SSC CHSL Tier-1 2021: Official Paper (Held On : 8 June 2022 Shift 3) (Previous Year Paper)

The correct answer is 'Many books have been written by her.'

Key Points

• The given sentence is in Present Perfect Tense and in Active voice.
• The rule for changing a sentence in Present Perfect Tense From Active Voice to Passive Voice:
• Interchange the object and subject with each other, i.e. object of the active sentence become the subject of the passive sentence.
• The helping verb comes before the object in the passsive voice as the passive voice starts with the object of the active voice.
• Convert the verb into past participle or 3^rd form of the verb.
• Structure: Subject + has/have + V3 + Object (Active Voice)

                                       Object + has/have + been + V3 + by + Subject (Passive Voice)

• Solution: She has written many books. (Active Voice)

                           Many books have been written by her. (Passive Voice)

• Therefore the correct answer is 'Many books were written by her.'

The correct answer is 'Satisfy'.

Key Points

• The most appropriate synonym for the given word 'Satiate' is 'Satisfy'.
• The meaning of the word 'Stiate' is 'satisfied to the full; sated'.
• The meaning of the word 'Satisfy' is 'meet the expectations, needs, or desires of (someone)'.
• Hence, the appropriate answer is 'Satisfy'.

Additional Information

• Let's understand the other options.
• State- the particular condition that someone or something is in at a specific time.
• Saturate- cause (something) to become thoroughly soaked with water or other liquid so that no more can be absorbed.
• Starve- suffer or die or cause to suffer or die from hunger.

The correct answer is BADC.

Key Points

• The most appropriate order to put the given sentences in, to form a meaningful and coherent paragraph is BADC.
• A paragraph should always start from the time and the situation and then about the people and their experiences and in the end how they co-operate with the situation.
• So, Sentence B shows the time and the situation, Sentence A shows the time and people, sentence D shows their experience and Sentence C shows the method of co-operation that Anne Frank used.
• Hence the appropriate answer is BADC.
• Solution: In 1942, with the Nazis occupying Holland, a thirteen-year-old Jewish girl, Anne, and her family fled their home in Amsterdam and went into hiding. For the next two years the Franks and another family lived cloistered in the ‘Secret Annex’ of an old office building.Cut off from the outside world, they faced hunger, boredom, the constant cruelties of living in confined quarters, and the ever-present threat of discovery and death. In her diary Anne Frank recorded vivid impressions of these experiences during this period.

The correct answer is To fulfill the required norms.

Key Points

• The most appropriate meaning for the given idiom 'Tick all the boxes' is 'To fulfill the required norms'.
• Tick all the boxes: fulfil all the necessary requirements
• If you're after summer thrills then this ticks all the boxes.

Additional Information

• Let's look at some synonyms for the given idiom.
• Pass inspection
• Satisfy requirements
• Suffice

The correct answer is Very rarely.

Key Points

• The most appropriate meaning for the given idiom 'Once in a blue moon' is 'Very rarely'.
• Once in a blue moon: something extremely rare in occurrence
• E.g. That company puts on a good performance only once in a blue moon.

Additional Information

• Let's look at some similar idioms.
• A snowball's chance in hell: Your proposal is too radical — it has a snowball's chance in hell of being approved.
• Like getting blood from a stone: Convincing fussy eaters to try new foods is like trying to get blood out of a stone.
• Like finding needle in a haystack: Trying to find a lost child during the IT fair would be like searching for a needle in a haystack!

The correct answer is 'The General have ordered'.

Key Points

• The part that contains the error is 'The General have ordered'.
• The word 'Have' is used with the pronouns I, you, we, and they.
• 'Has' is used with he, she, it and other names or designations.
• Hence, considering the above-mentioned points, the correct answer is The General have ordered.
• The correct sentence would be: The General has ordered that the troops should move immediately.

The correct answer is 'Ignision'.

Key Points

• The correct spelling of 'Ignision' according to the English dictionary is 'Ignition'.
• The word 'Ignition' means 'the electrical system in an engine that causes the fuel to burn or explode in order to start the engine'.
• Hence, the most appropriate answer is 'Ignision'.

Additional Information

• Let's understand the meaning of the other options.
• Copulate- to engage in sexual intercourse.
• Portion- an individual's part or share of something.
• Horticulture- the art or practice of garden cultivation and management.

The correct answer is 'An easy task'.

Key Points

• The most appropriate meaning for the given idiom 'Child's play' is 'An easy task'.
• Child's play: to be very easy
• E.g. Using computers nowadays is child's play.

Additional Information

• Let's look at some similar idioms.
• It's a piece of cake: something easily done
• For him, taking tests is a piece of cake.
• A walk in the park: something that is very easy to accomplish
• As any director will tell you, doing Shakespeare isn't a walk in the park.
• It's a cinch: an easy thing to do
• It sounds difficult, but compared to full-time work it was a cinch.

The correct answer is 'cannot help getting angry'.

Key Points

• The most appropriate option to improve the underlined part of the given sentence is 'cannot help getting angry'.
• The expression 'can't help getting angry' is used to talk about becoming angry about something.
• The underlined sentence is grammatically incorrect.
• The words do/does are not used with words that are in continuous tense, i.e. verbs following 'ing'.
• Hence, the most appropriate answer is 'can't help getting angry'.

The correct answer is 'The children ate the sweet mangoes.'

