Territorial Army Official Paper I (Conducted on 26 Sep 2021) (Previous Year Paper)

Ass is considered as Timid, i.e. lacking confidence and being shy. So,

Timid : Ass

similarly,

Fox is considered as Cunning, i.e. clever in nature. So,

Fox : Cunning

Hence, the correct answer is "Option (2)".

Antonym and Synonym are sharing opposite relation with each other. So,

Antonym : Synonym

similarly,

Synthesis is opposite to Analysis. Therefore,

Synthesis : Analysis

Hence, the correct answer is "Option (1)".

The content provided to a Newspaper is done by an Editor. So,

Editor : Newspaper

similarly.

The content provided to a Poem is done by a Poet. Therefore,

Poet : Poem

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

Amnesia is a disease which affects the memory of a person. So,

Amnesia : Memory

similarly,

Paralysis is a condition of a body in which it affects the movement of that person. Therefore,

Paralysis : Movement

Hence, the correct answer is "Option (2)".

The logic followed here is:

8 = 8 + 1 = 9

9² = 81**,** so

8 : 81

similarly,

10 = 10 + 1 = 11

11² = 121, therefore

10 : 121

Hence, the correct answer is "Option (2)".

Spoke is a part of the Wheel. So,

Spoke : Wheel

similarly,

Word is a part of a sentence. Therefore,

Word : Sentence

Hence, the correct answer is "Option(1)".

The logic followed here is;

(First number × Third Number)  2 = Second Number

(16, 64, 8)

= (16 × 8) ÷ 2

= 128 ÷ 2

= 64

Checking options:

1) (15, 60, 8)

**= (**15 × 8) ÷ 2

= 120 ÷ 2

= 60 (equal to second number)

2) (18, 90, 6)

= (18 × 6) ÷ 2

= 108 ÷ 2

= 54 (NOT equal to second number)

3) (10, 80, 8)

= (10 × 8) ÷ 2

= 80 ÷ 2

= 40 (NOT equal to second number)

4. (14, 62, 4)

= (14 × 4) ÷ 2

= 56 ÷ 2

= 28 (NOT equal to second number)

Hence, the correct answer is "Option(1)".

The pattern followed here is:

Given:

Similarly,

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

The pattern followed here is → Opposite letters as the given alphabet sequence.

Given:

Similarly,

Hence, the correct answer is "Option (2)".

Preparing the family tree using the following symbols:

Given:

A is B's sister, C is B's mother-

 

D is C's father, E is D's mother

Therefore, A will be D's Granddaughter.

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

Preparing the family tree using the following symbols:

Given:

A and B are brother and sister respectively-

C is A's father-

D is C's sister and E is D's mother-

Therefore, B will be E's Granddaughter.

Hence, the correct answer is "Option (1)".

Preparing a family tree using the following set of symbols:

Given:

Raghu and Babu are twins- 

Babu's sister is Reema-

Reema's husband is Rajan-

Raghu's mother is Laxmi-

Laxmi's husband is Rajesh-

Therefore, Rajesh will be Rajan's Father-in-Law.

Hence, the correct answer is "Option (1)".

Preparing a family tree using the following set of symbols:

Given: Gopal said, pointing at Govind, "His father is my father's only son-

Therefore, Gopal will be Govind's Father.

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

BODMAS rule;

Solving options:

1) 70 – 2 + 4 ÷ 5 × 6 = 44

⇒ 70 ÷ 2 - 4 × 5 + 6 = 44

⇒ 35 - 4 × 5 + 6 = 44

⇒ 35 - 20 + 6 = 44

⇒ 41 - 20 = 44

⇒ 21 ≠ 44 (INCORRECT)

2) 70 – 2 + 4 ÷ 5 × 6 = 21

⇒ 70 ÷ 2 - 4 × 5 + 6 = 21

⇒ 35 - 4 × 5 + 6 = 21

⇒ 35 - 20 + 6 = 21

⇒ 41 - 20 = 21

⇒ 21 = 21 (CORRECT)

3) 70 – 2 + 4 ÷ 5 × 6 = 341

⇒ 70 ÷ 2 - 4 × 5 + 6 = 341

⇒ 35 - 4 × 5 + 6 = 341

⇒ 35 - 20 + 6 = 341

⇒ 41 - 20 = 341

⇒ 21 ≠ 341 (INCORRECT)

4) 70 – 2 + 4 ÷ 5 × 6 = 36

⇒ 70 ÷ 2 - 4 × 5 + 6 = 36

⇒ 35 - 4 × 5 + 6 = 36

⇒ 35 - 20 + 6 = 36

⇒ 41 - 20  = 36

⇒21 ≠ 36 (INCORRECT)

Hence, the correct answer is "Option(2)".

BODMAS Rule table;

 

Given:

8 ÷ 4 – 6 + 3 × 4 = ?

⇒ 8 × 4 + 6 ÷  3 - 4 = ?

⇒ 8 × 4 + 2 - 4 = ?

⇒ 32 + 2 - 4 = ?

⇒ 30

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

BODMAS rule table;

Given equation:

36 × 12 + 4 ÷ 6 + 2 – 3 = ?

⇒ 36 - 12 ÷ 4 + 6 ÷ 2 × 3 = ?

⇒ 36 - 3 + 3 × 3 = ?

⇒ 36 - 3 + 9 = ?

⇒ 45 - 3 = ?

