, 11}. So U(5) ⨁ U(12)?

  1. AOption (a)

    (2, 3)

  2. BOption (b)

    (5, 3)

  3. COption (c)

    (1, 5)

  4. DOption (d)

    (4, 3)

View answer & explanation

Correct answer

Answer: (c)C.

(1, 5)

Why this answer is correct

Calculation:

U(5) represents the group of units modulo 5, consisting of all integers less than 5 and relatively prime to 5.

⇒ U(5) = {1, 2, 3, 4}

U(12) represents the group of units modulo 12, consisting of all integers less than 12 and relatively prime to 12.

⇒ U(12) = {1, 5, 7, 11}

The direct product U(5) ⊕ U(12) contains all ordered pairs (a, b)

such that a ∈ U(5) and b ∈ U(12)

Thus, an ordered pair is valid only if:

⇒ First element belongs to {1, 2, 3, 4}

⇒ Second element belongs to {1, 5, 7, 11}

Checking the given pairs:

(2, 3): 2 ∈ U(5) and  3 ∉ U(12) ⇒ valid element

(5, 3): 5 ∉ U(5) ⇒ not valid

(1, 5): 1 ∈ U(5) and 5 ∈ U(12) ⇒ valid element

(4, 3): 3 ∉ U(12) ⇒ not valid

The correct answer is (1, 5)

Question 165

Find out the incorrect statement for the interiors of A and B given by A° and B° in (X, d)? 

  1. AOption (a)

    A ⊆ B → A° ⊆ B°

  2. BOption (b)

    A° ∪ B° ⊆ (A ∪ B)° 

  3. COption (c)

    A° ∪ B° = (A ∪ B)° 

  4. DOption (d)

    A° ∩ B° = (A ∩ B)°

View answer & explanation

Correct answer

Answer: (c)C.

A° ∪ B° = (A ∪ B)° 

Why this answer is correct

Concept use:

A point s S is called interior point of S if there exists a neighborhood of s completely contained in S. The set of all interior points of S is called the interior, denoted by int(S).

Explanation:

The interior of a set A, denoted by A°, is the set of all points in A that have a neighborhood entirely contained within A. In other words, a point x belongs to A° if there exists an open ball centered at x that lies entirely inside A.

Take Option 1:

A ⊆ B → A° ⊆ B°

This statement is correct. If A is a subset of B, then any point in A° (which has a neighborhood entirely contained in A) must also have that neighborhood contained in B, implying that A° is a subset of B°.

Take Option 2:

A° ∪ B° ⊆ (A ∪ B)° This statement is correct. If a point x belongs to either A° or B°, then it has a neighborhood entirely contained in either A or B, respectively. Since the union of A and B contains both sets, this neighborhood must also be contained in (A ∪ B), implying that x belongs to (A ∪ B)°.

Take Option 3:

A° ∪ B° = (A ∪ B)° This statement is incorrect. The interior of the union of two sets can be strictly larger than the union of their interiors. This is because a point x may have a neighborhood that intersects both A and B, but is not entirely contained in either set. In such a case, x would belong to (A ∪ B)°, but not to A° or B°.

For Example Take Q as A and Qc as B then their Interior is ϕ 

Q ∪ Qc = R and Ro = R but The interior of Union of Set is R

Take Option 4:

A° ∩ B° = (A ∩ B)° This statement is correct. If a point x belongs to both A° and B°, then it has a neighborhood entirely contained in both A and B. This implies that the neighborhood is also contained in the intersection of A and B, making x a point in (A ∩ B)°. Conversely, if x belongs to (A ∩ B)°, then it has a neighborhood entirely contained in both A and B, implying that x belongs to both A° and B°.

Hence, the incorrect statement is statement 3: A° ∪ B° = (A ∪ B)°.

Question 166

Which one is not in the automorphism group?

  1. AOption (a)

    identity

  2. BOption (b)

    associativity

  3. COption (c)

    closure 

  4. DOption (d)

    decomposition

View answer & explanation

Correct answer

Answer: (d)D.

decomposition

Why this answer is correct

Explanation -

An automorphism group of a mathematical structure is the set of all automorphisms of the structure, with the operation of composition of functions.

An automorphism is an isomorphism from a mathematical structure to itself.

For a group to be considered an automorphism group, it must satisfy the following properties:

  1. Identity: There must be an identity element.

  2. Associativity: The group operation must be associative.

  3. Closure: The group must be closed under the operation.

  4. Inverse: Every element must have an inverse.

Among these, "decomposition" is not a property required for an automorphism group. 

Therefore, the correct answer is decomposition.

Question 167

Find out the all values of Z from this equation : ez = 1 + √3i.

  1. AOption (a)

    ln(2) + (π/3 + 2πn) i; n ϵ Z

  2. BOption (b)

    ln(2) + (π/2 + πn) i; n ϵ Z

  3. COption (c)

    ln(2) + (π/3 + πn) i; n ϵ Z

  4. DOption (d)

    ln(2) + (π/2 + 2πn) i; n ϵ Z

View answer & explanation

Correct answer

Answer: (a)A.

ln(2) + (π/3 + 2πn) i; n ϵ Z

Why this answer is correct

Formula used:

eiθ = cosθ + isinθ      Where i = √-1 

e2πni = 1 always  ∀ n where n = 1, 2, 3.......

elnx = x

Calculation:

Given that

ez = 1 + √3i

⇒ ez =2(12 + √32)⇒ \ e^z \ = 2(\frac{1}{2}\ +\ \frac{√ 3}{2})

⇒ ez =2(cosπ3 + sinπ3)⇒ \ e^z \ = 2(cos\frac{π}{3}\ +\ sin\frac{π}{3})                  [�� cosπ/3 = 1/2 & sinπ/3 = √3/2]

⇒ ez = 2eiπ/3                                           [∵ eiθ = cosθ + isinθ]

⇒ ez = eln2.eiπ/3.e2nπi                              [∵ elnx = x]

 ⇒ (ez) = e(ln2 + iπ3 + 2nπi)\Rightarrow\ (e^z)\ = \ e^{( ln2\ +\ i\frac{\pi}{3}\ +\ 2n\pi i )}                      [xaxb = xa +b]

Taking 'ln' on both side

 ⇒ ln⁡(ez) =ln e(ln2 + iπ3 + 2nπi)\Rightarrow\ \ln(e^z)\ = ln\ e^{( ln2\ +\ i\frac{\pi}{3}\ +\ 2n\pi i )}

⇒ Z = ln2 + iπ3 + 2nπi\Rightarrow\ Z\ = \ ln2\ +\ i\frac{\pi}{3}\ +\ 2n\pi i

Hence, the general solution of z is  Z = ln2 + iπ3 + 2nπi  ∀ n ϵ ZZ\ = \ ln2\ +\ i\frac{\pi}{3}\ +\ 2n\pi i \ \ \forall \ n\ \epsilon \ Z

Question 168

In which condition T is not disconnected?

  1. AOption (a)

    T has a decomposition into two nonempty closed sets

  2. BOption (b)

    T has a subset which is open, closed and is neither Φ nor T.

  3. COption (c)

    the only decompositions of T into open sets are T = Φ ∪ T and T = T ∪ Φ   

  4. DOption (d)

    T has a decomposition into two nonempty open sets

View answer & explanation

Correct answer

Answer: (c)C.

the only decompositions of T into open sets are T = Φ ∪ T and T = T ∪ Φ   

Why this answer is correct

Concept:

A topological space T is said to be disconnected if there exist two nonempty, disjoint open sets U and V such that

T=U∪V T = U \cup V

If no such decomposition exists, then T is called connected.

