SSC CHSL Tier-1 2021: Official Paper (Held On : 24 May 2022 Shift 3) (Previous Year Paper)

The correct answer is 'very sweet'

Key Points

• ​Here 'sweet' is an adjective which qualifies a subject (Noun/ Pronoun)
• However, an 'adverb' is required to modify a Verb
• Example- Phillip sings loudly in the shower.
• Thus, 'sweet' will be replaced by 'sweetly'.

Correct Sentence- Rahul sings very sweetly when he is in a good mood.

The correct answer is 'cultivated'.

Key Points

• Cultivated means to prepare and use land for growing plants for food or to sell.
• Ex-Olives have been cultivated for centuries in the Mediterranean countries.
• ​In this question, the use of the verb 'cultivated' fits the context of the sentence.
• Therefore, the correct sentence is: My grandfather was a farmer and he cultivated three acres of land.

Additional Information

• Advanced means to move forward.
• Grown means(used about plants) to exist and develop in a particular place; to make plants grow by giving them water, etc.
• Cultured means well educated, showing a good knowledge of art, music, literature, etc.

The correct answer is 'Not being as successful as previously'.

Key Points

• 'Lose your touch' is an idiom that means to no longer be able to do something as well as you could before.(पहले की तरह सफल नहीं हो पाना)
• Ex- After years of not practicing the piano, she found she had lost her touch, hitting the wrong notes more often than she used to.
• Hence, the correct answer is 'option 3'.

The correct answer is "she had resign"

Key Points

• When two actions happen in the past, it is necessary to emphasize which action happened before the other.
• For example: I had gone (3^rd form of verb) to Goa before my father came.
• Similarly, in this context, the person putting in the resign before the other apologised. So here 3^rd form of verb will be used i.e. resigned.
• The correct sentence is: She had resigned from the post before he apologised.

Therefore, the correct answer is "Option 1".

The correct answer is 'Confusion'.

Key Points

• The meaning of the word 'Bafflement' is the condition of being completely unable to understand or explain something. (किसी चीज को समझने या समझाने में पूरी तरह असमर्थ होने की स्थिति)
• For eg:- The man was left in bafflement when he witnessed the accident.
• Similarly, the word Confusion means the state of not being able to think clearly or not understanding something. (स्पष्ट रूप से सोचने या कुछ समझने में सक्षम न होने की अवस्था)

Hence the correct synonym to Bafflement is Confusion.

 Additional Information

• Pleasure: the feeling of being happy or satisfied. (आनंद)
• Clarity: the quality of being clear and easy to understand. (स्पष्टता)
• Cleanliness: being clean or keeping things clean. (स्वच्छता)

The correct answer is 'Condemn'.

Key Points

• The correct meaning of 'Praise' is to say somebody/something is good and should be admired.
• Condemn to say strongly that you think somebody/something is very bad or wrong.
• Hence the appropriate antonym for Praise is Condemn.​

Additional Information

• Celebrate: to mark a special day, event, or holiday.
• Hail: to proclaim that somebody/something is very good or very special
• Secure: free from worry or doubt.

The correct answer is 'among'.

Key Points

• We use 'among' to divide things among groups when the number of people is more than two.
• Whereas 'between' is used when divided among two.
• Between can also be used for times and places that indicate two end points.

Additional Information Let's look at the other option;

• At: when we refer to an address and
• We use as adjectives 'at' to talk about things that we do well or badly.
• For eg:- He is good at english.
• From: to refer to the place where someone or something starts or originates.
• For: to talk about a purpose or a reason for something.

The correct answer is 'Next to someone'.

Key Points

• The meaning of the idiom 'At one's elbow' is 'Next to someone'.
• The meaning of the idiom is at someone's side
• For eg:-His assistant is always at his elbow.

The correct answer is 'Challenge'.

Key Points

• Confront: To stand in front of somebody (किसी के सामने खड़ा होना)
• ​E.g.- The unarmed demonstrators were confronted by a row of soldiers.
• Challenge: Something new and difficult that forces you to make a lot of effort (चुनौती)
• ​E.g.- The company will have to face many challenges in the coming months.

Hence, the correct answer is 'Challenge'.

Additional Information

• Mingle: To mix with other things or people (अन्‍य वस्‍तुओं या व्‍यक्तियों में घुल-मिल जाना)
• ​E.g.- His excitement was mingled with fear.
• ​Conceal: To hide something/somebody [(किसी से कुछ) छिपाना]
• ​E.g.- She tried to conceal her anger from her friend.
• Scheme: An official plan or system for doing something (आधिकारिक योजना या प्रणाली)
• ​E.g.- A new scheme to provide houses in the area has been announced.

The correct answer is 'slept themselves peacefully'.

Key Points

• 'Themselves' is a reflexive pronoun.
• A reflexive pronoun is used with these words like absented, pride, acquitted, etc.
• Example: I pride myself on being his son.
• The verb 'slept' doesn't take a reflexive pronoun.
• Hence, the use of 'reflexive pronoun' is incorrect after 'slept'.

Therefore, the correct sentence is: After a long and fun-filled day, the children slept peacefully.

The correct answer is 'Laxuryous'.

Key Points

• 'Laxuryous' is the incorrect spelling of the word 'Luxurious'.
• Luxurious: Very comfortable and expensive (बहुत आरामदेह और मँहगी)
• ​E.g.- They have a very luxurious house.

Hence, the correct answer is 'Laxurious'.

Additional Information

• Innocent: Not having done wrong (जिसने ग़लत काम नहीं किया; निर्दोष)
• E.g.- An innocent man was arrested by mistake.
• Zealous: Using great energy and enthusiasm (ऊर्जावान या ऊर्जापूर्ण उत्‍साही, उत्‍साहपूर्ण)
• E.g.​- He is a zealous supporter of the government's policies.
• Judicial: Connected with a court of law or a legal judgment (न्‍यायाधीश या क़ानूनी निर्णय से संबंधित)
• E.g.- Several reflect critically on prosecution rates and judicial reactions.

The correct answer is 'Novice'.

Key Points

• Novice: A person who is new and without experience in a certain job (किसी विशेष काम के लिए नया और अनुभवहीन व्यक्ति)
• E.g.- I’m just a novice at making videos.

Hence, the correct answer is 'Novice'.