⇒ 42

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

Given:

8 × 20 ÷ 3 + 9 – 5 = 38

Interchanging numbers as given in options:

1) (8, 9)

⇒ 9 × 20 ÷ 3 + 8 – 5 = 38

⇒ 60 + 8 - 5 = 38

⇒ 63 ≠ 38

2) (3, 5)

⇒ 8 × 20 ÷ 5 + 9 – 3 = 38

⇒ 8 × 4 + 9 – 3 = 38

⇒ 32 + 9 - 3 = 38

⇒ 38 = 38

3) (3, 9)

⇒ 8 × 20 ÷ 9 + 3 – 5 = 38

⇒ 8 × 2.22 + 3 – 5 = 38

⇒ 17.76 + 3 - 5 = 38

⇒ 15.76 ≠ 38

4) (3, 8)

⇒ 3 × 20 ÷ 8 + 9 – 5 = 38

⇒ 3 × 6.66 + 9 – 5 = 38

⇒ 19.98 + 9 - 5 = 38

⇒ 23.98 ≠ 38

Hence, the correct answer is "Option(2)".

Given equation;

9 ? 27 ? 3 ? 4 ? 3

Checking options:

1) +, +, < and ×

⇒ 9 + 27 + 3 < 4 × 3

⇒ 39 < 12 (NOT correct)

2) +, ÷, > and ×

⇒ 9 + 27 ÷ 3 > 4 × 3

⇒ 9 + 9 > 12

⇒ 18 > 12 (Correct)

3) ×, ÷, = and ×

⇒ 9 × 27 ÷ 3 = 4 × 3

⇒ 9 × 9 = 12

⇒ 81 = 12 (NOT correct)

4) + ÷, < and +

9 + 27 ÷ 3 < 4 + 3

9 + 9 < 7

18 < 7 (NOT correct)

Hence, the correct answer is "Option (2)".

The actual meaning of the symbol '#' will be:

# ⇒ ×

Given:

1 # 3 # 7 = 21

⇒ 1 × 3 × 7 = 21

⇒  21 = 21

and,

12 # 1 # 3 = 36

⇒ 12 × 1 × 3 = 36

⇒ 36 = 36

Similarly,

14 # 2 # 0 = ?

⇒ 14 × 2 × 0 = ?

⇒ 0 × 0 = 0.

Hence, the correct answer is "Option (2)".

As per the given information, the path followed by the man will be:

Therefore, the man will be in the South-West direction from his original position.

Hence, the correct answer is "Option (1)".

The pattern followed here is:

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

The logic followed here is:

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

The pattern followed here is:

Hence, the correct answer is "Option (1)".

The logic followed here is:

Hence, the correct answer is "Option (1)".

The logic followed here is:

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

Logic:

And,

Similarly,

Here, 'NINE' is code as '6463' as per logic.

Hence, the correct answer is "6463".

Logic:

Similarly,

Here, TIDE is coded as 1983 as per logic.

Hence, the correct answer is "1983".

From the given information,

 

The positions of I in the word and the positions of 2 in the code are the same.

Similarly, the positions of O in the word and the positions of 0 in the code are the same.

So, the code for 'I' is '2' and the code for 'O' is '0'.

Similarly look for 'B' is '1'

Hence, the code for the given letters are:-

 

So, the code for INHIBITION is:-

 

Hence, the correct answer is "2532124205".

Here we use the table given below:

Logic:

E = 5

PEN = 16 + 5 + 14 = 35

Similarly,

PAGE = 16 + 1 + 7 + 5 = 29

Here, 'PAGE'  is coded as '29' as per logic.

Hence, the correct answer is "29".

Logic:

Similarly,

Here, 'BALE' is coded as 'EALB' as per logic.

Hence, the correct answer is "EALB".

Here, we use the table given below:

Logic:

Similarly,

Here, 'GEL' is coded as 'FDK' as per logic.

Hence, the correct answer is "FDK".

As per the given information:

Let, A = x

B = 2x

F = 4x

C = x/2

and, D = x/4

Thus, the arrangement is:

F > B > A > C > D

Here, we can see that 'B' is the second oldest among all.

Hence, the correct answer is "B".

As per the given information:

1) Five friends are sitting on the bench.

2) A is to the left of B but on the right of C.

Case (1)

Case (2)

Case (3)

3) D is to the right of B but on the left of E. So case (2) and case (3) are eliminated.

thus, the final arrangement is,

Here, "C and E" are at extremes in line as per arrangement.

Hence, the correct answer is "CE".

As per the given information:

1) The salaries of B, R, G, A and M are compared with one-another.

2)  M receives more salary than that of A & G.

Case (1)

M > A > G

Case (2)

M > G > A

3) The salary of R is less than the three persons.

Case (1)

M > A > G > R

Case (2)

M > G > A > R

4) The salary of B is least.

Case (1)

M > A > G > R > B

Case (2)

M > G > A > R > B

Here, in Both cases, "R" is the second lowest as per the given information.

Hence, the correct answer is "R".

Logic:

And,

Similarly,

Here, '91' is the correct answer as per logic.

Hence, the correct answer is "91".

Logic:

In

⇒ {(3 × 4) × (2)} = 24

In

⇒ {(2 × -2) × (-1)} = 4

Similarly,

In

⇒ {(6 × 0) × (5)} = 0

Here, '0' is the correct answer as per logic.

Hence, the correct answer is "0".

Given:

Logic:

Here, "8" is the correct answer as per logic.

Hence, the correct answer is "8"

Given:

Logic:

Here, "104" is the correct answer as per logic.

Hence, the correct answer is "104"

Given:

Logic:

Here, "250" is the correct answer as per logic.

Hence, the correct answer is "250"

Given:

1) The age of Ram is double as that of Shyam and half as that of Suresh.

2. The sum of their age is 70.

Calculations:

Let the age of Shyam = x years

Age of Ram = 2x years

Age of Suresh = 4x years

According to the question,

⇒ x + 2x + 4x = 70

⇒ 7x = 70

⇒ x = 10

Age of Ram = 2x = 2 × 10 = 20 years

Hence, the correct answer is "20".

Given:

1) Anil is as much younger to Vivek as he is older to Tarun.