Thus, T is not disconnected if and only if it is connected.

Calculation:

Assume that the only decompositions of T into open sets are

T=∅∪T T = \varnothing \cup T and T=T∪∅ T = T \cup \varnothing   (1)

This implies that there do not exist two nonempty, disjoint open sets whose union is T T .

Hence, by definition, T is connected.

Now, consider the other possibilities:

If T has a nontrivial subset which is both open and closed, say A, then

T=A∪Ac T = A \cup A^c

where both A and Ac are nonempty open sets.

This gives a separation of T, implying that T is disconnected.

Similarly, if T can be decomposed into two nonempty open sets, then it is disconnected.

Therefore, the condition under which T is not disconnected is exactly that given in (1).

∴  option (3) is correct

Question 169

Find the radius of convergence for the power series: ∑n = 1∞[7nn(3x − 1)n − 1] \sum_{n\ =\ 1}^{\infty}[\frac{7^n}{n(3x\ -\ 1)^{n\ -\ 1}}]

  1. AOption (a)

    3!x|4x - 9|

  2. BOption (b)

    (2x + 1) ÷ 6

  3. COption (c)

    1/7(|3x - 1|)

  4. DOption (d)

    5|x + 1|

View answer & explanation

Correct answer

Answer: (c)C.

1/7(|3x - 1|)

Why this answer is correct

Concept:

Power series:

A series of the form ∑n=0∞an(z − z0)n\sum_{n = 0}^{\infty}a_n(z\ -\ z_0)^n is called power series about a point z = z0.

If z0 = 0, then power series about z0 = 0 become ∑n=0∞anzn\sum_{n = 0}^{\infty}a_nz^n

The radius of convergence: 

Let ∑n=0∞an(z − z0)n\sum_{n = 0}^{\infty}a_n(z\ -\ z_0)^n  be a power series and let R be the radius of circle in which power series is convergence and 

1R = lim⁡n→∞∣an∣1[n = lim⁡n→∞∣an+1an∣\frac{1}{R}\ =\ \mathop {\lim }\limits_{n \to {\rm{\infty}}}|a_n|^{\frac{1}[n}\ =\ \mathop {\lim }\limits_{n \to {\rm{\infty}}}|\frac{a_{n+1}}{a_n}|

Calculation:

Given power series is

∑n = 1∞[7nn(3x − 1)n − 1] \sum_{n\ =\ 1}^{\infty}[\frac{7^n}{n(3x\ -\ 1)^{n\ -\ 1}}]

Let, R is the radius of convergence. 

⇒ 1R = lim⁡n→∞∣7n + 1(n + 1)(3x − 1)n + 1 − 17nn(3x − 1)n − 1∣⇒ \ \frac{1}{R}\ =\ \mathop {\lim }\limits_{n \to {\rm{\infty}}}{|\frac{\frac{7^{n\ +\ 1}}{{(n\ +\ 1)}(3x\ -\ 1)^{^{n\ +\ 1\ -\ 1}}}}{\frac{7^n}{n(3x\ -\ 1)^{n\ -\ 1}}}|}

⇒ 1R = lim⁡n→∞∣7n+1(3x − 1)n−17n(3x − 1)n∣⇒ \ \frac{1}{R}\ =\ \mathop {\lim }\limits_{n \to {\rm{\infty}}}|\frac{7^{n + 1}(3x\ -\ 1)^{n-1}}{7^{n }(3x\ -\ 1)^{n}}|

⇒ 1R = lim⁡n→∞∣7n(n + 1)(3x − 1)∣⇒ \ \frac{1}{R}\ =\ \mathop {\lim }\limits_{n \to {\rm{\infty}}}|\frac{7n}{(n\ +\ 1)(3x\ -\ 1)}|

⇒ 1R = lim⁡n→∞∣7(1 + 1n)(3x − 1)∣⇒ \ \frac{1}{R}\ =\ \mathop {\lim }\limits_{n \to {\rm{\infty}}}|\frac{7}{(1\ +\ \frac{1}{n})(3x\ -\ 1)}|

⇒ 1R = ∣7(3x − 1)∣⇒ \ \frac{1}{R}\ =\ |\frac{7}{(3x\ -\ 1)}|      (∵ 1/n = 0)

⇒ R = 17\frac{1}{7}|3x - 1|

Question 170

What is the representation of an open disk D(x, y) of radius r centered at origin in R2?

  1. AOption (a)

    D = {(x, y)|x2 + y2 < r}

  2. BOption (b)

    D = {(x, y)|x2 + y2 < r2}

  3. COption (c)

    D = {(x, y)|x2 + y2 > r2}

  4. DOption (d)

    D = {(x, y)|x2 + y2 = r2}

View answer & explanation

Correct answer

Answer: (b)B.

D = {(x, y)|x2 + y2 < r2}

Why this answer is correct

Concept 

D = {(x, y)|x2 + y2 < r} represents open disk centered at (0, 0) with radius √r.

D = {(x, y)|x2 + y2 < r2} represents open disk centered at (0, 0) with radius r.

D = {(x, y)|x2 + y2 > r2} represents hole in R2 plane with radius r centered at (0, 0).

D = {(x, y)|x2 + y2 = r2} represents circle in R2 plane with radius r centered at (0, 0).

∴ The correct option is D = {(x, y)|x2 + y2 < r2}

Question 171

Calculate the number of elements of order 7 in Z(49) ⊕ Z(7).

  1. AOption (a)

    48

  2. BOption (b)

    49

  3. COption (c)

    56

  4. DOption (d)

    51

View answer & explanation

Correct answer

Answer: (a)A.

48

Why this answer is correct

Concept:

Some properties of external direct products

  1. The order of (g1, g2, ..., gn) ∈ G1 ⊕ G2 ⊕ .......⊕ Gn is LCM of (|g1|, |g2|, ... , |gn|).
  2. Let G and H be cyclic groups. Then G ⊕ H is cyclic if and only if |G| and |H| are relatively prime.
  3. The external direct product G1 ⊕ G2 ⊕ .......⊕ Gn of a finite number of a finite cyclic groups is cyclic if and only if |Gi | and |Gj | are relatively prime for each i ≠ j.

 

Calculations:

For each such element, 7 = |(a, b)| = lcm(|a|, |b|)

There are three mutually exclusive cases

Case: (1) |a| = 7 and |b| = 7. There are 6 choices for a (7, 14, 21, 28, 35, 42) and 6 for b (1, 2, 3, 4, 5, 6) for a total of 36 choices.

Case: (2) |a| = 7 and |b| = 1. We have 6 choices for a and 1 for b, a total of 6 choices .

Case: (3) |a| = 1 and |b| = 7. Another 6 choices.

Thus Z49 ⊕ Z7 has 48 elements of order 7.

Hence, the correct answer is option 2)

Question 172

Let X be a set and G be a group. A group action of G on X is a map * ∶ G × X → X given by (g, x) -→ g * x such that

  1. (gh) * x = g * (h * x) for all g, h ϵ G and x ϵ X.
  2. e * x = x for all x ϵ X

If this happens then we say that G acts on X and X is a

  1. AOption (a)

    G-homomorphism

  2. BOption (b)

    G-isomorphism

  3. COption (c)

    G-Bijection

  4. DOption (d)

    G-set

View answer & explanation

Correct answer

Answer: (d)D.