Additional Information

• Professional: Doing something in a way that shows skill, training or care (निपुणता, प्रशिक्षण या सावधानी के साथ कुछ करते हुए)
• E.g.- The police are trained to deal with every situation in a calm and professional manner.
• Expert : A person who has a lot of special knowledge or skill (विषय का विशेषज्ञ)
• ​E.g.- She’s a leading expert in the field of genetics.
• ​Ace: A person who is very skilled at something (वह व्यक्ति जो किसी चीज में बहुत कुशल हो)
• ​E.g.- They were the one who could prove to me that I wasn’t ace.

The correct answer is 'DACB'.

Key Points

• The given question is an example of a Para Jumble.
• We know that the first statement of a para jumbled question is usually an independent general statement, a noun, a universal fact, etc.
• Sentence D will be the first sentence because it is an introductory sentence and independent of all the other sentences.
• Sentence A will be the next sentence because 'credited with saying' is directly connected with the first sentence.
• Sentence C will be the next sentence because it tells the 'action' connect with the previous one.
• Lastly, Sentence B will be used.

Correct Sentence: Dr. APJ Abdul Kalam, former President of India is credited with saying If you want to leave your footprint on the sands of time, do not dry your feet.

The correct answer is 'Arguement'.

Key Points

• The correct spelling of 'Arguement' is 'Argument'.
• Argument - an angry discussion between two or more people who disagree with each other.
• Example - Kavya had an argument with her mother about politics.

Hence, the correct answer is option 1.

Additional Information 

Let's have a look at other given options:

• Performance - the act of performing something in front of an audience.
• Valuable - very useful, worth a lot of money.
• Secretary - an official of a club or society who is responsible for keeping records, writing letters, etc.

The correct answer is 'Robbing'.

Key Points

• The context tells us that the author claims that depriving kids of their youth is wrong.
• In this context, the word that must fit the blank should be a gerund.
• A gerund is a part of speech that acts as a noun, but has a 'V+ ing' form.
• For example: He said that smoking is bad for health. (Here, 'smoking' is a gerund)
• Similarly, in this context, 'smoking' must fit the blank as it is used after 'that' (which introduces a restrictive clause).

Therefore, the correct answer is 'The writer declares that robbing children of their childhood is a criminal act'.

Additional Information

• Let us understand the participles:
• Participles are made of the verb which works as an adjective. Participles qualify the noun and pronoun.
• Present Participle - V⁴ (V¹+ing)
• Past Participle - V³
• Perfect Participle - having + V³

The correct answer is 'Harmless'.

Key Points

 Let's understand the meaning of the given words:

• Detrimental - Harmful, tending to cause harm.(हानिकारक, नुक़सानदेह)
• Example: Moving her could have a detrimental effect on her health.
• Harmless - not able or not likely to cause damage or injury.(हानिरहित, नुक़सान न देनेवाला)
• Example: These insects are completely harmless.

After seeing the above explanation, we can say that 'Harmless' is the opposite word of 'Detrimental'.

Hence, the correct answer is option 2.

Additional Information

 Let's understand the meaning of other given options:

• Baseless - not based on facts or good reasons; unfounded(बेबुनियाद, निराधार)
• Senseless - having no meaning or purpose.(बेमतलब, अर्थहीन, निरर्थक)
• Meaningless - without meaning, reason or sense.(बेमतलब, अर्थहीन)

The correct answer is 'to destroy by an explosion'.

Key Points

• Blow up means to be destroyed and killed by a bomb.
• E.g. -The army used precision-guided munitions to blow up enemy targets.

Hence, the correct answer is 'to destroy by an explosion'.

The correct answer is 'Giant'.

Key Points

• Let us understand the meaning of Dwarf:
• Dwarf - a person, animal, or plant that is much smaller than the usual size.(बौना)
• Example - The skyscraper dwarfs all the other buildings around.
• Giant - an extremely large, strong person.(विशाल)
• Example - The giant pumpkin is also on its travels.

 After seeing the above explanation we can say that 'Giant' is the opposite of 'Dwarf'.

 Hence, the correct answer is 'option 4'.

Additional Information

• Let us understand the meanings of other given options:
• ​Measurable - That can be measured.(नापने योग्य)
• Medium - in between two sizes, lengths, temperature, etc.(मध्यम)
• Tiny - very small.(बहुत छोटा)

The correct answer is 'cannopies'.

Key Points

• The correct spelling would be 'canopies'.
• Canopy:  A cover that hangs or spreads above something (चँदवा, छत्र, वितान)
• ​E.g.- The highest branches in the rainforest form a dense canopy.

​Hence, the correct answer is 'option 1'.

Additional Information

• Tassels: A group of short threads or ropes held together at one end, used as a hanging decoration on hats, curtains, furniture, etc.
• Dampened: to make something slightly wet
• Trickle: To go somewhere slowly and gradually.

The correct answer is 'multiple, myriad, alternating, succession'.

Key Points

• Here, 'multiple' should be used in the first blank because after that a 'plural object' has been used.
• In the second blank, 'myriad' should be used because it means- A very large number of something.
• In the third blank, 'alternating' should be used.
• In the fourth blank, 'succession' should be used.

Hence, the correct answer is 'multiple, myriad, alternating, succession'.

The correct answer is 'Distinguishing'.

Key Points

• The word 'Distinguishing' works as the present participle here.
• The Present Participle of the verb is formed by adding the suffix '-ing' to the base form of the verb. These are used to form continuous tenses.
• Present Participle shows that action is incomplete.
• For Example - The telecom sector is the backbone of the digital India vision and it deserves an enabling environment.

Therefore, the correct answer is 'option 3'.

The correct answer is 'put'.

Key Points

• Put: To move somebody/something into a particular place or position (किसी को किसी स्‍थान या स्थिति में रखना या डालना)
• E.g.​- She put the book on the table.

Therefore, according to the meaning of the word, 'put', the correct answer is 'put'.

The correct answer is 'overcome'.

Key Points

• Overcome: To manage to control or defeat somebody/something (व्‍यक्ति या वस्‍तु को पराजित या नियंत्रित कर लेना)
• ​E.g.- She tried hard to overcome her fear of flying.

Therefore, according to the meaning of the word, 'overcome', the correct answer is 'overcome'.

The correct answer is-' Arrive'.

Key Points

• Arrive means to come or happen

E.g. -The day of the wedding had finally arrived.