2. The total age of Vivek & Tarun is 48 years.

Calculation:

Vivek - Anil = Anil - Tarun (given) ........(1)

Vivek + Tarun = 48 years (given)  .....(2)

From equations (1) and (2), we get:

⇒ Vivek - Anil = Anil - Tarun 

⇒ Vivek + Tarun = 2 (Anil)

From equation (2)

⇒ 48 = 2 (Anil)

⇒ Anil = 24

Hence, the correct answer is "24".

Formula used:

(x1 × y1) / z1 = (x2 × y2) / z2

Here, x = cats

y = minutes

z = rats

Calculations:

Put all values in the formula,

⇒ (4 × 4) / 4 = (8 × y2) / 8

⇒ y2 = 4

Thus, '4' minutes will take 8 cats to kill 8 rats.

Hence, the correct answer is "4".

Concept:

Decimal to Binary conversion

Step 1: Divide the decimal number n by 2. Use the integer quotient obtained in this step as the dividend for the next step.

Step 2: Write the remainder from bottom to top i.e. in reverse chronological order.

Calculation:

The given number is 85432578 

By using the above concept,

 

Now, write the remainder from bottom to top 

The last three-digit will be 010.

Calculations:

Perimeter of rectangular area = 2(Lenght + Breadth) = 6 km  ........(1)

⇒ (Lenght + Breadth) = 3 km

Then,

Area of rectangular area = Lenght × Breadth = 2 sq Km

⇒ Length × Breadth = 2

⇒ Length = 2/ Breadth

Putting in equation (1),

⇒ {2/ Breadth) + Breadth} = 3 km

⇒ Breadth = 2 , 1

⇒ Length = 1, 2.

Since Length is greater than breadth, so

Length = 2 Km and Breadth = 1 Km

Here, the difference between length & breadth is = 2 - 1 = 1 km

Hence, the correct answer is "1 Km".

Logic:

When one comes into contact with fever, he visits a doctor, who diagnoses first to understand which type of disease it can be, and on the basis of his diagnosis, he prescribes medicines.

Thus, the correct arrangement is,

(b) Fever

(a) Doctor

(d) Diagnose

(c) Prescribe

(e) Medicine

Here, the meaningful order is 'b, a, d, c, e' as per logic.

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

Logic:

1. When we are born we became a son.

2. For our siblings we are brothers.

3. When we marry we become husband and then son-in-law.

4. When we give birth to a child, we become a father.

Thus, The meaningful order of the following words is,

(4) Son

(1) Brother

(2) Husband

(5) Son-in-law

(3) Father

Hence, the correct answer is "Option (2)".

The relationship between Sociology, Psychology, and Humanities is given below:

Sociology and Psychology are studies in Humanities.

Hence, the correct answer is "Option (1)".

Given:

As per arrangement:

Here, the '7' member represents the women doctor who is not employed as per arrangement.

Hence, the correct answer is "7".

Given:

As per the given information:

Here, "74" WOMEN are ENTREPRENEURS but not MOTHER as per arrangement.

Hence, the correct answer is "74".

The logic followed here is:

All fathers are males.

All brothers are males.

Here, there is possibilities that a father can be a brother to someone. some relation can be formed between these two as well.

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

Given:

Number of apples which are rotten = 5%

He sells 75% of the remainder apples.

Remaining apples = 190

Calculation:

Let the original number of apples be x.

Number of apples which were not rotten = [\(\frac{100-5}{{100}}\) × (x)] = 0.95x

Number of apples which were unsold = (\(\frac{100-75}{{100}}\) × 0.95x) = (0.25 × 0.95x)

Now, according to the question

⇒ 190 = (0.25 × 0.95x)

⇒ 760 = 0.95x

⇒ x = 800

∴ He had 800 apples originally.

Given:

The ratio of two numbers = 3 : 5

If 13 is subtracted from each, the new number are in the ratio = 10 : 21

Calculation:

Let the ratio of two numbers be 3x and 5x respectively

According to the question

⇒ \(\frac{3x-13}{{5x-13}}\) = \(\frac{10}{{21}}\)

⇒ 63x - 273 = 50x - 130

⇒ 13x = 143

⇒ x = 11

First number = 3x = (3 × 11) = 33

Second number = 5x = (5 × 11) = 55

Now, 15 is added to each of the original number then the ratio becomes

⇒ (33 + 15) : (55 + 15)

⇒ 48 : 70 = 24 : 35

∴ The required ratio is 24 : 35.

Shortcut Trick

Net Profit % = Markup % − Discount % − (Markup × Discount) ÷ 100

Net Profit % = 30 − 15 − (30 × 15) ÷ 100 = 15 − 4.5 = 10.5%

Selling Price = 100% + 10.5% = 110.5% of Cost Price

110.5% → 910

100% (Cost Price) = (910 ÷ 110.5) × 100 ≈ 823.5

∴ The correct option is A.

Alternate Method

Given:

Markup percentage = 30%

Discount percentage = 15%

Selling Price (SP) = 910/-

Formula Used:

Marked Price (MP) = CP × (1 + Markup%/100)

Selling Price (SP) = MP × (1 − Discount%/100)

CP MP SP

+30% Markup

−15% Discount 910/-

Calculations:

⇒ Let Cost Price be x

⇒ Marked Price (MP) = x + 30% of x = 1.30x

⇒ Selling Price (SP) = MP − 15% of MP = 85% of MP

⇒ 910 = 0.85 × 1.30x

⇒ 910 = 1.105x

⇒ x = 910 ÷ 1.105

⇒ x ≈ 823.529

∴ The correct option is A.

Additional Information

Successive Percentage Change

When a value is changed by a% and then b%, the net effective change is (a + b + ab/100)%.

Relationship between CP and MP

The ratio of Cost Price to Marked Price is CP/MP = (100 − Discount%)/(100 + Profit%).

Markup Percentage

Markup % = [(Marked Price − Cost Price) ÷ Cost Price] × 100.