G-set

Why this answer is correct

Explanation:

The given map * : G × X → X satisfies the defining properties of a group action.

The condition (gh) * x = g * (h * x) shows that the action is compatible with the group operation of G.

The condition e * x = x shows that the identity element of G leaves every element of X unchanged.

When both these conditions hold, the set X is said to be equipped with an action of the group G.

Such a set X is called a G-set.

The correct answer is Option (4) G-set.

Question 173

Let G be a finite group acting on a set X. Then which of the statement is not satisfied the action of G:

  1. AOption (a)

    The stabilizer Sx of x is a subgroup of G for every x ϵ X

  2. BOption (b)

    On(R) (group of orthogonal matrices) and S (unit sphere in Rn)

  3. COption (c)

    ker(φ) = ∩xϵXSx 

  4. DOption (d)

    If y ϵ 0x then there exists h ϵ G such that Sx = hSyh ^ -1

View answer & explanation

Correct answer

Answer: (b)B.

On(R) (group of orthogonal matrices) and S (unit sphere in Rn)

Why this answer is correct

Concept:

Let a finite group G act on a set X. We examine which statement is not satisfied by the action of G.

Statement (1):

The stabilizer of an element x∈X x \in X is defined as Sx=g∈G∣g⋅x=x S_x = { g \in G \mid g\cdot x = x } .

This set contains the identity, is closed under inverses and products, hence Sx S_x is a subgroup of G.

Statement (2):

The action of O(n) (the group of orthogonal matrices) on S (the unit sphere in Rn \mathbb{R}^n ) is considered.

However, O(n) is an infinite group, while the given condition requires G to be a finite group.

Hence this statement does not satisfy the hypothesis.

Statement (3):

Let φ:G→Sym(X) \varphi : G \to \text{Sym}(X) be the homomorphism induced by the action.

The kernel consists of elements fixing every point of X, so

ker⁡(φ)=⋂x∈XSx \ker(\varphi) = \bigcap_{x \in X} S_x .

This statement is true.

Statement (4):

If y∈Gx y \in Gx , then there exists h∈G h \in G such that

Sx=hSyh−1 S_x = h S_y h^{-1} .

This follows from the fact that stabilizers of elements in the same orbit are conjugate.

The statement that is not satisfied is option (2)

Question 174

The Metric space is both complete and totally bounded is known as _______.

  1. AOption (a)

    Scalar

  2. BOption (b)

    Discrete

  3. COption (c)

    Complete

  4. DOption (d)

    Compact

View answer & explanation

Correct answer

Answer: (d)D.

Compact

Why this answer is correct

Concept -

1. Complete Metric Space:

    A metric space is complete if every Cauchy sequence (a sequence where the elements get arbitrarily close to each other as the sequence progresses) converges to a point within the space. 

2. Totally Bounded Metric Space:

   A metric space is totally bounded if for every ϵ > 0, the space can be covered by a finite number of open balls of radius ϵ. In simpler terms, you can cover the entire space with a finite number of small "balls."

3. Compact Metric Space:

   A metric space is compact if every sequence has a subsequence that converges to a point within the space. Compactness is a stronger condition that implies both completeness and total boundedness.

Explanation -

The metric space that is both complete and totally bounded is known as compact.

In topology and analysis, a metric space is considered compact if every sequence has a subsequence that converges to a point within the space.

This property combines the notions of completeness (every Cauchy sequence converges) and total boundedness (the space can be covered by finitely many balls of any given radius).

Therefore, the correct answer is Compact.

Question 175

Let Q be the field of rational numbers. Find the Galois group order of f(x) = x2 - 2 over Q.

  1. AOption (a)

    1

  2. BOption (b)

    3

  3. COption (c)

    2

  4. DOption (d)

    4

View answer & explanation

Correct answer

Answer: (c)C.

2

Why this answer is correct

Solution:

f(x) = x2 - 2

The roots of the polynomial f(x) are ±2 \pm \sqrt{2}. Since all the roots of f(x) are distinct, f(x)=x2-2 is separable.

The Galois group of the separable polynomial f(x)=x^2-2 is the Galois group of the splitting field of f(x) over Q\mathbb{Q}. Since the roots of f(x) are ±2\pm \sqrt{2}, the splitting field of f(x) is Q(2) \mathbb{Q}(\sqrt{2}). Thus, we want to determine the Galois group

Gal⁡(Q(2)/Q).\operatorname{Gal}(\mathbb{Q}(\sqrt{2}) / \mathbb{Q}) .

Let σ∈Gal⁡(Q(2)/Q)\sigma \in \operatorname{Gal}(\mathbb{Q}(\sqrt{2}) / \mathbb{Q}). Then the automorphism σ\sigma permutes the roots of the irreducible polynomial f(x) = x2-2. Thus σ(2)\sigma(\sqrt{2}) is either 2\sqrt{2} or −2-\sqrt{2}. Since σ\sigma fixes the elements of Q\mathbb{Q},thisdeterminesσcompletelyaswehave, this determines \sigma completely as we have

σ(a+b2)=a+bσ(2)=a±2\sigma(a+b \sqrt{2})=a+b \sigma(\sqrt{2})=a \pm \sqrt{2} .

The map 2↦2\sqrt{2} \mapsto \sqrt{2} is the identity automorphism 1 of Q2.\mathbb{Q} \sqrt{2}.

The other map 2↦−2 \sqrt{2} \mapsto-\sqrt{2} gives non-identity automorphism τ \tau. Therefore, the Galois group Gal⁡(Q(2)/Q)=1,τ \operatorname{Gal}(\mathbb{Q}(\sqrt{2}) / \mathbb{Q})={1, \tau} is a cyclic group of order 2 .

Question 176

If G be a group of order 57. Assume that G is not a cyclic group. Then what is the number of elements in G of order 3?

  1. AOption (a)

    36

  2. BOption (b)

    35

  3. COption (c)

    32

  4. DOption (d)

    38

View answer & explanation

Correct answer

Answer: (d)D.

38

Why this answer is correct

Solution:

Observe the prime factorization 57 = 3 ×19. 

Let n19 be the number of Sylow 19-subgroups of G.

By Sylow's theorem, we know that

n19≡1( mod 19)n_{19} \equiv 1 (\bmod 19)  and  n19|3 .

It follows that n19 = 1.

Now, observe that if g∈Gg \in G, then the order of g is 1, 3, or 19.

Note that since G is not a cyclic group, the order of g cannot be 57 . As there is exactly one Sylow 19 -subgroup P, any element that is not in P must have order 3.

Therefore, the number of elements of order 3 is 57 - 19 = 38.

Question 177

If G is a finite group of order 217. Find out the number of generators of the group G.

  1. AOption (a)

    181

  2. BOption (b)

    188

  3. COption (c)

    1182

  4. DOption (d)

    180

View answer & explanation

Correct answer

Answer: (d)D.

180

Why this answer is correct

Concept Used:

Number of generators of a finite group of order n is given by ϕ(n).

where ϕ(n) is the Euler totient function. i.e., a number of integers between 1 and n which are relatively prime to n.

Also, ϕ(mn) = ϕ(m) ⋅ ϕ(n) 

Calculation:

G is a finite group of order 217

number of generators of G is ϕ(217) = ϕ(7⋅31) = ϕ(7)⋅ϕ(31)

Since 7 and 31 both are primes, all the integers before then that is relatively prime to it.