Hence, the correct answer is 'Arrive' according to the context of the given passage.

Additional Information 

• Awake means to wake up; to make somebody/something wake up.
• Arise means to begin to exist; to appear.
• Abase means to behave in a way that belittles or degrades (someone).

The correct answer is 'accordingly'.

Key Points

• After reading the passage we get an idea that the passage is based on competition and how a student tackles it.
• The suitable word which best suits is 'accordingly'.
• After adding the word in sentence Students must be aware of the tough environment they are entering and prepare accordingly in order to improve their chances.

​ 

Additional Information

Let's look at the other words,

• Reluctantly: not willing to do something.
• Consequently: as a result.
• Exponentially: more and more rapidly.

The logic followed here is: 

Here, two series are going simultaneously, 

One is, 15, 12, 9 and other is 27, ?, 33

In the first series, 3 is subtracted from the number to get the next number of the series.

15 - 3 = 12,  12 - 3 = 9

And in the second series, 3 is added to the number to get the next number of the series.

27 + 3 = 30,  30 + 3 = 33

Hence, '30' will replace the question mark in the above series.

The logic followed here is: 

Addition of the square of the odd number to previous number to get next number in the series.

13 + 1² =13 + 1 = 14

14 + 3²= 14 + 9 = 23

23 + 5² = 23 + 25 = 48 

48 + 7² = 48 + 49 = 97 

97 + 9² = 97 + 81 = 178

178 + 11² = 178 + 121 = 299 

Hence, the correct option is 299

The logic followed here is:

Hence, the correct answer is option (2)

The logic followed here is:

The middle number comes by adding the cube of first number and the square of the third number.

The pattern as follows

3³ + 4² = 27 + 16 = 43

6³ + 5²  = 216 + 25 = 241

Similarly

2³  + 11² = 8 + 121 = 129

Hence, '?' will be replaced by '129'

The figure is given below:

There are 20 triangles.

Hence, 'option 3' is the correct answer.

The logic followed here is:

Logic : (First number × 5) - 11 = Second number

• 16 : 69

⇒ (16 × 5) - 11

⇒ 80 - 11 = 69

• 31 : 144

⇒ (31 × 5) - 11 

⇒ 155 - 11 = 144

Similarly, 24 : ?

⇒ (24 × 5) - 11

⇒ 120 - 11 = 109

Hence, '109' is the correct answer.

The pattern followed here is:

Hence, ‘QWMN’ is the correct answer.

The logic followed here is: 

Let us go through the options: 

option 1) 

 

Here 32 x 32 is  equal to 1024.

option 2) 

Here 28 x 28 is  equal to 784. 

Option 3)

 

Here 36 x 36 is not equal to 1620.It is 1296.

option 4) 

Here 24 x 24 is equal to 576.

Hence, "18:1620" is the correct answer.

The logic followed here is:

The BODMAS rule :

Decoding the information, 

The given equation is -

(13 B 9) D 3 A (14 D 7) B 6 C 21 A (32 B 2) 

A denotes ‘+’, B denotes ‘×’, C denotes ‘−’ and D denotes ‘÷’

(13 × 9) ÷ 3 + (14 ÷ 7) × 6 - 21 + (32 × 2) 

= (13 × 9) ÷ 3 + (14 ÷ 7) × 6 - 21 + (32 × 2)

= 117 ÷ 3 + 2 × 6 - 21 + 64 

= 39 + 2 × 6 - 21 + 64 

= 39 + 12 - 21 + 64 

= 51 - 21 + 64

= 115 - 21

= 94

Hence, 94 is the correct answer.

The pattern followed here is as follows:

Hence, value of '?' is 140.

The logic followed here is:

Hence the correct answer is IFMTRE

The positional values of the English alphabet is as follows:

The pattern followed here is:

'ADVISORY' is written as 'BFYMRMOU'

'CHAMPION' is written as 'DJDQOGLJ'

Similarly, 'DESIGNER' will be written as:

Hence, 'EGVMFLBN' is the correct answer.

The pattern followed here is:

Logic : First number × (- 8) = Second number

• 24 : -192

⇒ 24 × (- 8) = -192

• -31 : 248

⇒ -31 × (- 8) = 248

Similarly, -18 : ?

⇒ -18 × (- 8) = 144

Hence, '144' is the correct answer.

The correct mirror image of the given figure when the mirror is placed at XY:

Hence, "option 1" is the correct answer.

The positional values of English alphabet is as given below:

The pattern followed here is as follows:

Hence, "ONM" is the correct answer.

The correct mirror image of the given figure when the mirror is placed at MN:

Hence, 

is the correct answer.

The positional values of English alphabet is as follows:

The followed here is

'KITCHEN' is written as 'LHUBIDO'.

'HUSBAND' is written as 'ITTABME'.

Similarly, 'ELEMENT' is written as:

Hence, "FKFLFMU" is the correct answer.

Option 4 is the correct answer.

The positional values of English alphabet is as follows:

“FAMOUS” is written as “AFOMSU”.

“FINGER” is written as “IFGNRE".

Similarly, “INVEST” is written as:

Hence,"NIEVTS" is the correct answer.

The least possible Venn diagram is as follows:

Conclusions:

I. Some jams are honeys → True (Since all breads are jams and some honeys are breads)

II. All honeys are butters → False (It is possible but not definite)

III. Some jams are butters → False (It is possible but not definite)

Hence, only conclusion 1 follows.

Given : 21^st June 2007 was a Thursday

• Total number of days in a normal year = 365 days
• Number of odd days in a normal year = 365/7 = 1 day
• Total number of days in a leap year = 366 days
• Number of odd days in a leap year = 366/7 = 2 days
• 21^st June 2011 = Thursday + 2 (2008) + 1(2009) + 1(2010) + 1(2011) = Thursday + 5 odd days =  Tuesday

Hence, 21^st June 2011 held on Tuesday.

The correct mirror image of the given figure when the mirror is placed at MN:

Hence, 

is the correct answer.

The correct order of the given words as they would appear in an English dictionary is as follows: 

4. Abduction
5. Addiction
6. Adequate
7. Adoption
8. Advertisement

Hence, "4,3,1,2,5" is the correct answer.

The logic followed here is:

• It is an ordinary dice as the sum of right and left side is 7.
• Since 4 and 6 are common in both dices, so opposite number of 3 is 1.

Hence, 1 is opposite to the number 3.