Given:

Time taken by A to work 30% = 30 days

Time taken by A and B to finish the work together = 20 days

Concept used:

Work = Efficiency × Time

Calculation:

Let total work to be done = 100 units

Work done by A in 30 days = (\(\frac{30}{{100}}\) × 100) = 30 units

A's efficiency = (\(\frac{30}{{30}}\)) = 1 unit/day

Remaining work = (100 - 30) = 70 units

Let B's efficiency = x units/day

Now, A and B complete remaining work in 20 days

⇒ (1 + x) × 20 = 70

⇒ 20 + 20x = 70

⇒ 20x = 50

⇒ x = 2.5

Time taken by B to complete the whole work alone = \(\frac{100}{{2.5}}\) = 40 days

∴ The required time taken by B is 40 days.

Given:

The given expression ax3 + 3x2 – 8x + b is exactly divisible by (x + 2) and (x – 2)

Calculation:

P(x) = ax3 + 3x2 – 8x + b

P(x) is divisible by (x + 2) and (x – 2)

⇒ P(-2) = a × (-2)³ + 3 × (-2)² - 8 × (-2) + b = 0

⇒ -8a + 12 + 16 + b = 0

⇒ -8a + b + 28 = 0 ....(1)

Again, P(2) = a × (2)3 + 3 × (2)2 - 8 × (2) + b = 0

⇒ 8a + 12 - 16 + b = 0

⇒ 8a + b - 4 = 0 ...(2)

Now, from equation (1) and (2)

⇒ 2b + 24 = 0

⇒ 2b = -24

⇒ b = -12

Now, from equation (1)

⇒ -8a + 12 + 16 + b = 0

⇒ -8a + 12 + 16 - 12 = 0

⇒ -8a = -16

⇒ a = 2

∴ The required value is a = 2, b = -12

Given:

The diameter of a wire is decreased by 10%

Concept used:

The volume of cylindrical shape = πr²l

Calculation:

Since, r = \(\frac{d}{{2}}\), so, r2 = \(\frac{d^2}{{4}}\)

Volume of cylindrical shape = πr2l = \(\frac{\pi d^2l}{{4}}\)  

Let the diameter of the wire = d

On a 10% decrease,

The diameter will be = d(\(\frac{90}{{100}}\)) = \(\frac{9d}{{10}}\)

Let the new length = L

Now, Old volume = New volume

So, \(\frac{\pi d^2l}{{4}}\) = \(\frac{\pi (9d)^2× L}{{4× 100}}\)

⇒ \(\frac{L}{{l}}\) = \(\frac{100}{{81}}\)

⇒ L = (\(\frac{100}{{81}}\) × l)

Percentage increase in length required 

⇒ [(\(\frac{100}{{81}}\) × l - l) / l] × 100

⇒ [l(\(\frac{100}{{81}}\) - 1) / l] × 100

⇒ (\(\frac{19}{{81}}\) × 100) 

⇒ 23.46% ~ 23%

∴ The required value is 23%

Given:

The ratio of the outer and inner perimeters of a circular path = 23 : 22

The path is 5m wide

Concept used:

Circumference of circle = 2πr

Diameter of the circle = 2r

Calculation:

According to the question

⇒ \(\frac{2\pi R}{{2\pi r}}\) = \(\frac{23}{{22}}\)

⇒ \(\frac{R}{{r}}\) = \(\frac{23}{{22}}\)

Let R = 23x and r = 22x

So, (R - r) = 5

⇒ (23x - 22x) = 5

⇒ x = 5

So, r = 22x = (22 × 5) = 110

Now, Diameter of the inner circle = 2r = (2 × 110) m = 220 m

∴ The require diameter of the inner circle is 220 m.

Concept used:

Pie chart: A diagram consisting of a circle divided into parts to show the size of particular parts in relation to the whole.

Explanation:

According to the above figure, we can clearly see that

In statistics, a suitable graph for representing the partitioning of the

total into sub-parts is a pie chart.

∴ The required answer is a pie chart.

Given:

The sum of twice a number and thrice its reciprocal  = \(\frac{25}{{2}}\)

Calculation:

According to the question

⇒ 2x + \(\frac{3}{{x}}\) = \(\frac{25}{{2}}\)

⇒ 4x² + 6 = 25x

⇒ 4x² - 25x + 6 = 0

⇒ 4x² - 24x - x + 6 = 0

⇒ 4x(x - 6) - 1(x - 6) = 0

⇒ (4x - 1)(x - 6) = 0

⇒ x = \(\frac{1}{{4}}\), 6

∴ The required number is 6.

Given:

The ratio of three number = 3 : 4 : 5

LCM = 4200

Concept used:

HCF: The highest common factor between one and more numbers.

LCM: The least common multiple is defined as the smallest multiple that two or more numbers have in common.

Calculation:

Let the ratio of three numbers be 3x : 4x : 5x respectively.

Now, according to the question, LCM of 

⇒ [(3x), (4x) ,(5x)] = 4200

⇒ 60x = 4200

⇒ x = 70

∴ The required HCF is 70.

Given:

2.8 + (5.2 ÷ 1.3 × 2) – 6 × 3 ÷ 8 + 2

Concept used:

Calculation:

2.8 + (5.2 ÷ 1.3 × 2) – 6 × 3 ÷ 8 + 2

⇒ 2.8 + 8 - 6 × 3 ÷ 8 + 2

⇒ 2.8 + 8 - 2.25 + 2

⇒ 12.8 - 2.25 

⇒ 10.55

∴ The required value is 10.55

Given:

x = 2 + √3

Concept used:

(a + b)² = a² + b² + 2ab

Calculation:

x = 2 + √3

Squaring both sides, we get

⇒ (x)² = (2 + √3)²

⇒ x² = (4 + 3 + 2 × 2 × √3)

⇒ x² = 7 + 4√3

Now, the value of \(\rm \frac{x^2-x+1}{x^2+x+1}\)

⇒ \(\rm \frac{7+4\sqrt 3-2-\sqrt 3+1}{7+4\sqrt 3+2+ \sqrt 3+ 1}\) = \(\frac{6 + 3\sqrt 3}{{10 + 5\sqrt 3}}\)

⇒ \(\frac{3(2 + \sqrt 3)}{{5(2 + \sqrt 3)}}\) = \(\frac{3}{5}\)

∴ The required value is \(\frac{3}{5}\).