Hence, ϕ(7) = 6 and ϕ(31) = 30

⇒ ϕ(217) = 6 ⋅ 30 = 180

Hence, the number of generators of G are 180.

Question 178

We call a topological space T sequentially compact if _______.

  1. AOption (a)

    T is not compact, it has a maximal open cover V without a finite subcover.

  2. BOption (b)

    U be the collection of their complements. By De Morgan's rules, ∪ is a cover of T.

  3. COption (c)

    It T compact, s ϵ S and ∪ is an open cover of T × S, there is open V ⊂ S containing s such that T × V can be covered by a finite subfamily of ∪.

  4. DOption (d)

    every sequence in T has a convergent subsequence

View answer & explanation

Correct answer

Answer: (d)D.

every sequence in T has a convergent subsequence

Why this answer is correct

Concept -

Sequentially Compact Spaces:

A topological space T is said to be sequentially compact if every sequence in T has a convergent subsequence. This property is crucial in analysis and topology because it provides a way to extract additional information from sequences within the space.

Explanation -

Sequential compactness is a concept in topology that generalizes the notion of compactness in a way that is particularly useful for sequences. In a sequentially compact space, no matter how you pick a sequence of points, there will always be a subsequence that converges to a point within the space. This is a powerful property because it implies that the space is, in a certain sense, limited or bounded, even if it may not be compact in the traditional topological sense.

For example, in the real numbers with the usual topology, the closed interval [0, 1] is sequentially compact because any sequence of numbers within this interval will have a subsequence that converges to a limit within the interval. This property is due to the Bolzano-Weierstrass theorem, which states that every bounded sequence in RnRnRn

has a convergent subsequence.

However, not all compact spaces are sequentially compact, and not all sequentially compact spaces are compact, depending on the topology. The notion of sequential compactness is particularly useful in spaces where the notion of sequences is more natural or easier to work with than the general notion of open covers used in the definition of compactness.

Therefore, the correct definition of a sequentially compact space is that every sequence in T has a convergent subsequence, which corresponds to option 4.

Key Points

Sequential compactness is a property that ensures the existence of convergent subsequences for every sequence in a space, making it a fundamental concept in the study of topological spaces and their properties.

Question 179

Let G be a group we know that G is the internal direct product of two subgroups of H and K. It is not true when:

  1. AOption (a)

    H ∩ K = {e}

  2. BOption (b)

    H and K are normal subgroups of G

  3. COption (c)

    HK = G

  4. DOption (d)

    H ∪ K = G

View answer & explanation

Correct answer

Answer: (d)D.

H ∪ K = G

Why this answer is correct


**Concept -**

For a group \\(G\\) to be the internal direct product of two subgroups \\(H\\) and \\(K\\), certain conditions must be satisfied:

- \\(H \cap K = \{e\}\\), where \\(e\\) is the identity element of \\(G\\). This implies that \\(H\\) and \\(K\\) intersect trivially.
- \\(H\\) and \\(K\\) are normal subgroups of \\(G\\). Being normal means that each subgroup is invariant under conjugation by elements of \\(G\\).
- Every element of \\(G\\) can be uniquely expressed as a product of an element from \\(H\\) and an element from \\(K\\), which is denoted by \\(HK = G\\).

Given these conditions, the statement that is **not** true when asserting that \\(G\\) is the internal direct product of \\(H\\) and \\(K\\) is:

**Option 4: \\(H \cup K = G\\)**. The union of \\(H\\) and \\(K\\) does not necessarily cover all elements in \\(G\\) unless every element of \\(G\\) can be expressed as either an element of \\(H\\) or \\(K\\), which is not a requirement for the internal direct product. The correct condition is \\(HK = G\\), meaning every element of \\(G\\) is a product of elements from \\(H\\) and \\(K\\), not merely their set union.

**Solution Statement -**

The correct answer is option 4 because the union of two subgroups \\(H\\) and \\(K\\) does not ensure that every element of the group \\(G\\) can be represented as a product of elements from \\(H\\) and \\(K\\). For \\(G\\) to be the internal direct product of \\(H\\) and \\(K\\), it is essential that \\(HK = G\\) and that \\(H\\) and \\(K\\) intersect trivially and are both normal in \\(G\\).

Question 180

What is the analytic region of f(z) = (x - y)2 + 2i(x + y)?

  1. AOption (a)

    x + y = 2

  2. BOption (b)

    x - y = -1

  3. COption (c)

    x + y = -2

  4. DOption (d)

    x - y = 1

View answer & explanation

Correct answer

Answer: (d)D.

x - y = 1

Why this answer is correct

Concept:

Function of a Complex Variable: ω = f (z) = u (x, y) + i v (x, y) Where x, y ϵ R  and  z = x + iy

u (x, y), v (x, y) are real valued functions. 

Analytic function: A function f (z) is said to be Analytic in a region R of z-plane if it is differentiable at every point of R. 

Necessary Condition for function f (z) to be Analytic: 

1. ∂u∂x, ∂u∂y, ∂v∂x, ∂v∂y\frac{\partial u}{\partial x},\ \frac{\partial u}{\partial y}, \ \frac{\partial v}{\partial x}, \ \frac{\partial v}{\partial y}  exist 2. f (z) to satisfy the Cauchy-Riemann (C-R) equation.

 ∂u∂x =  ∂v∂y, ∂u∂y = − ∂v∂x\frac{\partial u}{\partial x}\ =\ \ \frac{\partial v}{\partial y}, \ \frac{\partial u}{\partial y}\ =\ - \ \frac{\partial v}{\partial x}

Calculation: 

Given that,

f(z) = (x - y)2 + 2i(x + y) 

On compairing it from f (z) = u (x, y) + i v (x, y), we will get

u = (x - y)2   and  v =  2(x + y) 

Now, differentiating u & v with respect to x and y, we will get

∂u∂x =  ∂(x −y)2∂x = 2(x − y)\frac{\partial u}{\partial x}\ =\ \ \frac{\partial (x\ - y)^2}{\partial x} \ = \ 2(x\ -\ y)    

∂u∂y =  ∂(x −y)2∂y = − 2(x − y)\frac{\partial u}{\partial y}\ =\ \ \frac{\partial (x\ - y)^2}{\partial y} \ = \ -\ 2(x\ -\ y)

∂v∂x =  ∂ 2(x + y)∂x = 2\frac{\partial v}{\partial x}\ =\ \ \frac{\partial \ 2(x\ +\ y)}{\partial x} \ = \ 2

∂v∂y =  ∂ 2(x + y)∂y = 2\frac{\partial v}{\partial y}\ =\ \ \frac{\partial \ 2(x\ +\ y)}{\partial y} \ = \ 2  

According to the question, a function is analytic already, so its differential exists. Therefore by using the C-R equation,

∂u∂x =  ∂v∂y,    and   ∂u∂y = − ∂v∂x\frac{\partial u}{\partial x}\ =\ \ \frac{\partial v}{\partial y}, \ \ \ \ and\ \ \ \frac{\partial u}{\partial y}\ =\ - \ \frac{\partial v}{\partial x}

⇒ 2(x - y) = 2

⇒ x - y = 1

Hence, analytic region will be x - y = 1.