Logic : (First number + second number + 1) × 4  = Third number 

The pattern followed here is as follows:

• First row- 9, 21, 124

⇒ (9 + 21 + 1) × 4 

⇒ 31 × 4 = 124 

• Second row- 11, 25, 148

⇒ (11 + 25 +1) × 4 

⇒ 37 × 4 = 148

Similarly, third row- 17, ?, 220

⇒ (17 + ? + 1) × 4 = 220

⇒ (18 + ?) × 4 = 220

⇒ (18 + ?) = 220 ÷ 4 

⇒ (18 + ?) = 55

⇒ ? = 55 - 18 = 37

Hence, the value of '?' is 37.

The logic followed here is:

Logic : Difference between first and second number is 29.

1) 361 : 332 ⇒ 361 - 332 = 29

2) 533 : 504 ⇒ 533 - 504 = 29

3) 440 : 411 ⇒ 440 - 411 = 29

4) 520 : 481 ⇒ 520 - 481 = 39 ≠ 29

Hence, "520 : 481" is the odd one out.

GIVEN:

 $(4^3+4)$ ÷ [$5^2 - (7^2 -41)$ )]

FORMULA USED : 

CALCULATION :

⇒ $(4^3 +4 )$ ÷ [$5^2 - (7^2 -41)$)]

⇒ (64 + 4) ÷ [ 25 -  8]

⇒ 68 ÷ 17

⇒ 4

∴ The value is 4.

Shortcut Trick

For three numbers a, b, and c, the fourth proportional (x) is given by x = (b × c) ÷ a.

Given numbers: 5, 6, and 8.

⇒ Fourth proportional = (6 × 8) ÷ 5

⇒ 48 ÷ 5 = 9.6

∴ The correct answer is 9.6.

Alternate Method

Given:

First number (a) = 5

Second number (b) = 6

Third number (c) = 8

Formula Used:

If a, b, c, and d are in proportion, then a : b = c : d or a/b = c/d, where d is the fourth proportional.

Calculations:

Let the fourth proportional be x.

⇒ 5 : 6 = 8 : x

⇒ 5/6 = 8/x

⇒ 5 × x = 6 × 8

⇒ 5x = 48

⇒ x = 48 ÷ 5

⇒ x = 9.6

∴ The correct answer is 9.6.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is √(a × b).

Third Proportional

The third proportional to two numbers a and b is c such that a : b = b : c, which means c = b² ÷ a.

Continued Proportion

Three numbers a, b, and c are in continued proportion if a/b = b/c.

Shortcut Trick

Convert the given volume directly into cubic inches by multiplying by 12³.

Volume in inches = 22.5 × 1728 = 38880 inch³

Since Volume = Length × Width × Height

Height = 38880 ÷ (48 × 27)

Height = 38880 ÷ 1296 = 30 inches

∴ The correct answer is 30 inches.

Alternate Method

Given:

Length (L) = 48 inches

Width (B) = 27 inches

Volume (V) = 22.5 ft³

1 ft = 12 inches

Formula Used:

Volume of a cuboid = L × B × H

Calculations:

⇒ Volume in inch³ = 22.5 × 12 × 12 × 12

⇒ Volume = 22.5 × 1728 = 38880 inch³

⇒ L × B × H = 38880

⇒ 48 × 27 × H = 38880

⇒ 1296 × H = 38880

⇒ H = 38880 ÷ 1296

⇒ H = 30 inches

∴ The correct answer is 30 inches.

Additional Information

Volume of a Cuboid

V = L × B × H, where L, B, and H denote length, breadth, and height respectively.

Surface Area of a Cuboid

Total Surface Area (TSA) = 2(LB + BH + HL). Lateral Surface Area (LSA) = 2H(L + B).

Diagonal of a Cuboid

The length of the longest space diagonal inside a cuboid is √(L² + B² + H²).

Unit Conversions

Since 1 foot = 12 inches, calculating volume requires taking the cube of 12. Thus, 1 cubic foot = 12³ = 1728 cubic inches.

GIVEN:

The amount of wheat is 5 kg and the price is Rs18/kg

The amount of wheat is 2kg

The amount of wheat is 7 kg and the price is Rs20/kg

FORMULA USED:

Quantity in kg × amount per kg = Price in Rs.

CALCULATION :

Let the price of 2 kg wheat be Rs. y/kg then,

5 × 18 + 2 × y  = 7 × 20 

⇒ 90 + 2y = 140 

⇒ 2y = 50 

⇒ y = 25

∴ The price of costlier wheat is Rs25.

Shortcut Trick

We know the complementary angle property: tan θ = cot(90° − θ).

⇒ tan 15° = cot(90° − 15°) = cot 75° = 2 − √3

We also know that cot θ = 1 ÷ tan θ.

⇒ cot 15° = 1 ÷ (2 − √3)

Rationalizing the denominator by multiplying with its conjugate (2 + √3):

⇒ cot 15° = (2 + √3) ÷ (2² − (√3)²) = (2 + √3) ÷ (4 − 3) = 2 + √3

∴ The correct answer is 2 + √3.

Alternate Method

Given:

cot 75° = 2 − √3

Formula Used:

cot(A − B) = (cot A × cot B + 1) ÷ (cot B − cot A)

Calculation:

We can write 15° as (45° − 30°).

⇒ cot 15° = cot(45° − 30°)

⇒ cot 15° = (cot 45° × cot 30° + 1) ÷ (cot 30° − cot 45°)

Substitute the standard trigonometric values (cot 45° = 1 and cot 30° = √3):

⇒ cot 15° = (1 × √3 + 1) ÷ (√3 − 1)

⇒ cot 15° = (√3 + 1) ÷ (√3 − 1)

To rationalize, multiply both the numerator and the denominator by (√3 + 1):

⇒ cot 15° = [(√3 + 1) × (√3 + 1)] ÷ [(√3 − 1) × (√3 + 1)]

⇒ cot 15° = (3 + 1 + 2√3) ÷ (3 − 1)

⇒ cot 15° = (4 + 2√3) ÷ 2

⇒ cot 15° = 2 × (2 + √3) ÷ 2

⇒ cot 15° = 2 + √3

∴ The correct answer is 2 + √3.

Additional Information

Trigonometric Ratios of Complementary Angles

Two angles are complementary if their sum is exactly 90°. Key relations include: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, and sec(90° − θ) = cosec θ.