Key Points

• The given sentence is in Past tense and in Passive Voice.
• The rule for changing a sentence in Past Tense from Passive Voice to Active Voice:
• ​Interchange the object and the subject with each other, i.e. object of the passive sentence becomes the subject of the active sentence.
• The helping verb get's removed as it is a simple past sentence.
• Convert the verb from the 3^rd form of the verb to the 2^nd form of the verb.
• Structure: Subject + V2 + Object (Active Voice)

​                                       Object + was/were + V3 +by + Subject (Passive Voice)

• Solution: The sweet mangoes were eaten by the children. (Passive Voice)

​                           The children ate the sweet mangoes. (Active Voice)

• Therefore, the correct answer is 'The children ate the sweet mangoes.'

The correct answer is: salty.

Key Points

• The word "savoury" refers to food that is salty or spicy rather than sweet. (स्वादिष्ट)
• Example: Her savoury dishes are always a hit at family gatherings.
• "Salty" is a term that describes food seasoned with or tasting of salt. (नमकीन)
• Example: The soup was a bit too salty for my taste.
• Hence, we can infer that a synonym for "savoury" is "salty".

Therefore, the correct answer is: salty.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Stale (बासी): Food that is no longer fresh.
• Example: The bread has gone stale, so we should throw it away.
• Stinky ( दुर्गंधयुक्त): Something that has a strong, unpleasant smell.
• Example: The garbage bin is stinky, it needs to be emptied.
• Stenchy (दुर्गंधित): Having a very strong, unpleasant smell.
• Example: The stenchy locker room made it hard to breathe.

The correct answer is 'The boy helped the old woman while crossing the road'.

Key Points

• The given sentence is in Past tense and in Passive Voice.
• The rule for changing a sentence in Past Tense from Passive Voice to Active Voice:
• ​Interchange the object and the subject with each other, i.e. object of the passive sentence becomes the subject of the active sentence.
• The helping verb get's removed as it is a simple past sentence.
• Convert the verb from the 3^rd form of the verb to the 2^nd form of the verb.
• Structure: Subject + V2 + Object (Active Voice)

​                                       Object + was/were + V3 +by + Subject (Passive Voice)

• Solution: The old woman was helped by the boy while crossing the road. (Passive Voice)

​                           The boy helped the old woman while crossing the road. (Active Voice)

• Therefore, the correct answer is 'The boy helped the old woman while crossing the road'.

The correct answer is 'Lexicography'.

Key Points

• The most appropriate one-wor substitute for 'The activity of writing dictionaries' is 'Lexicography'.
• Let's understand the other options.
• ​Biography- an account of someone's life written by someone else.
• Bibliography- a list of the books referred to in a scholarly work, typically printed as an appendix.
• Theology- the study of the nature of God and religious belief.

The correct answer is Polyglot.

Key Points

• The most appropriate one-word substitute for the given group of words 'One who is well-versed in many languages' is 'Polyglot'.
• Let's understand the other options.
• Verbose- using or expressing in more words than are needed.
• Linguist- a person who studies linguistics.
• Stalwart- loyal, reliable, and hard-working.

The correct answer is 'The careful crafted personality'.

Key Points

• The segment that contains a grammatical error is 'The careful crafted personality'.
• The word 'careful' is an adjective and the word 'carefully' is an adverb of manner.
• The adverb 'carefully' is used when the mannerisms of a person are being talked about.
• Since the mannerisms of Anahita are being talked about, therefore the error lies in the segment 'The careful crafted personality'.
• So the correct sentence is: The carefully crafted personality of Anahita did not help here much in the electoral campaign.

Therefore, the correct answer is "Option 2".

The correct answer is 'over such a triviale matter'.

Key Points

• The correct spelling of 'triviale' according to the English dictionary is 'trivial'.
• The word 'trivial' means 'of little value or importance.'
• Hence, the most appropriate answer is 'over such a triviale matter'.

The correct answer is 'shy'.

Key Points

• The most appropriate synonym for the given word 'modest' is 'shy'.
• The meaning of the word 'modest' is 'unassuming in the estimation of one's abilities or achievements'.
• The meaning of the word 'shy' is 'nervous or timid in the company of other people.'
• Hence, the most appropriate answer is 'shy'.

Additional Information

• Let's understand the other options.
• Courageous- not deterred by danger or pain; brave.
• Dry- free from moisture or liquid; not wet or moist.
• Horrific- very bad and shocking.

The correct answer is 'go out'.

Key Points

• The given sentence is grammatically incorrect.
• Here, 'go out' should be used instead of 'are going out'.
• The given sentence is an example of a conditional sentence.
• The type 1 conditional is used to refer to the present or future where the situation is real.
• Type 1 conditional refers to a possible condition and its probable result.
• Let's look at the structure of these kinds of sentences: If + simple present, simple future
• Example: If you don't hurry, you will miss the train.
• Thus, the correct answer is Option 2.

Correct Sentence: If they go out, surely we will follow them.

The correct answer is Stunning.

Key Points

• The most appropriate synonym for the given word 'Stupendous' is 'Stunning'.
• The meaning of the word 'Stupendous' is 'extremely impressive'.
• The meaning of the word 'Stunning' is 'extremely impressive or attractive'.
• Hence, the most appropriate answer is 'Stunning'.

Additional Information

• Let's understand the other options.

• Magnifying- make (something) appear larger than it is, especially with a lens or microscope.

• Shining- giving off or reflecting light; radiant; bright.

• Amplifying- increase the volume of (sound), especially using an amplifier.

The correct answer is Allimony.

Key Points

• The correct spelling of 'Alimony' according to the English dictionary is 'Alimony'.
• The word 'Alimony' means 'financial support that a person is ordered by a court to give to their spouse during separation or following divorce; maintenance.'
• Hence, the most appropriate answer is 'Allimony'.