Discipline 5 (10 questions)

Question 181

To prove the Fundamental Theorem of Algebra, we will need the

  1. AOption (a)

    Extreme value theorem for two real variables

  2. BOption (b)

    Real valued functions for two real variables

  3. COption (c)

    Neither 1 or 2

  4. DOption (d)

    Both 1 & 2

View answer & explanation

Correct answer

Answer: (d)D.

Both 1 & 2

Why this answer is correct

Fact:

To prove the Fundamental Theorem of Algebra, we need an Extreme value theorem for a real-valued function for two real variables.

Hence, we need both 1 and 2.

Question 182

Find the residues of a function 

f(z) = 1(z − 2)(z + 1)3f(z) \ = \ \frac{1}{(z\ -\ 2)(z\ +\ 1)^3}

  1. AOption (a)

    −127  and  −1125\frac{-1}{27}\ \ and\ \ \frac{-1}{125}

  2. BOption (b)

    1125  and  −1125\frac{1}{125}\ \ and\ \ \frac{-1}{125}

  3. COption (c)

    127  and  −127\frac{1}{27}\ \ and\ \ \frac{-1}{27}

  4. DOption (d)

    127  and  1125\frac{1}{27}\ \ and\ \ \frac{1}{125}

View answer & explanation

Correct answer

Answer: (c)C.

127  and  −127\frac{1}{27}\ \ and\ \ \frac{-1}{27}

Why this answer is correct

Residue Theorem:

If f(z) is analytic in a closed curve C except at a finite number of singular points within C, then

\mathop \smallint \limits_C f\left( z \right)dz = 2\pi i \times \left[ {{\rm{sum;of;residues;at;the;singualr;points;within;C}}} \right]

Formula to find residue:

  1. If f(z) has a simple pole at z = a, then

Res;f(a)=lim⁡z→a[(z−a)f(z)]Res;f\left( a \right) = \mathop {\lim }\limits_{z \to a} \left[ {\left( {z - a} \right)f\left( z \right)} \right]

  1. If f(z) has a pole of order n at z = a, then

Res;f\left( a \right) = \frac{1}{{\left( {n - 1} \right)!}}{\left{ {\frac{{{d^{n - 1}}}}{{d{z^{n - 1}}}}\left[ {{{\left( {z - a} \right)}^n}f\left( z \right)} \right]} \right}_{z = a}}

Calculation:

Given, f(z) = 1(z − 2)(z + 13)f(z) \ = \ \frac{1}{(z\ -\ 2)(z\ +\ 1^3)}

For a simple pole at z = 2

Residue of {f(z)}z=2 = \mathop {\lim }\limits_{z \to 2} \left{ {\left( {z - 2} \right).\frac{1}{{\left( {z - 2} \right){{\left( {z + 1} \right)}^3}}}} \right}

=lim⁡z→21(z+1)3⇒133=127 = \mathop {\lim }\limits_{z \to 2} \frac {1}{\left( {z + 1} \right)^3} \Rightarrow \frac{1}{{{3^3}}} = \frac{1}{{27}}

For multiple pole (n = 3) at z = -1

Residue will be  

\Rightarrow \frac{1}{{\left( {3 - 1} \right)!}} \mathop {\lim }\limits_{z \to -1} \left{ {\frac{{{d^2}}}{{d{z^2}}}{{\left( {z + 1} \right)}^3}.\frac{1}{{\left( {z - 2} \right){{\left( {z + 1} \right)}^3}}}} \right}

⇒ 12lim⁡z→−1ddz[ddz1(z − 2)]\Rightarrow \ \frac{1}{2}\mathop {\lim }\limits_{z \to -1}\frac{d}{dz}[\frac{d}{dz}\frac{1}{(z\ -\ 2)}]

⇒ 12lim⁡z→−1ddz−1(z − 2)2\Rightarrow \ \frac{1}{2}\mathop {\lim }\limits_{z \to -1}\frac{d}{dz}\frac{-1}{(z\ -\ 2)^2}      [∵ d/dx(xn) = nxn-1]

 ⇒ 12lim⁡z→−12(z − 2)3\Rightarrow \ \frac{1}{2}\mathop {\lim }\limits_{z \to -1}\frac{2}{(z\ -\ 2)^3}

⇒1(−3)3=−127\Rightarrow \frac{1}{{{{\left( { - 3} \right)}^3}}} = - \frac{1}{{27}}

Hence, the residue of function f(z) is 127  and  −127\frac{1}{27}\ \ and\ \ \frac{-1}{27}

Question 183

What is the value of xx if x = √-1

  1. AOption (a)

    eπ2e^\frac{\pi}{2}

  2. BOption (b)

    0

  3. COption (c)

    e−π2e^{-\frac{\pi}{2}}

  4. DOption (d)

    1

View answer & explanation

Correct answer

Answer: (c)C.

e−π2e^{-\frac{\pi}{2}}

Why this answer is correct

Concept:

Complex number:

Z = x + iy is a complex number where i = √-1  where x and y are real parts and imaginary parts of Z and represented by Re(Z) and Img(Z) respectively.

|Z| = √(x2 + y2)√{(x^2 \ +\ y^2)}

We can represent the Z in polar form also, which is

Z = r(cosθ + i sinθ)     where r =  √(x2 + y2)√{(x^2 \ +\ y^2)}

The argument of Z:

arg(x + iy) = θ  = tan−1yxtan^{-1}\frac{y}{x}

Hence, we can represent Z in another from

Z = r(cosθ + i sinθ) = r.eiθ 

Calculation:

Let, Z = √-1 = i

⇒ Z = 0 + i                                      .......(1)

Hence, Re(z) = 0 and Img(Z) = 1 

Hence, by using above formula, 

|Z| = r = 0+  12 = 1\sqrt{0 +\ \ 1^2}\ =\ 1

Therefore, 

arg(Z) = θ = tan−1yxtan^{-1}\frac{y}{x}

⇒ θ = tan−110 = π2\theta \ =\ tan^{-1}\frac{1}{0}\ = \ \frac{π}{2}                    (∵ tan(π/2) = ∞)  

⇒ Z = i = eiπ2e^{i\frac{\pi}{2}}                                  (∵ Z = r(cosθ + i sinθ) = r.eiθ )    

⇒ XX =   (eiπ2)i(e^{i\frac{\pi}{2}}) ^i

⇒ XX = e−π2     ∵i2 = −1e^{-\frac{\pi}{2}}\ \ \ \ \ \because i^2\ = \ -1

Hence, the value of XX is e−π2e^{-\frac{\pi}{2}}

Question 184

Let A be a 5 × 5 matrix with real entries and x ≠ 0. Then, the vectors x, Ax, A2x, A3x, A4x, A5x are

  1. AOption (a)

    linearly independent

  2. BOption (b)

    linearly dependent

  3. COption (c)

    linearly independent if and only if A is symmetric

  4. DOption (d)

    linear dependence/independence cannot be determined from given data

View answer & explanation

Correct answer

Answer: (b)B.

linearly dependent

Why this answer is correct

Calculation:

The vectors  x, Ax, A2x, A3x, A4x, A5x  are all vectors in R5\mathbb{R}^5,

since A is a 5×55\times 5 real matrix and x∈R5x \in \mathbb{R}^5

But dim⁡(R5)=5\dim(\mathbb{R}^5)=5. Therefore, any set of more than 5 vectors in R5\mathbb{R}^5 must be linearly dependent

Here we have 6 vectors:  x, Ax, A2x, A3x, A4x, A5x

Hence they are always linearly dependent (independent of whether AA is symmetric or not, and for any x≠0x\neq 0)

The correct answer is linearly dependent

Question 185

_______ is regular element of Z(20)

  1. AOption (a)

    15

  2. BOption (b)

    13

  3. COption (c)

    16

  4. DOption (d)

    14

View answer & explanation

Correct answer

Answer: (b)B.