Given:

A taxi average 25 km per hr for 50% of the distance.

20 km pr hr for 40% of it.

10 km per hr for remaining.

Concept used:

Average speed = Total distance/Total time

Calculation:

Total distance = 100 km

Total time = \(\frac{50}{{25}}\) + \(\frac{40}{{20}}\) + \(\frac{10}{{10}}\)

⇒ (2 + 2 + 1) = 5 hours

Average speed = (\(\frac{100}{{5}}\)) = 20 km/hr

∴ The required average speed is 20 km/hr.

Given:

The given number = 5, 10, 17, 26, 38, 50, 65

Calculation:

According to the question

⇒ 5 + 5 = 10

⇒ 10 + 7 = 17

⇒ 17 + 9 = 26

⇒ 26 + 11 = 37

⇒ 37 + 13 = 50

∴ The required wrong number in the sequences is 38.

Given:

x + y + z = 3 and x2 + y2 + z2 = 101

Concept used:

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

a3 + b3 + c3 - 3abc = (a + b + c) [(a + b + c)2 - 3(ab + bc + ca)]

Calculation:

According to the above concept

(x + y + z)2 = x2 + y2 + z2 + 2(xy + yz + zx)

⇒ (3)² = 101 + 2(xy + yz + zx)

⇒ 2(xy + yz + zx) = (9 - 101) = -92

⇒ xy + yz + zx = -46

Again, as we know

a3 + b3 + c3 - 3abc = (a + b + c) [(a + b + c)2 - 3(ab + bc + ca)]

⇒ x3 + y3 + z3 - 3xyz = 3[(3)² - 3(-46)]

⇒ x3 + y3 + z3 - 3xyz = 3(9 + 138)

⇒ x3 + y3 + z3 - 3xyz = 441

So, \(\rm \sqrt{x^3+y^3+z^3-3xyz}\) = 21

∴ The required value is 21.

Shortcut Trick

Ratio of Girls to Boys = 40% : 60% = 2 : 3.

Using Alligation: Let average weight of boys be x kg.

(x − 59.2) / (59.2 − 55) = 2 / 3

⇒ (x − 59.2) / 4.2 = 2 / 3

⇒ x − 59.2 = 2.8 ⇒ x = 62 kg.

∴ The correct answer is 62 kg.

Alternate Method

Given: Total students = 60, Average weight = 59.2 kg, Girls = 40%, Girls average = 55 kg.

⇒ Number of Girls = 40% of 60 = 24

⇒ Number of Boys = 60 − 24 = 36

⇒ Total weight of Class = 60 × 59.2 = 3552 kg

⇒ Total weight of Girls = 24 × 55 = 1320 kg

⇒ Total weight of Boys = 3552 − 1320 = 2232 kg

⇒ Average weight of Boys = 2232 ÷ 36 = 62 kg

∴ The correct answer is 62 kg.

Additional Information

Weighted Average Formula

The average of a combined group is given by A_avg = (n₁a₁ + n₂a₂) ÷ (n₁ + n₂), where n is the quantity and a is the average.

Deviation Method

Net deviation from the average is always zero. If one group is below the average, the other must be above it to balance the total.

Given:

The ratio of age of two boys = 5 : 6

The ratio of age of two boys after two years = 7 : 8

Calculation:

The ratio of present age of A and B be 5x and 6x respectively.

Now, according to the question

⇒ \(\frac{5x+2}{{6x+2}}\) = \(\frac{7}{{8}}\)

⇒ 40x + 16 = 42x + 14

⇒ 2x = 2

⇒ x = 1

Now, present age of A = 5x = (5 × 1) = 5

Present age of B = 6x = (6 × 1) = 6

The ratio of their age after 12 yrs = (5 + 12) : (6 + 12)

⇒ 17 : 18 = \(\frac{17}{18}\)

∴ The required age is \(\frac{17}{18}\)

Given:

Mixture of milk and water in the canister = 36 litres

The ratio of milk and water = 3 : 1

Calculation:

Quantity of milk in the mixture = (\(\frac{3}{{4}}\) × 36) = 27

Quantity of water in the mixture = (\(\frac{1}{{4}}\) × 36) = 9

On adding 15 litres of milk,

Required ratio = (27 + 15) : 9

⇒ 42 : 9 = 14 : 3

∴ The new ratio of the mixture is 14 : 3

Given:

The sum of two positive numbers = 20% of the sum of their square.

And 25% of the difference of their squares.

Calculation:

According to the question

⇒ x + y = \(\frac{20}{{100}}\) × (x2 + y2)

⇒ x + y = \(\frac{1}{{5}}\) × (x² + y²)

⇒ x² + y² = 5(x + y) ....(1)

Also,  x + y = \(\frac{25}{{100}}\) × (x2 - y2)

⇒ x + y = \(\frac{1}{{4}}\) × (x2 - y2)

⇒ x2 - y2 = 4(x + y) ....(2)

Adding equation (1) and (2), we get

⇒ 2x² = 9(x + y)

⇒ \(\rm \frac{x+y}{x^2}\) = \(\frac{2}{9}\)

∴ The required value is \(\frac{2}{9}\).

Given:

The top of two poles of height 24 m and 36 m are connected by a wire. 