Additional Information

• Let's understand the other options.
• Hallucinate- experience an apparent sensory perception of something that is not actually present.
• Structural- relating to or forming part of the structure of a building or other item.
• Halloween- the night of 31 October, the eve of All Saints' Day, often celebrated by children dressing up in frightening masks and costumes. Halloween is thought to be associated with the Celtic festival Samhain when ghosts and spirits were believed to be abroad.

The correct answer is Create.

Key Points

• The most appropriate answer to fill the blank is 'create'.
• The meaning of the word 'create' is 'bring (something) into existence'.
• Hence, the most appropriate word is 'create'.

Additional Information

• Let's understand the other options.
• Design- the arrangement of the features of an artifact, as produced from following a plan or drawing.
• Construct- build or make (something, typically a building, road, or machine).
• Spread- open out (something) so as to extend its surface area, width, or length.

The correct answer is 'survive'.

Key Points

• The most appropriate word to fill the blank is 'survive'.
• The word 'survive' means 'continue to live or exist, especially in spite of danger or hardship'.
• Hence, the most appropriate answer is 'survive'.

Additional Information

• Let's understand the other options.
• Prevail- prove more powerful or superior.
• Exist- live, especially under adverse conditions.
• Bear- carry the weight of; support, endure.

The correct answer is 'right'.

Key Points

• The most appropriate answer is 'right'.
• The meaning of the word 'right' is 'true or correct as a fact'.
• Hence, the appropriate word to fill in the blank is 'right'.

Additional Information

• Let's understand the other options.
• Helpful- giving or ready to give help.
• Equal- being the same in quantity, size, degree, or value.
• Valid- (of an argument or point) having a sound basis in logic or fact; reasonable or cogent.

The correct answer is 'inside'.

Key Points

• The most appropriate answer is 'inside'.
• The meaning of the word 'inside' is 'the inner side or surface of something'.
• Hence, the appropriate word to fill in the blank is 'inside'.

Additional Information

• Let's understand the other options.
• Interior- situated on or relating to the inside of something; inner.
• Inner- situated inside or further in; internal.
• Inward- directed or proceeding towards the inside; coming in from outside.

The correct answer is 'dries'.

Key Points

• The most appropriate answer is 'dries'.
• The meaning of the word 'dries' is 'to become dry'.
• Hence, the appropriate word to fill in the blank is 'dries'.

Additional Information

• Let's understand the other options.

• Soaks- make or allow (something) to become thoroughly wet by immersing it in liquid.

• Wipes- clean or dry (something) by rubbing with a cloth, a piece of paper, or one's hand.

• Dehydrates- lose a large amount of water from the body.

Here, the alphabets and their positions are shown as below:

The logic here is: The reverse of each letter is taken. The 1^st two letters interchange their positions, similarly next two letters do and so on,

‘PALACE’ is written as ‘ZKZOVX’ 

And,

‘DEVINE’ is written as ‘VWREVM’

Similarly,

‘SLOWLY’ be written as '?'

Hence, the code language is 'OHDLBO' for the given word.

The Family chart is:

According to the given statements, we will get,

Hence, the wife of Jorge is 'sister in law' to Sheila.

Here, the alphabets and their positions are shown below:

The logic here is:

Hence, the code language is 'SGRB' for the missing place.

The pattern followed in the given series is:

   Sum of all the digits in the first term + 4 = Second term

Let's apply the patter on the given series:

6789 : 34 :: 1786 : 26 :: 4327 : ?

(6 + 7 + 8 + 9) + 4 = 34

30 + 4 = 34 

34 = 34

6789 : 34 :: 1786 : 26 :: 4327 : ?

(1 + 7 + 8 + 6) + 4 = 26

22 + 4 = 26 

26 = 26

6789 : 34 :: 1786 : 26 :: 4327 : ?

(4 + 3 + 2 + 7) + 4 = ?

16 + 4 = ?

20 = Answer

Hence the correct answer is option 2)

The least possible Venn diagram is given below:

Conclusion:

I. Some pencils are books → Follow → (As all pens are pencils and some books are pens. So, some pencils are books. This is true.

II. Some books are not pencils → Does not follow → (As some books are pens and all pens are pencils. So, some books are pencils. This is not true.

Hence, only conclusion I is true.

Here, we have two dices given and two common faces.

So, when we rotate the 2^nd dice in such a way that it looks like the 1^st one only as face 6 and face 1 are opposite to each other as shown below:

Next, we can see that numbers 2 and 3 are not there and so they will be the opposite of either number 4 or number 5.

Thus, number 2 will be opposite number 5 and number 3 will be opposite number 4.

So, sum of the numbers that are opposite to 4 and 5 = 3 + 2 = 5

Hence, the sum of the numbers that are opposite to '4 and 5' is '5'.

Here, the pattern is: Each 1^st term square is taken and then subtracted by 1 to get the 2^nd term. 

We will check all the options as:

1. 4 : 15 ⇒ (4)² - 1 = 16 -1 =15

2. 12 : 143 ⇒ (12)2 =  144 - 1 = 143

3. 8 : 66 ⇒ (8)2 = 64 - 1 ≠  65. This is an odd one

4. 3 : 8 ⇒ (3)2 = 9 - 1 = 8

Hence, the odd number pair is '8 : 66'.

Here the pattern followed is as below:

Hence, the missing number is '572'.

Here, we have a paper that is folded two times and then some design is made on it.

So, we get figure 2 below when the 1^st fold is opened.

Hence, we will get option 4 as the correct answer.

Family chart:

According to the given statement:

Hence, the woman is 'sister' to Noor.

Here, the pattern is: The word is written from right to left.