13

Why this answer is correct

Concept Used:

In ring theory, a nonzero element of a ring that is neither a left nor a right zero divisor is called a regular element.

Calculation:

The zero divisors in Z(20) are {2,4, 5,6,8, 10,12,14,15,16,18}

Clearly, 13 is not zero divisor of Z(20).

Hence, 13 is a regular element in Z(20).

Question 186

In Ring of real quaternion, (1 - 2i - 3j - 2k)-1 = _______

  1. AOption (a)

    (1 - 2i - 3j - 2k) ÷ 18

  2. BOption (b)

    (-1 + 2i + 3j + 2k) ÷ 18

  3. COption (c)

    (1 + 2i + 3j + 2k) ÷ 18

  4. DOption (d)

    (1 - 2i - 3j - 2k) ÷ 6

View answer & explanation

Correct answer

Answer: (c)C.

(1 + 2i + 3j + 2k) ÷ 18

Why this answer is correct


**Concept -**

In the ring of real quaternions, the inverse of a quaternion \\(q = a + bi + cj + dk\\) (where \\(i^2 = j^2 = k^2 = ijk = -1\\), and \\(a, b, c,\\) and \\(d\\) are real numbers) is given by \\(q^{-1} = \frac{a - bi - cj - dk}{a^2 + b^2 + c^2 + d^2}\\). This formula is derived from the fact that the product of a quaternion and its conjugate gives a scalar (real number) that is the sum of the squares of its components. This scalar is then used as the denominator to normalize the quaternion's conjugate, effectively giving its inverse.

**Explanation -**

Given the quaternion \\(q = 1 - 2i - 3j - 2k\\), its conjugate is \\(q^* = 1 + 2i + 3j + 2k\\). The norm of \\(q\\), denoted as \\(||q||^2\\), is \\(1^2 + (-2)^2 + (-3)^2 + (-2)^2 = 1 + 4 + 9 + 4 = 18\\).

Therefore, the inverse of \\(q\\) is \\(q^{-1} = \frac{q^*}{||q||^2} = \frac{1 + 2i + 3j + 2k}{18}\\).

**Hence Option (3) is true.**

**Key Points**

The inverse of a quaternion is found by dividing its conjugate by the sum of the squares of its components. This process effectively normalizes the quaternion's conjugate, providing the inverse quaternion.

Question 187

Let A be a complex n × n matrix. Let λ1, λ2, λ3 be three distinct eigenvalues of A, with corresponding eigenvectors z1, z2, z3. Then, which of the following statements is false? 

  1. AOption (a)

    z1 + z2, z1 - z2, z3 are linearly independent.

  2. BOption (b)

    z1, z2, z3 are linearly independent 

  3. COption (c)

    z1, z1 + z2, z1 + z2 + z3 are linearly independent

  4. DOption (d)

    z1, z2, z3 are linearly independent if and only if A is diagonolizable

View answer & explanation

Correct answer

Answer: (d)D.

z1, z2, z3 are linearly independent if and only if A is diagonolizable

Why this answer is correct


**Concept -**

The question involves understanding the properties of eigenvalues and eigenvectors of a complex matrix and their implications for linear independence and diagonalizability of the matrix.

**Explanation -**

The correct statement that is false is that "z1, z2, z3 are linearly independent if and only if A is diagonalizable". This statement is not always true because:

- The linear independence of a set of eigenvectors is related to the distinctness of their corresponding eigenvalues. If a matrix has n distinct eigenvalues, then it will have n linearly independent eigenvectors, making the matrix diagonalizable.
- However, a matrix being diagonalizable does not solely depend on the linear independence of any three eigenvectors but rather on the existence of a complete set of n linearly independent eigenvectors for an n x n matrix.
- Therefore, while z1, z2, z3 being linearly independent is necessary for A to be diagonalizable, it is not a sufficient condition unless A is a 3 x 3 matrix specifically. For an n x n matrix where n > 3, A being diagonalizable would require n linearly independent eigenvectors.

Thus, the statement given in option 4 is misleading and false in the general context, making it the correct answer.

**Key Points**

The diagonalizability of a matrix depends on the existence of a complete basis of linearly independent eigenvectors, which is directly related to the matrix having enough distinct eigenvalues. However, having a set of three linearly independent eigenvectors does not guarantee diagonalizability unless the matrix is specifically 3 x 3.

Question 188

Find out the Taylor series expansion of 3 sin x + 2 cos x:

  1. AOption (a)

    2 - 3x + x2 + (x3 ÷ 2) + ....

  2. BOption (b)

    2 + 3x - x2 - (x3 ÷ 2) + ....

  3. COption (c)

    2 + 3x - x2 + (x3 ÷ 2) + ....

  4. DOption (d)

    2 - 3x - x2 - (x3 ÷ 2) + ....

View answer & explanation

Correct answer

Answer: (b)B.

2 + 3x - x2 - (x3 ÷ 2) + ....

Why this answer is correct

Formula Used:

The Taylor series expansion of f(x) about c :

f(c)+f′(c)(x−c)+f(2)(c)2!(x−c)2+f(3)(c)3!(x−c)3+… f(c)+f^{\prime}(c)(x-c)+\frac{f^{(2)}(c)}{2 !}(x-c)^2+\frac{f^{(3)}(c)}{3 !}(x-c)^3+\ldots

or Taylor Series  =∑k=0∞1k!f(k)(c)(x−c)k =\sum_{k=0}^{\infty} \frac{1}{k !} f^{(k)}(c)(x-c)^k

and sin⁡x=x−x33!+x55!−x77!+… cos⁡x=1−x22!+x44!−x66!+… \begin{aligned} & \sin x=x-\frac{x^3}{3 !}+\frac{x^5}{5 !}-\frac{x^7}{7 !}+\ldots \ & \cos x=1-\frac{x^2}{2 !}+\frac{x^4}{4 !}-\frac{x^6}{6 !}+\ldots \end{aligned}

Calculation: 

sin⁡x=x−x33!+x55!−x77!+… \sin x=x-\frac{x^3}{3 !}+\frac{x^5}{5 !}-\frac{x^7}{7 !}+\ldots

⇒ 3sin⁡x=3x−3x33!+3x55!−3x77!+… 3\sin x =3x-\frac{3x^3}{3 !}+\frac{3x^5}{5 !}-\frac{3x^7}{7 !}+\ldots  (i)

cos⁡x=1−x22!+x44!−x66!+… \cos x=1-\frac{x^2}{2 !}+\frac{x^4}{4 !}-\frac{x^6}{6 !}+\ldots

⇒ 2cos⁡x=2−2x22!+2x44!−2x66!+… 2\cos x=2-\frac{2x^2}{2 !}+\frac{2x^4}{4 !}-\frac{2x^6}{6 !}+\ldots(ii)

add equations (i) and (ii)

3sin⁡x+2cos⁡x=2+3x−x2−(x3÷2)+… 3\sin x + 2\cos x = 2+3x - x^2 -(x^3÷2)+\ldots

Hence, option (2) is the correct answer.

Question 189

What is the magnitude of the expression below: ez, where z is a complex number.