The wire makes an angle of 60° with the horizontal.

Concept used:

sinθ = perpendicular/Hypotenuse

sin 60° = \(\frac{√3}{{2}}\)

Calculation:

From the above figure,

The length of the wire is AD,

In ΔADE,

sin 60° = \(\frac{DE}{{AD}}\)

⇒ \(\frac{√3}{{2}}\) = \(\frac{(36-24)}{{AD}}\)

⇒ \(\frac{√3}{{2}}\) = \(\frac{12}{{AD}}\)

⇒ AD = \(\frac{24}{{√3}}\)

On multiply and divide by √3

⇒ (\(\frac{24}{{√3}}\) × \(\frac{√ 3}{{√3}}\)) = \(\frac{24√ 3}{{3}}\) = 8√3 m

∴ The required value is 8√3 m.

Concept used:

Rectangle

• A rectangle is a quadrilateral.
• The opposite sides are parallel and equal to each other.
• Each interior angle is equal to 90 degrees.
• The sum of all the interior angles is equal to 360 degrees.
• The diagonals bisect each other
• Both diagonals have the same length.

Explanation:

According to the question, we draw a figure of a rectangle ABCD

AB = CD

BC = AD

AC = BD

It will be a rectangle and each angle will be a right angle.

∴ The required angle is 90°

Given:

Central angle = 45°

Radius of a circle = 28 cm

Concept used:

Length of an arc = 2πr × (\(\frac{θ }{{360}}\))

Where θ is a central angle and r is a radius

Calculation:

According to the above concept

Length of an arc = 2πr × (\(\frac{θ }{{360}}\))

⇒ (2 × \(\frac{22}{{7}}\) × 28 × \(\frac{45}{{360}}\)) = 22 cm

∴ The length of the arc is 22 cm

Given:

Total number of heads of cows and hens = 180

Total number of legs of cows and hens = 420

Calculation:

Let the number of cows be x.

Then, number of hens would be (180 - x).

According to the question

⇒ (180 - x) × 2 + 4x = 420

⇒ 360 - 2x + 4x = 420

⇒ 2x = 60

⇒ x = 30

∴ The required number of cows in the farm is 30.

Given:

Total number of children = 20

Total number of sweets distributed = 114

Number of sweets each girls gets = 6

Number of sweets each boys gets = 5

Concept used:

Average = Sum of observations/Number of observations

Calculation:

Average sweets per children = (114/20) = 5.7

By allegation method,

The ratio of girls to boys = 7 : 3

The number of girls to boys is 7x and 3x respectively.

Number of boys = (\(\frac{3x}{{10x}}\) × 20) = 6

∴ The required number of boys is 6.

Given:

A sum of money on compound interest becomes three times in 4 years

Concept used:

A = P(1 + \(\frac{r}{{100}}\))ⁿ

Where, A = Amount, P = Principal, r = Rate of interest and n = Time period

Calculation:

According to the above concept

A = P(1 + \(\frac{r}{{100}}\))ⁿ

⇒ A = P(1 + \(\frac{r}{{100}}\))⁴ = 3P

⇒ (1 + \(\frac{r}{{100}}\))4 = 3

⇒ P(1 + \(\frac{r}{{100}}\))ⁿ = 27P

⇒ (1 + \(\frac{r}{{100}}\))n = 27

⇒ (1 + \(\frac{r}{{100}}\))n = (3)³

⇒  (1 + \(\frac{r}{{100}}\))n =  (1 + \(\frac{r}{{100}}\))⁴]³ 

⇒ n = (4 × 3) = 12

∴ The required sum is 12 years.

Given:

Simple interest on a sum for 10 years is equal to 5% of the principle.

Concept used:

SI = \(\frac{P \times R \times T}{{100}}\)

Where, P = Principle, R = Rate, T = Time 

Calculation:

Let the principal be Rs. P

According to the above concept

SI = \(\frac{P \times R \times 10}{{100}}\) = 5% of P

⇒ \(\frac{P \times R \times 10}{{100}}\) = \(\frac{5P}{{100}}\)

⇒ R = 0.5%

Now, let in t years the simple interest is equal to the principle

⇒ SI = \(\frac{P \times 0.5 \times t}{{100}}\)

⇒ P = \(\frac{P \times t}{{200}}\)

⇒ t = 200 years

∴ The required time is 200 years.

Given:

The sum of three consecutive integers = 81

Calculation:

Let the three consecutive numbers be x - 1, x, x + 1.

Now, according to the question

⇒ x - 1 + x + x + 1 = 81

⇒ 3x = 81

⇒ x = 27

The three consecutive numbers = 26, 27, 28

So, the middle one number = 27

∴ The required value is 27.

Given:

The sum of two numbers = 1215

HCF = 81

Concept used:

HCF: The highest common factor between one and more numbers.

Prime number: A number that has only two factors one and itself.

Explanation:

Let the number be 81a and 81b where a and b are two numbers prime to each other.

So, 81a + 81b = 1215

⇒ a + b = 15

Now, find two numbers, whose sum is 15,

The possible pairs are

(14, 1) , (13, 2), (12, 3), (11, 4), (10, 5), (9, 6 ), (8, 7)

Of these the only pairs of numbers that are prime to each other are (14, 1), (13, 2), (11, 4), and (8, 7).

Hence, the required number is

⇒ (14 × 81, 1 x 81), (13 x 81, 2 x 81), (11 x 81, 4 x 81), (8 x 81, 7 x 81)

or (1134, 81), (1053, 162), (891, 324), (648, 567)

∴ There are 4 possible pairs.