BRIGHT : THGIRB

And,

DANGER : REGNAD

Similarly,

FOREST : ?

Hence, the code language is 'TSEROF' for the missing place.

The pattern followed here is:

 3, 5, 7, 11, 13, 17, 19 form a series of consecutive prime numbers.

Hence, the missing number is 362.

Here, we are given the word and we need to find the mirror image of it.

First keep in mind that, in mirror image the first letter will be the last letter of the given word. Also, the size of the letters will not change i.e. capital letters will appear capital only.

The correct mirror image of the given combination when the mirror is placed at MN is as shown below :-

Detailed Explanation :- 

• The line XY is the mirror.
• The word "

"  ends with "

" . So, the mirror image will be "

" → Option 4 is eliminated.

• Also, the letters '

' and '

' are used. So, the mirror image will be '

'  and '

' respectively. → Options 1  and 3 are eliminated.

Hence, the mirror image of the given word is 'option 2'.

Additional Information

• In a mirror image Left changes to rightand theright changes to leftwheretop and bottom remains the same.
• When placement of mirror not given, by default we will assume it in vertical direction.
• The starting letter word will be at last after the mirror image drawn with left changes to right and right changes to left.

Logic : Sum of digits of first number - Number of digits in first number = Second number

9876 : 26

9 + 8 + 7 + 6 = 30 

30 - 4 = 26

8671 : 18

8 + 6 + 7 + 1 = 22

22 - 4 = 18 

Similarly, 3427 : ?

3 + 4 + 2 + 7 = 16 

16 - 4 = 12

Hence, 'option 4' is the correct answer.

Here, we can see that number 9 and 10 are common in both the dices.

So, we will rotate the dice in such a way that, the positions of both the numbers are the same.

From this, we can say that the number 12 is opposite to number 8.

Hence, the number '12' will be on the face opposite the one showing ‘8’.

Here, the alphabets and their positions are shown below:

So, for the word TABLE and CHAIR, we have:

 'TABLE' is written as '2012125'

And,

 'CHAIR' is written as '381918'

Similarly, 

'STAND' be written as '?'

Hence, the code language is '19201144' for the given word.

Here, we have two dice and a common face number 6 on it.

So, we will start writing numbers from 6 of the dice in the clockwise direction

1^st dice: 6 - 4 - 3

2^nd dice: 6 - 5 - 2

As we can see all the numbers are present except number 1, and so number 1 is opposite to number 6.

Hence, the number 1 will be on the face opposite the one showing the number 6.

Here the pattern is:

The position of the last letter is same. But the 1^st letter will go to 3^rd position, 2^nd letter will move to 1^st position and 3^rd letter will go to 2^nd position.

“SPOT” is written as “POST”

“PLOT” is written as “LOPT”

Similarly,

“SINK” be written as '?'

Hence, code language is 'INSK' for the given word.

Here, the alphabets and their positions are shown as below:

The logic here is:

Hence, the missing place is "SJKU".

Here, the pattern is:

Each term is divided by 0.4 to get the next consecutive term.

So, we have,

Hence, the next number is '23437.50' in the series.

Here, the paper is folded two times and some designs are made.

So, when we open the 1^st fold we will get,

Hence, the correct answer is 'option 3'.

Here, the pattern is:

The 2^nd term is the number we get when the 1^st term is divided by 8. And one zero is written after that number to get the final answer.

So, from the options, we can say that, 

1. 48 ÷  8= 6, it should be 60 and not 62. This is not true.

2. 32 ÷  8 = 4 , it is 40

3. 64 ÷  8 = 8, it is 80

4) 56 ÷  8 = 7, it is 70

Hence, the number pair '48 - 62' is the odd one out.

Here, we need to apply the combination of signs to get the correct equation.

Given: 75 * 85 * 17 * 12 * 6 * 8 

BODMAS Rule is:

First we will check options one by one as below:

1) ×, –, ÷, +, =

$\Rightarrow$ 75 × 85 - 17 ÷ 12 + 6 = 8

Applying BODMAS rule, we will get, 

$\Rightarrow$ 75 × 85 - 1.41 + 6 = 8 

$\Rightarrow$ 6375 - 1.41 + 6 = 8

$\Rightarrow$ 6379.59 ≠  8, this is not possible

2) +, ×, ÷, –, =

$\Rightarrow$ 75 + 85 × 17 ÷ 12 - 6 = 8

Applying BODMAS rule, we will get, 

$\Rightarrow$ 75 + 85 × 1.41 - 6 = 8

$\Rightarrow$ 75 + 119.85 - 6 = 8

$\Rightarrow$ 188.85 ≠  8, this is not possible

3) ×, ÷, –, +, =

$\Rightarrow$ 75 × 85 ÷ 17 - 12 + 6 = 8

Applying BODMAS rule, we will get,  

$\Rightarrow$ 75 × 5 - 12 + 6 = 8

$\Rightarrow$ 375 - 12 + 6 = 8

$\Rightarrow$ 369 ≠  8, this is not possible

4) +, ÷, –, ×, =

$\Rightarrow$ 75 + 85 ÷ 17 - 12 × 6 = 8

Applying BODMAS rule, we will get,

$\Rightarrow$ 75 + 5 - 12 × 6 = 8

$\Rightarrow$ 75 + 5 - 72 = 8

$\Rightarrow$ 8 = 8, this is possible

Hence, the combination of signs '+, ÷, –, ×, =' is used to get the correct balanced equation.