  1. AOption (a)

    √(x2 + y2)

  2. BOption (b)

    1

  3. COption (c)

    ex

  4. DOption (d)

    √(x2 - y2)

View answer & explanation

Correct answer

Answer: (c)C.

ex

Why this answer is correct

Concept:

Complex number: A complex number z = x + iy can be expressed in the polar form z = reiθ, where,

 r=x2+y2r = \sqrt {{x^2} + {y^2}}  is its length and θ the angle between the vector and the horizontal axis.

The fact x = r cos θ, y = r sin θ are consistent with Euler’s formula

eiθ = cos θ + isin θ.

Let z = x + iy be a complex number.

1.  ∣z∣=x2+y2=Re(z)2+Im;(z)2\left| {\rm{z}} \right| = {\rm{}}\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} = {\rm{}}\sqrt {{\rm{Re}}{{\left( {\rm{z}} \right)}^2} + {\rm{Im;}}{{\left( {\rm{z}} \right)}^2}} 2. |z1z2| = |z1||z2|

Calculation:

Let, z = x + iy

⇒ |ez| = |ex + iy| = |ex||eiy|

⇒ |ez| = |ex||eiy|

⇒ |ez| = ex|eiy|            (∵ ex > 0 for all x ∈ R)

But, we know that eiθ = cos θ + isin θ

⇒ |ez| = ex|cos y + i sin y|  

We know that, 

∣x + iy∣=x2+y2\left| {\rm{x\ +\ iy}} \right| = {\rm{}}\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}}

⇒ |ez| = excos2θ + sin2θ\sqrt{cos^2θ\ +\ sin^2θ} 

⇒ |ez| = ex    (∵ sin2θ + cos2θ = 1)

Hence, option 3 is correct.

Question 190

Let G be a connected open set and let : G → f be analytic function. Then which of the following is not an equivalent statements:

  1. AOption (a)

    f ≡ 0

  2. BOption (b)

    {z ϵ G ∶ f(z) = 0} has limit point in G

  3. COption (c)

    there is a point a in G such that fn(a) for each n ≥ 0;

  4. DOption (d)

    R > 0 is arbitrary, radius of convergence is infinite

View answer & explanation

Correct answer

Answer: (d)D.

R > 0 is arbitrary, radius of convergence is infinite

Why this answer is correct

Concept:

Let G be a connected open set and f:G→C f : G \to \mathbb{C} be an analytic function. The problem is based on the Identity Theorem for analytic functions.

If an analytic function vanishes on a set having a limit point in G G , then the function must be identically zero on G.

Statement (1):

f≡0 f \equiv 0

Statement (2):

The zero set z∈G:f(z)=0 { z \in G : f(z)=0 } has a limit point in G.

By the Identity Theorem, (2) implies f≡0 f \equiv 0 . Hence, (1) and (2) are equivalent.

Statement (3):

There exists a point a∈G a \in G such that f(n)(a)=0 f^{(n)}(a)=0 for all n≥0 n \ge 0 .

If all derivatives of f f vanish at a point, then its Taylor series about that point is identically zero. Hence f≡0 f \equiv 0 .

Therefore, statements (1), (2), and (3) are all equivalent.

Statement (4):

For arbitrary R>0, the radius of convergence is infinite.

This only implies that f f is an entire function. An entire function need not be identically zero (for example, f(z)=ez f(z)=e^z ).

Hence, statement (4) is not equivalent to the others.

The correct answer is option (4)

Teaching Methodology (10 questions)

Question 191

"People who have variations of sex characteristics that do not fit with typical male or female bodies." is called _______.

  1. AOption (a)

    Intersex

  2. BOption (b)

    Genderqueer

  3. COption (c)

    Binary Gendered

  4. DOption (d)

    Non-Binary Gendered

View answer & explanation

Correct answer

Answer: (a)A.

Intersex

Why this answer is correct

Key Points

  • Intersex is a general term used for a variety of conditions in which a person is born with sex characteristics (such as genitals, gonads, and chromosomes) that do not fit the typical definitions of male or female.
  • Intersex people may have biological features that are typically associated with both males and females.
  • Genderqueer is a non-binary gender identity that refers to someone who does not identify exclusively as a man or a woman. Genderqueer people may identify as both male and female, neither male nor female, or somewhere in between.
  • Binary gendered refers to people who identify as either male or female. This is the most common gender identity, and it is often assumed that everyone is binary gendered.
  • Non-binary gendered refers to people who do not identify as either male or female. This is a more recently recognized gender identity, and there is still a lot of misunderstanding about it.

Thus, it is concluded that People who have variations of sex characteristics that do not fit with typical male or female bodies is called Intersex.

Question 192

Which of these is not an origin of knowledge?

  1. AOption (a)

    Objectivism

  2. BOption (b)

    Pragmatism

  3. COption (c)

    Constructivism

  4. DOption (d)

    Interpretivism

View answer & explanation

Correct answer

Answer: (a)A.

Objectivism

Why this answer is correct

Key Points

  • Objectivism is not considered an origin of knowledge. It is a philosophical position that asserts that knowledge is based on objective reality and that truth exists independently of human perception or interpretation.
  • Objectivism argues that knowledge is derived from an objective, external reality that exists independently of our thoughts or beliefs.

Hint

  • Pragmatism is a philosophical approach that emphasizes the practical consequences and utility of beliefs and knowledge. It suggests that the value of knowledge is determined by its usefulness and effectiveness in achieving specific goals or solving practical problems.
  • Constructivism is a perspective that emphasizes the role of individual and social construction in the creation of knowledge. It posits that knowledge is actively constructed by individuals and is influenced by their experiences, interactions, and cultural context.
  • Interpretivism is a philosophical perspective that focuses on the interpretation of meaning and understanding in the study of human behavior and social phenomena. It suggests that knowledge is derived from interpreting and understanding the subjective experiences and meanings of individuals and groups.

Thus, it is concluded that Objectivism is not an origin of knowledge.

Question 193

The language to which a child is primarily exposed from infancy on is called _____.

  1. AOption (a)

    Father's language

  2. BOption (b)

    Mother tongue

  3. COption (c)

    National language

  4. DOption (d)

    Second language

View answer & explanation

Correct answer

Answer: (b)B.

Mother tongue

Why this answer is correct

Key Points

  • The mother tongue is the language that a child is first exposed to and learns from their primary caregiver, who is usually the mother. It is the language that the child will be most fluent in and that will have the strongest influence on their cognitive development.
  • The father's language, national language, and second language are all terms that can be used to refer to other languages that a child may learn. However, the mother tongue is the language that is most fundamental to a child's language development.
  • In some cases, a child may be exposed to two or more languages from infancy. This is known as bilingualism. Bilingual children can learn to speak both languages fluently, and there is evidence that bilingualism can have cognitive benefits.

Thus, it is concluded that the language to which a child is primarily exposed from infancy on is called mother tongue.

Question 194

Which of this is not a characteristic of evaluation?

  1. AOption (a)

    Evaluation is a continuous process

  2. BOption (b)

    It is a systematic process

  3. COption (c)

    It is not confined to classrooms

  4. DOption (d)

    It includes only academic subjects

View answer & explanation

Correct answer

Answer: (d)D.

It includes only academic subjects

Why this answer is correct

Evaluation is a systematic way to assess learner’s abilities, analyze performance, provide appropriate feedback to each learner and help them to progress. It is one of the crucial components of the teaching-learning process.