Important Values for 15° and 75°

sin 15° = cos 75° = (√6 − √2) ÷ 4

cos 15° = sin 75° = (√6 + √2) ÷ 4

tan 15° = cot 75° = 2 − √3

cot 15° = tan 75° = 2 + √3

Rationalization Rule

When a denominator contains a surd of the form (a − √b), it can be eliminated by multiplying both the numerator and the denominator by its conjugate pair (a + √b) based on the algebraic identity (x − y)(x + y) = x² − y².

Shortcut Trick

Reduction percentage = 5%

Reduction amount = ₹20

5% → ₹20

1% → ₹4

100% → ₹400

∴ The correct answer is ₹400.

Alternate Method

Given:

Reduction rate = 5%

Reduction amount = ₹20

Formula Used:

Total Amount = (Reduction Amount ÷ Reduction Percentage) × 100

Calculations:

⇒ Let the total electricity bill be x.

⇒ 5% of x = 20

⇒ (5 ÷ 100) × x = 20

⇒ x = (20 × 100) ÷ 5

⇒ x = 4 × 100

⇒ x = 400

∴ The correct answer is ₹400.

Additional Information

Percentage Change

To find the original value when a percentage part is known: Original Value = (Part Value ÷ Percentage) × 100.

Successive Percentage

If a value is changed by a% and then b%, the net change is [a + b + (a × b)/100]%.

Shortcut Trick

We know the direct ratio formula between Cost Price (CP) and Printed Price (MP):

CP : MP = (100 − Discount%) : (100 + Profit%)

⇒ CP : MP = (100 − 10) : (100 + 12)

⇒ CP : MP = 90 : 112

⇒ CP : MP = 45 : 56

∴ The correct answer is 45 : 56.

Alternate Method

Given:

Profit = 12%

Discount = 10%

Formula Used:

Selling Price (SP) = CP × (100 + Profit%) ÷ 100

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

Calculations:

Let the Cost Price (CP) be 100.

⇒ SP = 100 × (100 + 12) ÷ 100 = 112

Since the discount given is 10% on the Printed Price (MP),

⇒ SP = MP × (100 − 10) ÷ 100

⇒ 112 = MP × 90 ÷ 100

⇒ MP = 112 × 100 ÷ 90 = 1120 ÷ 9

Now, we find the ratio of CP to MP:

⇒ CP : MP = 100 : (1120 ÷ 9)

⇒ CP : MP = 900 : 1120

⇒ CP : MP = 45 : 56

∴ The correct answer is 45 : 56.

Additional Information

Relation between CP and MP

When Profit percentage (P%) and Discount percentage (D%) are given, the direct ratio is CP : MP = (100 − D) : (100 + P).

Marked Price & Discount

Discount is always calculated on the Printed or Marked Price. Selling Price (SP) = MP − Discount.

Profit & Loss

Profit and Loss are always calculated on the Cost Price (unless specifically stated). Profit = SP − CP.

Shortcut Trick

When the time taken by both moving objects is constant, the ratio of the distance covered is equal to the ratio of their speeds.

Speed ratio (Thief : Policeman) = 11 : 13

Distance ratio (Thief : Policeman) = 11 : 13

Difference in distance covered = 13 parts − 11 parts = 2 parts

Given the initial distance difference is 225 m ⇒ 2 parts = 225 m ⇒ 1 part = 112.5 m

Distance run by the thief = 11 parts = 11 × 112.5 = 1237.5 metres

∴ The correct answer is 1237.5 metres.

Alternate Method

Given:

Initial distance between policeman and thief = 225 m

Speed of thief = 11 km/h

Speed of policeman = 13 km/h

Formula Used:

Relative speed (same direction) = Speed₁ − Speed₂

Time = Distance ÷ Relative Speed

Distance = Speed × Time

Calculations:

⇒ Relative speed = 13 − 11 = 2 km/h

⇒ Relative speed in m/s = 2 × (5/18) = 5/9 m/s

⇒ Time taken to catch the thief = Distance ÷ Relative speed = 225 ÷ (5/9)

⇒ Time taken = 225 × (9/5) = 45 × 9 = 405 seconds

⇒ Speed of thief in m/s = 11 × (5/18) = 55/18 m/s

⇒ Distance covered by thief before getting caught = Speed of thief × Time taken

⇒ Distance = (55/18) × 405 = 55 × 22.5

⇒ Distance = 1237.5 m

∴ The correct answer is 1237.5 metres.

Additional Information

Relative Speed (Same Direction)

When two bodies are moving in the same direction, their relative speed is the difference between their individual speeds: S = S₁ − S₂.

Relative Speed (Opposite Direction)

When two bodies are moving in opposite directions towards each other, their relative speed is the sum of their individual speeds: S = S₁ + S₂.

Unit Conversion

To convert speed from km/h to m/s, multiply the value by 5/18. To convert speed from m/s to km/h, multiply the value by 18/5.

Shortcut Trick

Check the sum of digits for each number; if the sum is divisible by 3, the number is also divisible by 3.

Sum of digits of 7345932 = 7+3+4+5+9+3+2 = 33.

33 is divisible by 3 (33 ÷ 3 = 11).

∴ The correct option is A.

Alternate Method

Given: Numbers in options A, B, C, and D.

Formula: Divisibility rule of 3 states that a number is divisible by 3 if the sum of its digits is a multiple of 3.

⇒ Option A: 7345932 → Sum = 7+3+4+5+9+3+2 = 33

⇒ 33 ÷ 3 = 11 (Divisible by 3)

⇒ Option B: 5439763 → Sum = 5+4+3+9+7+6+3 = 37 (Not divisible by 3)

⇒ Option C: 3642589 → Sum = 3+6+4+2+5+8+9 = 37 (Not divisible by 3)

⇒ Option D: 3262735 → Sum = 3+2+6+2+7+3+5 = 28 (Not divisible by 3)

∴ The correct option is A.

Additional Information

Divisibility by 9

A number is divisible by 9 if the sum of its digits is divisible by 9.

Divisibility by 6

A number is divisible by 6 if it is divisible by both 2 (the number is even) and 3.

Divisibility by 4

A number is divisible by 4 if its last two digits are divisible by 4.

Divisibility by 11

A number is divisible by 11 if the difference between the sum of digits at odd positions and even positions is 0 or a multiple of 11.