The logic followed here is:

Third Number = (First number × 2) - (Second number ÷ 5)

First row - 12, 25, 19 → (12 × 2) - (25 ÷ 5) = 24 - 5 = 19

Second row - 35, 50, 60 → (35 × 2) - (50 ÷ 5) = 70 - 10 = 60

Similarly,

Third row - 24, 70, ? → (24 × 2) - (70 ÷ 5) = 48 - 14 = 34

Hence, 'option 4' is the correct answer.

Here, we have the pattern as:

The 1^st term is added by 3 to get the next term, and again the next term is added by double of 3 (i.e. 6) to get the next term, and so on.

So, we have,

Hence, the next term is 107 in the given series.

Given:- Priya + Renu can do the job in 12 hours, whereas Priya alone can do it in 18 hours

To find:- Time taken by Renu to complete the job

Formula used:- Work done = Time × Efficiency

Calculation:- Let the total work be the LCM(12 , 18)

Total work ( job ) = 72 units

Let the efficiency of Priya and Renu be P and R,

According to the question,

(P + R) ×  12 = 72

⇒(P + R) = 6

Now it is given that Priya takes 18 hours to complete the work alone

⇒P × 18 = 72

⇒P = 4

The efficiency of Priya is 4

⇒R = 6 - 4 = 2

The efficiency of Renu is 2,

Now by using the formula of work done

⇒72 = Time × 2

⇒ 72/2 = 36 = Time

Renu will take 36 hours to complete the job.

Given:- A data in a tabulated form where sales of car from 2015 to 2020 by different manufacturers are mentioned and all the figures are in thousands

To find:-  total sales of cars from 2015 to 2020

Concept used:- Basic addition of values

Calculation:- Total sales in year 2015 = 44 + 40 + 36 + 48 = 168

Total sales in year 2016 = 48 + 41 + 38 + 44 = 171 

Total sales of cars 2017 = 47 + 41 + 40 + 44 = 172

Total sales of cars 2018 = 50 + 41 + 42 + 42 = 175

Total sales of cars 2019 = 52 + 42 + 44 + 42 = 180

Total sales of cars 2020 =  51 + 43 + 50 + 43 =  187

Total sales from 2015 to 2020 = 168 + 171 + 172 + 175 + 180 + 187 

⇒Total sales from 2015 to 2020 = (170 - 2) + (170 +1)+ (170 + 2) + (170 + 5)  + (180) + (180 + 7) 

⇒Total sales from 2015 to 2020 = 170 × 4 + 180 × 2 + (8 - 2 + 7) = 1053

Since the figures were in thousands,

⇒Total sales from 2015 to 2020 = 1053 × 1000 = 1053000

The total sales of cars of the four manufacturers under consideration in the city in the entire period of 6 years is 1053000

Given:- sec2 θ = $\frac{4}{3}$

To find:- cosec (θ + 30°)

Formula used:- sec θ = $\frac{Hypotenuse}{Base}$ ,  Cosec θ =  $\frac{Hypotenuse}{Height}$

calculation:- It is given that sec2 θ = $\frac{4}{3}$

On taking square root both sides,

⇒sec θ = $\frac{2}{\sqrt3}$

Since θ is an acute angle then In a right-angled triangle, 

Base = $\sqrt3$ unit and hypotenuse = 2 units

Now by using the Pythagorean theorem in a right triangle,

⇒Hypotenuse² = Base² + height²

⇒ 2² = ($\sqrt3$)² + height²

⇒Height = 1 unit

Since Cosec θ =  $\frac{Hypotenuse}{Height}$

⇒ Cosec θ =  $\frac{2}{1}$ = 2

⇒ Sin θ = $\frac{Height}{Hypotenuse}$ = $\frac{1}{2}$

⇒Cos θ = $\frac{Base}{Hypotenuse}$ = $\frac{\sqrt3}{2}$

Now by using the conversion of Cosec θ , cosec θ = ${1\over \sinθ}$ 

⇒cosec (θ + 30°) = ${1\over \sin(30+θ)}$

Using the formula, sin(a + b) = sin a cos b + cos a sin b

⇒${1\over \sin(30+θ)}$ = $\frac{1}{\sin30^{\circ}.\cosθ\ +\ \cos30^{\circ}.sinθ}$

using the standard value of sin 30° and cos 30° as $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$ respectively, and using Sin θ and Cos θ by  $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$ respectively,

⇒${1\over \sin(30+θ)}$ = $\frac{1}{\frac{1}{2}.\frac{\sqrt3}{2}\ +\ \frac{\sqrt3}{2}.\frac{1}{2}}$

⇒${1\over \sin(30+θ)}$ = $\frac{4}{2\sqrt3}$

⇒${1\over \sin(30+θ)}$ = $\frac{2}{\sqrt3}$

The value of cosec (θ + 30°) is $\frac{2}{\sqrt3}$ 

Alternate Method

It is given that sec2 θ =  $\frac{4}{3}$ then ⇒ sec θ = $\frac{2}{\sqrt3}$ ⇒ sec 30° = $\frac{2}{\sqrt3}$, so θ = 30° 

Now, cosec (θ + 30°) ⇒ cosec (30° + 30°) ⇒ cosec 60° = $\frac{2}{\sqrt3}$

Given:- px3 + x2 + 3x + q is exactly divisible by (x + 2) and (x - 2)

To find:- The value of p and q

Concept used:- If f(x) is divisible by (x + a) and (x - b) then (-a) and (-b) are the factors of f(x) 

Calculation**:-** Since the equation is divisible by (x + 2) and (x - 2) then 2 and -2 are factors of the equation f(x) 

⇒f(-2) = f(2) = 0

f(x) = px3 + x2 + 3x + q

⇒f(-2) = 0 = p(- 2)³ + (2)² + 3(- 2) + q

⇒0 = -8p + 4 - 6 + q

⇒-8p + q - 2 = 0

⇒-8p + q = 2 ------------------(i)