  • Evaluation needs to be integrated with the process of teaching and learning. Language evaluation should measure the learner's language proficiency during the teaching-learning process

Key Points

Characteristic of evaluation:

  • Evaluation is a continuous process, it helps to know which kind of help the students want in learning a language as it helps in improving academic achievements and students' motivation level.
  • It is a systematic process, it is done to assess or evaluate someone's performance in a specified field. The teacher does the evaluation of the student's performance to get an idea of the achievement of a student.
  • It is not confined to classrooms and it doesn't include academic subjects only rather it emphasizes ensuring the all-round development of students including cognitive, psychomotor, and affective domains.

Hence, it could be concluded that 'It includes only academic subjects' is not a characteristic of evaluation.

Question 195

What is social science?

  1. AOption (a)

    Geography

  2. BOption (b)

    Astrology

  3. COption (c)

    Biology

  4. DOption (d)

    Sociology

View answer & explanation

Correct answer

Answer: (d)D.

Sociology

Why this answer is correct

Social science is the study of society and the manner in which people behave and influence the world around us. It is a broad field that encompasses a wide range of disciplines, including anthropology, economics, political science, psychology, and sociology.

Key Points

  • Geography is the study of the Earth's physical features and their relationship to human activity.
  • Astrology is the study of the movements of celestial bodies and their supposed influence on human affairs.
  • Biology is the study of living organisms.
  • So, the social science is sociology. It is the study of human society and social relationships. It is concerned with the structure, function, and development of societies, as well as the social interactions and behaviors of individuals and groups.

Thus, it is concluded that Sociology is social science.

Question 196

Which of the following is not a model of disability?

  1. AOption (a)

    rehabilitation model

  2. BOption (b)

    mental model

  3. COption (c)

    social model

  4. DOption (d)

    charity model

View answer & explanation

Correct answer

Answer: (b)B.

mental model

Why this answer is correct

Key Points

  • The mental model is not a model of disability. It is a cognitive framework that people use to understand and interact with the world around them.
  • It is not specific to disability, and it can be used to understand any kind of phenomenon.

The other three options, rehabilitation model, social model, and charity model, are all models of disability.

  • The rehabilitation model views disability as a problem that needs to be fixed or cured. It focuses on the individual with the disability and on providing them with services and support to help them overcome their disability.
  • The social model views disability as a product of society's barriers and limitations. It focuses on changing society to make it more accessible and inclusive for people with disabilities.
  • The charity model views disability as a misfortune that needs to be pitied and helped. It focuses on providing financial and material assistance to people with disabilities.

Thus, it is concluded that mental model is not a model of disability.

Question 197

Which of the following does not describe the nature of a school subject?

  1. AOption (a)

    serves as a yardstick by which we learn

  2. BOption (b)

    gives a practical view of the outside world

  3. COption (c)

    receives content from a wide range of sources

  4. DOption (d)

    have a specific code of conduct

View answer & explanation

Correct answer

Answer: (d)D.

have a specific code of conduct

Why this answer is correct

Key Points

  • "Have a specific code of conduct" does not typically describe the nature of a school subject. School subjects are primarily focused on the content, knowledge, and skills related to a specific academic area or field of study. While certain subjects may incorporate aspects of ethics, morality, or behavior, a "specific code of conduct" is not a defining characteristic of a school subject.

The other options are more relevant to the nature of school subjects:

  • Serves as a yardstick by which we learn suggests that school subjects provide a framework or standard against which learning and knowledge can be measured and assessed.
  • Gives a practical view of the outside world implies that school subjects offer insights and knowledge about various aspects of the real world, helping students understand and interact with their environment.
  • Receives content from a wide range of sources indicates that school subjects draw on a diverse set of information and sources to provide comprehensive knowledge within a specific domain.

Thus, it is concluded that have a specific code of conduct does not describe the nature of a school subject.

Question 198

James Banks discusses three sets of approaches to multicultural education. Find the odd one out from the following:

  1. AOption (a)

    Social action approach

  2. BOption (b)

    Summative approach

  3. COption (c)

    Curriculum/Contribution Approach

  4. DOption (d)

    Dominance Approach

View answer & explanation

Correct answer

Answer: (b)B.

Summative approach

Why this answer is correct

James Banks discusses three sets of approaches to multicultural education, and the other three approaches are commonly associated with his framework.

Key Points

  • Social Action Approach: This approach emphasizes addressing social issues, advocating for social justice, and promoting positive change in society through education. It encourages students to become active and engaged citizens who work to combat inequality and discrimination.
  • Curriculum/Contribution Approach: This approach focuses on incorporating diverse perspectives and contributions from various cultural groups into the curriculum. It aims to recognize and include the contributions of different cultural and ethnic groups to the overall body of knowledge.
  • Dominance Approach: The dominance approach examines how one group's culture and perspectives have historically dominated educational curricula and systems, often to the detriment of other cultural groups. It highlights the need to challenge and change these power imbalances.

Thus, the "summative approach" is not typically associated with James Banks' framework for multicultural education, making it the odd one out in this context.

Question 199

Gender identity is self-identified as _____ in almost all instances.

  1. AOption (a)

    It is one's own subjective experience.

  2. BOption (b)

    It is manifested by observable factors such as behavior and appearance within society.

  3. COption (c)

    individual feeling of being male or female

  4. DOption (d)

    the result of a combination of inherent and extrinsic or environmental factors

View answer & explanation

Correct answer

Answer: (d)D.

the result of a combination of inherent and extrinsic or environmental factors

Why this answer is correct

Key Points

  • Gender expression is the way in which a person expresses their gender identity. It can be manifested by observable factors such as behavior and appearance within society. However, gender expression does not always match a person's gender identity. For example, a person who is transgender may express their gender identity in a way that is different from the sex they were assigned at birth.
  • Gender identity is the result of a combination of inherent and extrinsic or environmental factors. Some of the factors that may influence gender identity include:
  • Biological factors, such as hormones and genes
  • Social factors, such as family, friends, and culture
  • Personal experiences, such as trauma or abuse
  • It is important to respect a person's gender identity, regardless of whether or not it is the same as the sex they were assigned at birth.

Thus, it is concluded that Gender identity is self-identified as the result of a combination of inherent and extrinsic or environmental factors in almost all instances.

Question 200

What stages of society have sociologists classified based on technology, tools, and modes of living?

  1. AOption (a)

    metaphysical stage

  2. BOption (b)

    theological stage

  3. COption (c)

    hierarchical stage

  4. DOption (d)

    positive phase

View answer & explanation

Correct answer

Answer: (d)D.

positive phase

Why this answer is correct

Sociologists have classified stages of society based on technology, tools, and modes of living as part of Auguste Comte's theory of social evolution.

 Key PointsComte's theory includes the following stages:

  • Theological Stage: In this stage, society is characterized by a belief in supernatural and religious explanations for natural phenomena. People rely on religion and mythology to explain the world around them.
  • Metaphysical Stage: In this stage, society shifts away from supernatural explanations but still relies on abstract and philosophical ideas to understand the world. People seek more abstract and metaphysical explanations for natural phenomena.
  • Positive Phase: Also known as the Scientific or Positivist Stage, this is the final stage in Comte's theory. In this stage, society relies on empirical observation, science, and the scientific method to explain and understand the world. It is characterized by the use of reason and empirical evidence rather than religious or philosophical explanations. It is based on technology, tools, and modes of living.

Thus, it is concluded that positive phase is the correct answer.

200 questions in the full paper