GIVEN:

The sum amounts to Rs10,890 in 2 years

The sum amounts to Rs11,979 in 3 years

FORMULA USED:

Amount = $P (1 +\frac{R}{100})^t$ 

where P = Principal, R = Rate of interest, and t = time.

CALCULATION :

10,890 = P× $( 1+\frac{R}{100})^2$  ...............(1)

11,979 = P × $(1 + \frac{R}{100})^3$ ................(2)

On dividing Eq (2) by Eq(1) we get,

⇒ 1.1 = $( 1 + \frac{R}{100})$ 

⇒ 0.1 = R/100  = 10

⇒ R = 10 p.a. 

Now put the value of R in eq1 we get,

⇒ 10,890 = P× $(1.1)^2$ 

⇒ P = 9000 .

⇒ The sum is Rs9000.

GIVEN:

Three companies namely P,Q and R producing electric wires over the years ( 2015 - 2019 )  respectively

FORMULA USED:

Average =  $( sum of all the values )\over ( number of all the values )$

CALCULATION : 

Company P producing electric wires in 2017 =  25 

⇒ P in 2018 = 50 

⇒ P in 2019 = 40 

Company Q producing electric wires in 2017 = 35

⇒ Q in 2018 =  40 

⇒ Q in 2019 = 50 

Now, 

⇒ sum of all the production in (2017-2019) done by P = ( 25 + 50 + 40 ) = 115 

⇒average production of company P = 115 ÷ 3 = 115/3.

⇒sum of all the production in ( 2017 -2019 ) done by Q = ( 35 + 40 + 50 ) = 125

⇒average production of company Q = 125 ÷ 3 = 125/3.

⇒ Ratio of average production  done by P and Q = 115/3 : 125/3 = 115 :125 = 23 : 25

∴ Ratio of average production done by P and Q is 23 : 25.

GIVEN:

Different  coloured dresses are worn by 60 students, which is represented by a pie graph 

CONCEPT  USED :

100 % =  Total number of students who wore different colour dresses for a college part 

100 % = 60

CALCULATION :

The sector of students who wore yellow colour dresses

= 10%

⇒ Now, 

⇒ 100% = 60

⇒ 10 % = 6.

∴ The number of students wearing yellow colour dresses = 6.

GIVEN:

39 students having average weight = 51 kg

39 students and teachers combined average weight = 51.2 kg 

FORMULA USED :

Average = $( sum of all the values)\over( number of all values )$

CALCULATION :

Average of students = $\frac{sum}{no.of students}$ =  51 

⇒ sum/39 = 51 

⇒ sum  of weight of  students = ( 39 × 51) = 1989.

⇒Average of students and teachers including  = $sum \over( no. of students and teachers)$ = 51.2

⇒ sum/40 = 51.2 

⇒ sum of the weight of students and teachers = 2048. 

weight of teacher = (sum of the weight of teacher and students - the sum of the weight of students )

⇒ ( 2048 - 1989) = 59 kg.

∴ The weight of the teacher is 59 kg.

Shortcut TrickAverage weight increases by .2 kg and .2 kg are increased in every 39 students.

so the weight of the teacher = .2× 39 + 51.2 = 59kg

GIVEN:

Amount at the end of 2 years = Rs 3,364

Rate = 16%.

FORMULA USED:

A =  P × $( 1+ \frac{R}{100})^t$ where, A = Amount, P = Principal,

R = Rate of interest and T= Time. 

CALCULATION:

A = P × $(1 +\frac{R}{100})^t$

⇒ 3,364 = P × $( 1+ \frac{16}{100})^2$

⇒ 3,364 = P × $29\over25$× $29\over25$ 

⇒ P = Rs 2500.

∴ The sum is Rs.2500.

Shortcut Trick R = 16% = 4/25

1^st year         2^nd year

25                 29

25                 29

29 × 29→ 3364,1→ 4

Sum→ 4 × 25 × 25 = 2500.

GIVEN:

The surface area of a sphere = 1386 $cm^2$ 

FORMULA USED:

The surface area of a sphere = 4πr²where r is the radius of the sphere.

CALCULATION:

The surface area of a sphere =4πr2 = 1386 

⇒  4 × (22/7) × r² = 1386      ....(value of  $\pi$ is $\frac{22}{7}$)

⇒ r2 = 110.25 

⇒ r2 = $\frac{11025}{100}$  

⇒ r = $\sqrt\frac{11025}{100}$ = $\frac{105}{10}$ = 10.5 cm.

∴ The radius of the sphere is 10.5 cm.

Shortcut Trick

Original Fraction = (Resulting Fraction) × (100 − %Decrease in Denom) / (100 + %Increase in Num)

Original Fraction = (25/36) × (72 / 150)

Simplify: (25/150) × (72/36) = (1/6) × 2

Original Fraction = 2/6 = 1/3

∴ The correct option is D.

Alternate Method

Given:

Increase in Numerator = 50%

Decrease in Denominator = 28%

Resulting Fraction = 25/36

Original x / y

Num: +50% Den: −28%

New Fraction 25 / 36

x × 150% / y × 72% = 25 / 36

Calculation:

Let the original fraction be x/y.

⇒ New Numerator = x + 50% of x = 1.5x

⇒ New Denominator = y − 28% of y = 0.72y

⇒ According to the question: 1.5x / 0.72y = 25/36

⇒ (x/y) × (150/72) = 25/36

⇒ x/y = (25/36) × (72/150)

⇒ x/y = (25/150) × (72/36)

⇒ x/y = (1/6) × 2

⇒ x/y = 2/6 = 1/3

∴ The correct option is D.

Additional Information

Percentage to Fraction Conversion

To increase a value by a%, multiply it by (100+a)/100. To decrease by b%, multiply by (100−b)/100.

Reciprocal Property

When moving a fraction from one side of an equation to another, it is multiplied by its reciprocal: (a/b) × x = c → x = c × (b/a).

Base 100 Method

Assuming the original parts as 100x and 100y simplifies the percentage calculation to 150x and 72y respectively.

Shortcut Trick Formula:

(a+b)2 - 4ab = (a-b)2

Calculation:

(957 + 932)2 - 4 × 957 × 932

= (957 - 932)2

= 252 = 625

Alternate Method GIVEN:

$( 957 + 932 )^2$ - 4 × 957 × 932.