Given:

√y = √4 – √6

Concept used:

(a - b)² = a² + b² - 2ab

Calculation:

√y = √4 – √6

Squaring both sides, we get

⇒ (√y)² = (√4 – √6)²

⇒ y = (√4)² + (√6)² - 2 × √4 × √6

⇒ y = 4 + 6 - 2√24

⇒ y = 10 - 4√6

Squaring the equation again,

⇒ y² = (10 - 4√6)² 

⇒ y² = (10)² + (4√6)² - 2 × 10 × 4√6

⇒ y² = 100 + 96 - 80√6

⇒ y² = 196 - 80√6

Now, we have to find the value of (y2 - 20y + 12)

So, put the value of y and y² in the given equation

⇒ 196 - 80√6 - 20 × (10 - 4√6) + 12

⇒ 196 - 80√6 - 200 + 80√6 + 12

⇒ 208 - 200 = 8

∴ The required value is 8.

Concept used:

Average = sum of observations/Number of observations

Calculation:

The prime numbers from 1 to 30 are

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Now, according to the above concept

⇒ Average = (2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29)/10

⇒ (129/10) = \(12\frac{9}{10}\)

∴ The required average is \(12\frac{9}{10}\).

Given:

Distance covered by man by 50 km/hr by running.

Same distance return back with 5 km/hr by walking.

Whole journey time = 5 hr 30 min

Concept used:

Time = Distance/Speed

Calculation:

Let the distance covered by train or walking be D.

Total time = 5 hr 30 min = (5 + \(\frac{30}{{60}}\)) 

⇒ (5 + \(\frac{1}{{2}}\)) = \(\frac{11}{{2}}\) hours

Time taken by train + Time taken by walking = \(\frac{11}{{2}}\)

⇒ \(\frac{D}{{50}}\) + \(\frac{D}{{5}}\) = \(\frac{11}{{2}}\)

⇒ \(\frac{D+10D}{{50}}\) = \(\frac{11}{{2}}\)

⇒ \(\frac{11D}{{50}}\) = \(\frac{11}{{2}}\)

⇒ D = 25 km

∴ The required distance is 25 km.

Alternate MethodTotal distance = \(\frac{x × y}{{x+y}}\)  [Total time]

Where x and y are the speed for going an returning.

The distance = [(\(\frac{50 × 5}{{50+5}}\)) × \(\frac{11}{{2}}\)]

⇒ (\(\frac{250}{{55}}\) × \(\frac{11}{{2}}\)) = 25 km

∴ The required distance is 25 km.

Given:

A certain number of men complete a work n 60 days. 

When 8 men join, work completed 10 days earlier = (60 - 10) = 50 days

Concept used:

Total work = Total manpower × days

M₁ × D₁ = M₂ × D₂

Calculation:

Let the number of men originally be N.

According to the question

⇒ N × 60 = (N + 8) × 50

⇒ 60N = 50N + 400

⇒ 10N = 400

⇒ N = 40

∴ The number of men originally is 40.

Calculation:

Total expenditure = 48 cr

The total expenditure of the company Q, R & T together = (22 + 18 + 20)% = 60%

Now, according to the question

The total expenditure of the company Q, R & T together = 60% of total expenditure

⇒ (60% of 48) = (\(\frac{60}{{100}}\) × 48)

⇒ \(\frac{144}{{5}}\) = 28.8

∴ The required expenditure is 28.8

Calculation:

Total expenditure = 48 cr

Expenditure of company Q & R together = (22 + 18)% = 40% of Total expenditure

⇒ (\(\frac{40}{{100}}\) × 48) = 19.2

Expenditure of company P = (24% of 48) = (\(\frac{24}{{100}}\) × 48) = 11.52

Percentage of expenditure of Q & R greater in terms of P

= [(19.2 - 11.52)/11.52] × 100

⇒ (\(\frac{7.68}{{11.52}}\) × 100) = (\(\frac{768}{{11.52}}\)) = 66.67%

∴ The required percentage is 66.67%

Given:

AB = 7 cm, CD = 4 cm, OB = 5 cm

Concept used:

We know that if two secant segments are drawn to a circle from the same external point, the product of the length of one secant segment and its external part is equal to the product of the length of the other secant segment and its external part.

Calculation:

Let the length of OD be x

According to the above concept

OA × OB = OD × OC

⇒ 12 × 5 = (x) × (4 + x)

⇒ 60 = 4x + x²

⇒ x² + 4x - 60 = 0

⇒ x² + 10x - 6x - 60 = 0

⇒ x(x + 10) - 6(x - 10) = 0

⇒ (x + 10) (x - 6) = 0

⇒ x = -10, 6

Negative Value of x is not acceptable.

∴ The required length of OD is 6 cm.

Shortcut Trick

Ratio of Volume (V) to Curved Surface Area (CSA) = r / 2

r / 2 = 1336.5 / 594 = 2.25

⇒ Radius (r) = 4.5 cm

Height (h) = CSA ÷ (2 × π × r)

h = 594 ÷ (2 × 22/7 × 4.5) = 21 cm

∴ The correct answer is 21 cm.

Alternate Method

Given: Curved Surface Area (CSA) = 594 cm², Volume (V) = 1336.5 cm³

Formula Used: CSA = 2πrh; Volume = πr²h

r

h CSA = 2πrh = 594 Vol = πr²h = 1336.5

⇒ Volume / CSA = (πr²h) / (2πrh) = 1336.5 / 594

⇒ r / 2 = 2.25

⇒ r = 4.5 cm

⇒ 2 × (22 / 7) × 4.5 × h = 594

⇒ (198 / 7) × h = 594

⇒ h = (594 × 7) / 198

⇒ h = 3 × 7 = 21 cm

∴ The correct answer is 21 cm.

Additional Information

Total Surface Area (TSA) of Cylinder

The total surface area includes the curved surface and the areas of the two circular bases: TSA = 2πrh + 2πr² = 2πr(h + r).

Volume of a Hollow Cylinder

If R is the outer radius and r is the inner radius, the volume is V = πh(R² − r²).