Similarly, 

⇒f(2) = 0 = p(2)3 + (2)2 + 3(2) + q

⇒ 8p + 4 + 6 + q = 0

⇒ 8p + q = -10 -----------------(ii)

On adding the eq(i) and (ii),

⇒(- 8p + q) + (8p + q) = 2 - 10

⇒ 2q = -8

⇒q = -4

Substituting the value of q in eq(ii),

⇒ 8p + (-4) = -10

⇒ 8p = -6

⇒ p = $\frac{-6}{8}$ = ${-3\over4}$

The value of p is ${-3\over4}$ and the value of q is -4

Given:- Pie chart with data of the distribution of 36 marbles

To find:- number of marbles Madhuri has

Concept used:- In the pie chart, 360° = 100%

Calculation**:-** The portion of Madhuri is of 180°,

In percentages,

⇒360° = 100%

then, 1° = $\frac{100\%}{360}$

⇒180° = $\frac{100\%}{360^{\circ}}\times 180^{\circ}$ = 50%

So the share of Madhuri is 50% of the total marbles

⇒Total number of marbles Madhuri has = 50% of 36

⇒Total number of marbles Madhuri has = $\frac{50}{100}\times 36$ = 18

Madhuri has 18 marbles

Given:- A data in tabulated form having marks of different subjects by 100 students

To find:- Number of students who scored above 60 in subjects B and C

Calculation**:-** As mentioned in the question, Students scoring at least 60% marks in B and C are eligible for higher studies in subjects B and C,

The total marks is 100

⇒60% of total marks = $\frac{60}{100}\times 100$ = 60 marks

So, the students who scored 60 and above are eligible for higher studies in subjects B and C

⇒Total number of students who scored 60 and above in B = 20

⇒Total number of students who scored 60 and above in C = 26

Total students who are eligible to pursue higher studies in B and C  = 20 + 26 = 46

46 students are eligible to pursue higher studies in subjects B and C.

Given:- The surface area of cube = 726 cm2

To find:- Volume of cube

Formula used:- Surface area of cube = 6 × (side)² , Volume of cube = (side)³

Calculation**:-** Let the side of cube be ' a cm '

 Surface area of cube = 726 cm² 

⇒6 × a² = 726

⇒a² = 121

⇒a= 11 cm

Volume of cube = (side)³ = (a)³

⇒Volume of cube = (11)³ 

⇒Volume of cube = 1331 cm³ 

The volume of cube is 1331 cm³.

Given:-  7P323Q is divisible by 88

To find:- The value of P and Q

Concept used:- Divisibility of 11 and divisibility of 8

Calculation**:-** It is given that the number 7P323Q is completely divisible by 88

Since the number 88 is divisible by 11 this means the number 7P323Q must be divisible by 11 and 8

⇒The divisibility of 8 states that if the last three digits are divisible by 8 then the number is divisible by 8,

The last three digits of the number are 23Q,

The last three digits can only be divisible by 8 if the value of Q is 2

⇒Divisibility of 11 states that the number is divisible by 11 if the difference between the sum of digits in odd places and the sum of digits in even places is divisible by 11 or 0.

 In the given number, 7P3232, the digits in odd places and even places are,

Digits of even places = P, 2, 2

Digits of Odd places = 7, 3, 3

Difference = (7 + 3 + 3) - (P + 2 + 2)

⇒Difference = 9 - P

The ' P ' can't be negative therefore, the difference should have to be 0

⇒9 - P = 0

⇒P = 9

The value of P is 9 and the value of Q is 2

Given:- a ∶ b = 2 ∶ 5 and b : c = 3 ∶ 4

To find:-  The value of a ∶ b ∶ c

Calculation**:-**  a ∶ b = 2 ∶ 5

On multiplying the ratio by 3,

⇒a : b = 6 : 15 

It is given that, b : c = 3 ∶ 4

On multiplying the ratio by 5,

⇒ b : c = 15 : 20

Now, a : b = 6 : 15  and  b : c = 15 ∶ 20 

⇒a ∶ b ∶ c = 6 : 15 : 20

 The value of a ∶ b ∶ c is  6 : 15 : 20

Given:- Length(L) = 13 m, Breadth(B) = 7 m, Height(H) = 3 m

To find:- Diagonal cuboid

Formula used:- Diagonal of cuboid = $\sqrt{L^{2}+B^{2}+H^{2}}$

Calculation**-** By using the formula,

Diagonal = $\sqrt{L^{2}+B^{2}+H^{2}}$

⇒Diagonal = $\sqrt{13^{2}+7^{2}+3^{2}}$

⇒Diagonal = $\sqrt{169+49+9}$

⇒Diagonal = $\sqrt{227}$

∴ The diagonal of cuboid is $\sqrt{227}$ m

Given:

The total profit obtained by selling all the sugar = Rs. 90

To find:

Cost price of the entire quantity of sugar

Formula used: 

Cost price × profit% = Selling price

Calculation**:** 

Let the cost price of sugar be Rs. x per Kg

Suppose there is 200 kg of sugar in the bag,

⇒Total price of the sugar in the bag = 200 × x = 200x

As per the question,

⇒ The profit on 100 kg of sugar = 20%

⇒ The profit on 50 kg of sugar = 16% [quarter of the quantity of the sugar, $\frac{1}{4}\times 200$]

⇒ The profit on 50 kg of sugar = 24% [Left amount is 50 kg]

Total profit on cost price = Rs. 90

⇒$\frac{20}{100}\times100x$ + $\frac{16}{100}\times 50x$ + $\frac{24}{100}\times 50x$ = 90

⇒ 20x + 8x + 12x = 90

⇒ x = $\frac{9}{4}$

The Cost price of sugar = 200x = 200 × $\frac{9}{4}$

⇒The Cost price of sugar = 450

The cost price of sugar is Rs. 450

Given:

Radius (R) = 28 cm

radius (r) = 18 cm

Formula Used:

Length of common tangent = 2√(R × r)

R & r are radii

Calculation:

Solution:

Length of common tangent = 2√(28 cm × 18 cm)

= 2√(504 cm²)

= 2 × √(504 cm²)

≈ 2 × 22.47 cm

≈ 44.90 cm (rounded to 2 decimal places)

Therefore, the length of the common tangent between the two circles is approximately 44.90 cm.