FORMULA USED:

BODMAS 

CALCULATION :

⇒ $(1889)^2$ - 4 × 957 × 932 

⇒ 3568321 - 4 × 957 × 932 

⇒ 3568321 - 3567696  =  625 

∴ The value is 625.

GIVEN:

The surface area of a sphere = 64πcm2  

FORMULA USED:

The surface area of a sphere = 4πr²    

The volume of a sphere = $\frac{4\pi r^3}{3}$ 

CALCULATION:

The surface area of a sphere = 64π

⇒ 4πr2 = 64π

⇒ r2 = 16 

⇒ r = 4cm

Now, volume = 4/3 $\pi$(r^3\)   = 4/3 ×π × 4 × 4 × 4 = $256 \pi\over3$ $cm^3$. 

∴ The volume of the sphere is $256 \pi\over3$ $cm^3$.

Shortcut Trick

Initial profit = 20% of CP = 20% of ₹560 = ₹112.

If no discount is given, the selling price equals the marked price.

New profit = Initial profit + Discount given = 112 + 28 = ₹140.

New Gain % = (New Profit ÷ CP) × 100% = (140 ÷ 560) × 100% = 25%.

∴ The correct answer is 25%.

Alternate Method

Given:

Cost Price (CP) = ₹560

Discount = ₹28

Gain % after discount = 20%

Formula Used:

Selling Price (SP) = CP × (100 + Gain%) ÷ 100

Marked Price (MP) = SP + Discount

Gain % = [(New SP − CP) ÷ CP] × 100

CP: ₹560 SP: ₹672 MP: ₹700

Profit = ₹112 (20%) Discount = ₹28 New Profit = ₹140 (25%)

Calculation:

⇒ SP = 560 × (100 + 20) ÷ 100

⇒ SP = 560 × 1.2 = ₹672

⇒ MP = 672 + 28 = ₹700

If the article is sold at the marked price, the new selling price is ₹700.

⇒ New Gain = 700 − 560 = ₹140

⇒ New Gain % = (140 ÷ 560) × 100 = 25%

∴ The correct answer is 25%.

Additional Information

Marked Price (MP)

It is the price printed on an article. Discount is always calculated on the Marked Price, not on the Cost Price.

Successive Discounts

If multiple discounts d₁% and d₂% are given, the equivalent single discount % = d₁ + d₂ − (d₁ × d₂) ÷ 100.

Relation between CP and MP

CP × (100 + Profit%) = MP × (100 − Discount%). This is useful for directly finding CP or MP without calculating the Selling Price.

GIVEN:

x + $\frac{1}{x}$ = -2√ 3 

FORMULA USED:

 (x + $\frac{1}{x}$)² = x2 + $\frac{1}{x^2}$ + 2 × x × $\frac{1}{x}$ = x2 + $\frac{1}{x^2}$ + 2

 (x + $\frac{1}{x}$)³ = x³ + $\frac{1}{x^3}$ + 3 × x × $\frac{1}{x}$ × (x + $\frac{1}{x}$)  = $x^3 + \frac{1}{x^3}$  + 3 × (- 2√3 )

CALCULATION:

(- 2√3 )² = x² + $\frac{1}{x^2}$  + 2

⇒ 10 = x2 + $\frac{1}{x^2}$       ----(1)

Now, (- 2√3)³ = x³ + $\frac{1}{x^3}$ - 6√3

⇒ - 24√3 + 6√3 = x3 + $\frac{1}{x^3}$

⇒ - 18√3 = x3 + $\frac{1}{x^3}$      ----(2)

Now, 

⇒ (x⁵ + $\frac{1}{x^5}$) = (x2 + $\frac{1}{x^2}$)( x3 + $\frac{1}{x^3}$  ) - (x + $\frac{1}{x}$ ) 

⇒ (x5 + $\frac{1}{x^5}$) = 10 × -18√3 - (-2√3) = - 178√3.

∴ The value of  (x5 + $\frac{1}{x^5}$) is -178√3.

Shortcut Trick If x + $\frac{1}{x}$ = p, then x⁵ + $\frac{1}{x^5}$ = (p² - 2)(p³ - 3p) - p

 If x + $\frac{1}{x}$ = -2√3, then x⁵ + $\frac{1}{x^5}$ = {(-2√3)² - 2}{(-2√3)³ -

3(-2√3)} - (-2√3) = (12 - 2)(-24√3 + 6√3) - (-2√3) =

10 × -18√3 + 2√3 = -180√3 + 2√3 = -178√3.

Shortcut Trick

Efficiency of Avi = 3, Bindu = 1 (Total Work = 12 units).

Work done in one 2-hour alternate cycle = 3 + 1 = 4 units.

Total time for 12 units = (12 ÷ 4) × 2 = 6 hours.

Completion time = 5 a.m. + 6 hours = 11 a.m.

∴ The correct answer is 11 a.m.

Alternate Method

Given: Avi completes in 4 hr, Bindu completes in 12 hr, start time is 5 a.m.

⇒ In the 1^st hour (5 a.m. − 6 a.m.), Avi works = 3 units.

⇒ In the 2^nd hour (6 a.m. − 7 a.m.), Bindu works = 1 unit.

⇒ Total work in 1 cycle (2 hours) = 3 + 1 = 4 units.

⇒ Number of cycles to complete 12 units = 12 ÷ 4 = 3 cycles.

⇒ Total duration = 3 × 2 = 6 hours.

⇒ Project completion time = 5 a.m. + 6 hours = 11 a.m.

∴ The correct answer is 11 a.m.

Additional Information

Alternate Work Principle

When people work alternately, we calculate the total work done in one full cycle. If work remains after full cycles, the person who started the project works next.

Efficiency and Time Relationship

Efficiency is inversely proportional to time. Total Work = Efficiency × Time. Using LCM of time as Total Work simplifies calculations.

Work and Wages

In alternate work scenarios, wages are always distributed in the ratio of the actual work done by each person, not just their efficiency.

Shortcut Trick For two circles touching externally, the length of the direct common tangent is: MN = 2 √(R × r) ⇒ MN = 2 √(12 × 8) = 2 √96 ⇒ MN = 2 × 4√6 = 8√6 cm ∴ The correct answer is 8√ 6 cm.