Diagonal of a Cylinder

The longest segment within a cylinder connects opposite points on the top and bottom circles, calculated as d = √(h² + (2r)²).

Shortcut Trick

Downstream Speed (u + v) = 18 + 6 = 24 km/h

Upstream Speed (u − v) = 18 − 6 = 12 km/h

Total Time = (Distance ÷ Downstream Speed) + (Distance ÷ Upstream Speed)

Total Time = (6 ÷ 24) + (6 ÷ 12) = 1/4 + 1/2 = 3/4 hours

In Minutes = 3/4 × 60 = 45 min

∴ The correct answer is 45 Min.

Alternate Method

Given:

Speed of Dinesh in still water (u) = 18 km/h

Speed of the stream (v) = 6 km/h

Total distance (D) = 6 km

Formula Used:

Time = Distance ÷ Speed

Downstream Speed (D_s) = u + v

Upstream Speed (U_s) = u − v

Calculations:

⇒ D_s = 18 + 6 = 24 km/h

⇒ U_s = 18 − 6 = 12 km/h

⇒ Time taken downstream = 6 ÷ 24 = 0.25 hours

⇒ Time taken upstream = 6 ÷ 12 = 0.5 hours

⇒ Total Time = 0.25 + 0.5 = 0.75 hours

⇒ Total Time in minutes = 0.75 × 60

⇒ Total Time = 45 minutes

∴ The correct answer is 45 Min.

Additional Information

Relative Speed in Water

When moving with the flow (downstream), speeds are added. When moving against the flow (upstream), the stream speed is subtracted from the boat speed.

Average Speed for Round Trip

Average speed in a round trip is given by [ (u² − v²) ÷ u ], where u is speed in still water and v is stream speed.

Speed Conversion

To convert hours into minutes, multiply the value by 60. To convert km/h to m/s, multiply by 5/18.

Given:

cos θ = \(\frac{3}{5}\)

Concept used:

By Pythagoras theorem 

Hypotenuse² = perpendicular² + base² 

cosθ = base/hypotenuse

sinθ = perpendicular/hypotenuse

Calculation:

cos θ = \(\frac{3}{5}\)

cosθ = base/hypotenuse = \(\frac{3}{{5}}\)

Now, by Pythagora's theorem 

Hypotenuse2 = perpendicular2 + base2 

⇒ 5² = perpendicular2 + 3²

⇒ 25 = perpendicular2 + 9

⇒ perpendicular² = 16

⇒ perpendicular = 4 

Now, sinθ = perpendicular/hypotenuse = \(\frac{4}{5}\)

∴ The required value is \(\frac{4}{5}\).

Given:

3√5 + \(√{125}\) = 17.88

Calculation:

3√5 + \(√{125}\) = 17.88

⇒ 3√5 + 5√5 = 17.88

⇒ 8√5 = 17.88

Now solving the given value

⇒ √80 + 6√5 = √(16 × 5) + 6√5

⇒ 4√5 + 6√5 = 10√5

So basically, we need to find out the value of 10√5.

Since 10√5 can be written as below;

⇒ 10√5 = 8√5 × \(\frac{10√ 5}{{8√5}}\)

⇒ 10√5 = (17.88 × \(\frac{10√ 5}{{8√5}}\))       (∵ 8√5 = 17.88)

⇒ 10√5 = (17.88 × \(\frac{5}{{4}}\)) 

⇒ 10√5 = 22.35

∴ The required value is 22.35.

Given:

Water contains hydrogen = \(14 \frac{2}{7}%\)% = \(\frac{100}{7}%\)%

Calculation:

According to the question​

Hydrogen in 350 g water = (\(\frac{100}{7}\) × \(\frac{1}{100}%\) × 350) = 50 g

Now, the oxygen in 350 g water = (350 - 50) g = 300 g

∴ The required oxygen is 300.

Given:

The area of smallest square which can be formed with joining of rectangles of dimension 6 cm × 4 cm

Concept used:

Area of square = side × side

Calculation:

LCM of 6, 4 is 12, which is the smallest possible side of the square.

So, area of square = (side × side)

⇒ (12 × 12) = 144 cm²

∴ The required value is 144 cm²

Given:

The mean proportion between \((8+√{15})\) and \((32-4√{15})\)

Calculation:

Required mean proportional = √\((8+√{15})\)((32-4√{15})\)

⇒ √4(8 - √15)(8 + √15)

⇒ √4[(64 - (√15)²]

⇒ √4(64 - 15)

⇒ √(4 × 49) 

⇒ 2 ×  7 = 14

∴ The required mean proportion is 14.

Concept:

Theorem: Angles opposite to equal sides are equal.

∠A + ∠B = ∠D  (Exterior angle property)

⇒ ∠A = ∠E   (Alternate angles)

Calculation:

According to the above figure,

∠HDC = ∠DCF = 10a + 5   [alternate angles]

⇒ 10a + 5 = 40 + 6a - 5   [exterior angle of ΔECF]

⇒ 4a = 30

⇒ a = 7.5

∴ The required value of a is 7.5

Given:

Height of well (h) = 20 m

Diameter of the well (d) = 7 m

Length of the rectangular plot = 22 m

Breadth of the rectangular plot = 14 m

Concept used:

Volume of well = πr²h

Volume of the rectangular slot = length × breadth × height

Calculation:

 

Radius of well (r) = (\(\frac{7}{{2 }}\)) = 3.5 m

Volume of well = (\(\frac{22}{{7}}\) × 3.5 × 3.5 × 20) = 770 m³

Volume of well = Volume of the rectangular slot 

Now, according to the above concept,

Volume of the rectangular slot = length × breadth × height

⇒ 770 = (22 × 14 × height)

⇒ Height = (\(\frac{770}{{22 \times 14}}\)) = (\(\frac{5}{{2 }}\)) = 2.5 m

∴ The required height is 2.5 m