Given:

Mixture of milk and water = 75%

Required percentage mixture of milk and water = 45%

Volume of mixture = 300 ml

Concept used:

Concentration before mixture = Concentration after mixture

Calculation:

⇒ Let the amount of water mixed is x ml.

⇒ The concentration of milk in the initial mixture = 75%

⇒ Volume of mixture = 300 ml

⇒ Final concentration of milk = 45%

⇒ So, after adding, Volume of mixture = 300 + x

⇒ The amount of milk in the mixture is remain unchanged.

⇒ 75% × 300 = 45% × (300 + x)

⇒ $\frac{75}{100}\times 300=\frac{45}{100}\times (300+x)$

⇒ $\frac{75\times 3\times 100}{45}=300+x$

⇒ 500 = 300 + x

⇒ x = 500 – 300 = 200 ml.

Hence, the amount of water mixed in the mixture is 200 ml to reduce the concentration from 75% to 45%.

Given:- Marked price = 1200 and two successive discounts of 25% and 20%

To find:- Selling price

Formula used:- Selling price = (1-${first\ discount\ \over 100}$)(1-${second\ discount\ \over 100}$) × Marked price

Calculation:- By using the formula,

⇒Selling price = $\left(1-\frac{25}{100} \right)\times \left(1-\frac{20}{100} \right)\times1200$

⇒Selling price = $\left(1-\frac{1}{4}\right)\times \left(1-\frac{1}{5} \right)\times1200$

⇒Selling price = $\left(\frac{3}{4}\right)\times \left(\frac{4}{5} \right)\times1200$

⇒Selling price = Rs. 720

The selling price of the shirt after two successive discounts is Rs. 720

Given:- Total students = 30, Number of students score 75 = 20 and Number of students scored 87 = 10

To find:- Average of the batch

Formula used:-  Average = $\frac{Sum\ of\ observation}{Total\ number\ of\ observation}$

Calculation**:-** The sum of marks who scored 75 = 20 × 75

Total sum of marks of students who scored 75 marks = 1500

The sum of marks of students who scored 87 marks = 10 × 87

Total sum of marks of students who scored 87 marks = 870 

Total marks scored by the batch = 1500 + 870 = 2370

By using the formula of average,

⇒Average = $\frac{2370}{30}$

.·. Average = 79

The average marks of the batch is 79

Given:- $\rm 1\frac{2}{3}\div\frac{4}{9}×\frac{p}{9}=1\frac{1}{4}×\frac{2}{3}\div\frac{1}{6}$

To find:- The value of p

Calculation**:-** It is given that $\rm 1\frac{2}{3}\div\frac{4}{9}×\frac{p}{9}=1\frac{1}{4}×\frac{2}{3}\div\frac{1}{6}$,

Solving the Left-hand side,

⇒$\rm 1\frac{2}{3}\div\frac{4}{9}×\frac{p}{9}$ = $\rm \frac{5}{3}\div\frac{4}{9}×\frac{p}{9}$

⇒$\rm 1\frac{2}{3}\div\frac{4}{9}×\frac{p}{9}$ = $\rm \frac{5}{3}×\frac{9}{4}×\frac{p}{9}$

⇒$\rm 1\frac{2}{3}\div\frac{4}{9}×\frac{p}{9}$ = $\frac{5p}{12}$

Solving the Right-hand side,

⇒$1\frac{1}{4}×\frac{2}{3}\div\frac{1}{6}$ = $\frac{5}{4}×\frac{2}{3}\div\frac{1}{6}$

⇒$1\frac{1}{4}×\frac{2}{3}\div\frac{1}{6}$ = $\frac{5}{4}×\frac{2}{3}×\frac{6}{1}$

⇒$1\frac{1}{4}×\frac{2}{3}\div\frac{1}{6}$ = $\frac{60}{12}$ = 5

Equating the Left-hand side and right-hand side,

⇒$\frac{5p}{12}$ = 5

⇒ p = 12

The value of p in the expression is 12

Alternate Method $\rm 1\frac{2}{3}\div\frac{4}{9}×\frac{p}{9}=1\frac{1}{4}×\frac{2}{3}\div\frac{1}{6}$

⇒ $\rm \frac{5}{3}×\frac{9}{4}×\frac{p}{9}=\frac{5}{4}×\frac{2}{3}×6$

⇒ Solving we get, P = 2 × 6 = 12

Given:- Principal amount = Rs.6000, time period = 2 years, compounded annually with rate = 10% p.a.