Alternate Method Given: Radius of the first circle (R) = 12 cm Radius of the second circle (r) = 8 cm The circles touch each other externally. Formula Used: Length of direct common tangent (MN) = √[d2 − (R − r)2] Where d is the distance between the centers of the circles.

Calculations: ⇒ Since the circles touch externally, the distance between centers (d) = R + r ⇒ d = 12 + 8 = 20 cm ⇒ MN = √[202 − (12 − 8)2] ⇒ MN = √[400 − 42] ⇒ MN = √[400 − 16] = √384 ⇒ MN = √(64 × 6) = 8√6 cm ∴ The correct answer is 8√ 6 cm.

Additional Information Direct Common Tangent (DCT) The length of the direct common tangent is given by √[d2 − (R − r)2], where d is the distance between centers. Transverse Common Tangent (TCT) The length of the transverse common tangent is given by √[d2 − (R + r)2]. It exists only if d > R + r. Touching Externally vs Internally For externally touching circles, d = R + r. For internally touching circles, d = |R − r|.

Given:

a + 2b = 27

a³ + 8b³ = 5427

Formula:

(A + B)³ = A³ + B³ + 3AB(A + B)

Calculation:

By using the above formula,

(a + 2b)3 = a3 + 8b3 + 3.a(2b)(a + 2b)

⇒ (27)³ = 5427 + 3 × 2ab(27) 

⇒ 19683 = 5427 + 81 × 2ab

⇒ 81 × 2ab = 14256

⇒ 2ab = 176

∴ The value of 2ab = 176.

Given:

Sales of an electronic chip over 5 years.

Concept used:

Average = $\frac{Sum\space of \space Observations}{No. \space of\space observations}$

Calculations:

Average of the sales over five years

⇒ $\frac{45+54+57+60+69}{5}$

⇒ 285/5

⇒ 57

Average of sales over five years = sales in year 2017

∴ The year in which sales is equal to average sales is 2017.

Given:

80% of the eligible voters cast their votes,

In which 5% were invalid and

10545 votes are 75% of total valid votes

Calculations:

Let Total eligible voters = z

Total casted votes = 80% of total eligible voters

Total casted votes = z × $\frac{80}{100}$

Total valid votes = 95% of total casted votes

Total valid votes = $\frac{95}{100}$ × $\frac{80}{100}$ × z

According to question,

10545 = 75% of total valid votes

⇒ 10545 = 75% of ( $\frac{95}{100}$ × $\frac{80}{100}$ × z)

⇒ 10545 = ($\frac{75}{100}$) × ($\frac{95}{100}$) × ($\frac{80}{100}$) × z

⇒ 10545 × 1000000 = 570,000 × z

z = Eligible voters = 18500

∴ No. of eligible voters is 18,500.

Shortcut Trick

Total votes = z

z ×  $\frac{80}{100}$ × $\frac{95}{100}$ × $\frac{75}{100}$ = 10545

z = $\frac{10545 × 100\times 100\times 100}{570,000}$ = 18500

∴ No. of eligible voters is 18,500.

The correct answer is 2003.

Key Points

• National Biodiversity Authority
• After signing the Convention on Biological Diversity (CBD) in 1992, India formed the National Biodiversity Authority (NBA), a statutory autonomous organisation under the Ministry of Environment, Forests, and Climate Change, in 2003.
• It was made to carry out the requirements of the Biological Diversity Act, 2002.
• It serves as facilitation, regulation, and advisory body to the Indian government "on issues of conservation, sustainable use of biological resources, and fair and equitable distribution of benefits emerging from the use of biological resources."
• Its main office is in Chennai, India.
• Additionally, it provides guidance to State Governments in designating biodiversity hotspots as heritage monuments.
• NBA held the first-ever National Biodiversity Congress (NBC) in Thiruvananthapuram, Kerala, in 2012.

Additional Information

• Since its founding, NBA has supported the development of State Biodiversity Boards (SBBs) in 29 States and aided the construction of approximately 1,39,831 Biodiversity Management Committees (BMCs).
• The National Biodiversity Authority is charged with controlling access to biological resources and/or related knowledge for research, bio-survey, and commercial use, as well as obtaining intellectual property rights, transferring research findings, and transferring accessed biological resources.
• On the website of the National Biodiversity Authority there are the details of the application forms for Access and Benefit Sharing (ABS) of particular activities.
• The current chairperson of NBA is Dr. V. B. Mathur.

The correct answer is Avani Lekhara.

Key Points

• Avani Lekhara
• Avani Lekhara, an Indian Paralympian and rifle shooter, was born on November 8, 2001.
• At the Tokyo 2020 Paralympics, she earned a Gold Medal in the 10m air rifle standing event and a Bronze Medal in the 50m rifle three positions event.
• In 2022, Avani Lekhara receives the Padma Shri award from former President Ram Nath Kovind.
• She is the first Indian woman to win two medals in the same Paralympics and the country's first gold medalist.
• At the Sportstar Aces Awards in Mumbai, Abhinav Bindra presents para-shooter Avani Lekhara with the Sportswoman of the Year (Parasports) award.
• She won Major Dhyan Chand Khel Ratna Award 2021 in Para Shooting discipline.

Additional Information

• Major Dhyan Chand Khel Ratna Award 
• The highest athletic honour in India in Sports and Games is the Khel Ratna Award, also known as the Major Dhyan Chand Khel Ratna Award and formerly the Rajiv Gandhi Khel Ratna Award.
• The Ministry of Youth Affairs and Sports distributes it every year.
• Chess Grandmaster Viswanathan Anand was the first recipient for this award, recognised for his efforts in the academic year 1991–1992.

The correct answer is Satish Vyas.

Key Points

• Satish Vyas
• India's Satish Vyas plays the santoor.
• His father is an Indian classical singer named C. R. Vyas.
• He's been a fantastic musician for more than 30 years.
• He received the esteemed Tansen Samman award in December 2020.
• Around the globe, he has given performances at well-known venues.
• At the Aga Khan Museum in Toronto, Canada, he gave a performance for the Raag-Mala Music Society of Toronto in 2019.
• In 2003, the Padma Shri award was given to him.​

Additional Information

• Tansen Samman
• Every year in the month of December, Behat village in the Gwalior district of Madhya Pradesh celebrates Tansen Samaroh or Tansen Sangeet Samaroh.
• A 4-day musical festival is taking place.
• Here, musicians and artists from all over the world come to honour the legendary Indian musical maestro Tansen.