To find:- Final amount Mahes has to give,

Formula used:- A = $P\left(1+\frac{r}{n} \right)^{nt}$

Calculation**:-** As it is mentioned that the interest will be compounded yearly,

.·. n = 1 [Means rate would be applied once in a year]

By using the formula of compound interest,

⇒Final amount = $P\left(1+\frac{r}{1} \right)^{1× t}$

⇒Final amount = $6000\left(1+\frac{10}{100}\right)^ 2$

⇒Final amount = $6000\left(\frac{11}{10}\right)^ 2$

⇒Final amount = $6000 × \frac{121}{100}$

⇒Final amount = $60 × 121$ = 7260 

At the end of 2 years, the final amount will be Rs. 7260

Shortcut Trick

⇒ Principle → Amount

⇒ 10 → 11

⇒ 10 → 11

⇒ Total → 100 → 121

∴ If Principle = 100x, then Amount = 121x

⇒ 100x = 6000

⇒ x = 60 

∴ Amount = 121x = 121 × 60 = Rs. 7260.

Given:- Discount of 19% on Marked price and Selling price = Rs. 453.60

To find:- Marked price of the book

Concept used:- Selling price = Marked price - Discount% on marked price

Calculation:- Let the marked price be M,

⇒M - 19% of M = Selling Price

⇒M - ${19M\over 100}$= 453.60

⇒81M = 45360

⇒M = 560

The marked price of the book is Rs. 560

Shortcut Trick

Total Salary = ₹30,000

Food & Rent (25% of Total) = 0.25 × 30,000 = ₹7,500

Education (20% of Remaining) = 0.20 × (30,000 − 7,500) = 0.20 × 22,500 = ₹4,500

Income Tax (5% of Total) = 0.05 × 30,000 = ₹1,500

Total Deductions = 7,500 + 4,500 + 1,500 = ₹13,500

Amount available after savings = Total Salary − Total Deductions − Savings = 30,000 − 13,500 − 8,000 = ₹8,500

∴ The correct answer is ₹8500.

Alternate Method

Given:

Monthly Salary = ₹30,000

Savings = ₹8,000

Calculations:

⇒ Expenditure on Food and Rent = 25% of 30,000 = (25/100) × 30,000 = ₹7,500

⇒ Remaining Salary after Food and Rent = 30,000 − 7,500 = ₹22,500

⇒ Expenditure on Education = 20% of 22,500 = (20/100) × 22,500 = ₹4,500

⇒ Income Tax = 5% of Total Salary = (5/100) × 30,000 = ₹1,500

⇒ Total Amount Deducted (Expenses + Tax + Savings) = 7,500 + 4,500 + 1,500 + 8,000 = ₹21,500

⇒ Amount available after savings = Total Salary − Total Deductions = 30,000 − 21,500 = ₹8,500

∴ The correct answer is ₹8500.

Additional Information

Percentage of Remaining

When a percentage is applied to the remaining amount, always subtract the previous expenditure from the total before calculating the next value.

Total Salary Components

Total Salary = Total Expenditure + Savings + Taxes. The amount available is what remains after specific deductions are made.

Given:- x3 + y3 = 416 and  x + y = 8

To find:- x4 + y4

Formula used:- (x² + y²)² = x4 + y4 + 2x2y2  

Calculation**:-** x + y = 8 --------(i)

On squaring both sides,

⇒(x + y)² = 8²

⇒x² + y² + 2xy = 64 ---- (ii)

On cubing the equation (i) on both sides,

⇒(x + y)³ = 8³

⇒x³ + y³ + 3xy(x + y) = 512

substituting the values, x3 + y3 = 416 and  x + y = 8

⇒416 + 3xy(8) = 512

⇒ 24xy = 512 - 416 = 96

⇒ xy = 4

Putting the value xy = 4 in the equation (ii),

⇒x2 + y2 + 2(4) = 64

⇒x2 + y2 = 56

Now squaring the above equation both sides,

⇒(x2 + y2)² = 56²

⇒ x4 + y4 + 2x2y2  = 3136

⇒ x4 + y4 = 3136 - 2x2y2

⇒ x4 + y4 = 3136 - 2(xy)²

Putting the value, xy = 4 in the above equation

⇒ x4 + y4 = 3136 - 2(4)²

⇒ x4 + y4 = 3136 - 32 = 3104

.·.The value of x4 + y4  is 3104

Given:- 323 - 3(322)(12) + 3(32)(122) - 123

To find:- The value of expression, 323 - 3(322)(12) + 3(32)(12²) - 123

Formula used:- (a - b)³ = a³ - 3(a)²b + 3a(b)² - b³

Calculation**:-**  The given expression is an expanded form of (a - b)3 

⇒a3 - 3(a)2b + 3a(b)2 - b³ ---------(i)

⇒323 - 3(322)(12) + 3(32)(122) - 123 -----------(ii)

On comparing eq(i) and (ii),

we can observe that,  a = 32 and b = 12

⇒(32 - 12)³ = 323 - 3(322)(12) + 3(32)(122) - 123 

⇒323 - 3(322)(12) + 3(32)(122) - 123 = 20³ = 8000

.·.The value of the expression is 8000

Given:- Sailing out speed:- 15 km/h sailing back speed:- 10km/h

To find:- The average speed of sailing of ship

Formula used:- Average speed = $\frac{2ab}{a+b}$, a and b are the going and returning speed

Calculation**:-** By using the formula, the Average speed = $\frac{2ab}{a+b}$

⇒Average speed = $\frac{2\times 15\times 10}{15+10}$

⇒Average speed = $\frac{300}{25}$

⇒Average speed = 12 km/h

The average speed of sailing of the ship is 12 km/h.

Given:- 25% of income is spent by the women and expenditure = Rs.1250

To find:- Her income

Concept used:- Income - expenditure = Spent amount

Calculation**:-** Let the income of women be Rs. x

According to the question,

25% of x = 1250

⇒$\frac{25}{100}\times x$ = 1250