SSC CHSL Tier-1 2021: Official Paper (Held On : 3 June 2022 Shift 2) (Previous Year Paper)

The correct option is Let a glass of water be brought to me, please.

Key Points

• Given sentence "Please bring me a glass of water" is in Simple Present Tense and in Active voice.
• The sentence given in the above question is an imperative sentence as it refers to a request or command as seen in the given sentence, where the speaker is asking someone to bring a glass of water.
• The rule for changing a Simple Present Tense sentence into a Passive voice is as follows:
• ​Interchange the object and subject with each other, i.e. object of the active sentence become the subject of the passive sentence.
• We need to change the verb 'bring' down to passive, which will become ‘be brought’.
• Structure (for the sentence that involves requests): Let + Subject + Third form of verb + Object.
• Example:  Sit down, Please. (Active Voice)

                                       Let me be seated. (Passive Voice)

The correct answer is Surprise.

Key Points

• Startle refers to take an action that surprises and occasionally disturbs a person or an animal.
• Example: She was reading the novel but my voice startled her.
• Surprise refers to an unexpected event. 
• Example: He gave his mother a surprise on her birthday.
• "Startle" and "Surprise" are similar in meaning.

Therefore, the correct option is Surprise.

Additional Information

• Tumble refers to drop suddenly and uncontrollably.
• Impress refers to earn someone's regard or admiration.
• Delight means considerable enjoyment, satisfaction, or pleasure.

The correct answer is "Talked over."

Key Points

• "Talked of" is grammatically incorrect in this sentence.
• The correct usage is "Talked over" instead of "Talked of."
• We use **"of" when we want to show that people or things relate to other things or people.
• ​Example: She was staring at the wall of her office.
• ​We use the "over" when describing movement or position that is higher than another and "on something."
• Example: We were talking over phone.
• In the given sentence, the delegates talked on the matter.

The correct sentence is: The delegates talked over the matter for an hour but without any result.

The correct answer is At 2% interest rate per month.

Key Points

• The sentence can be improved by replacing the "In 2% interest rate per month." with "At 2% interest rate per month."
• We use the preposition "in" when describing a location in relation to a bigger area.
• Example: The novel is in my locker.
• We use the preposition "at" when we talk about prices and numbers.
• Example: We booked the hotel at 2500 rupees.
• Thus, the correct option is At 2% interest rate per month**.**

The correct sentence is: Sahukaars provide loan at 2% interest rate per month which is higher than the bank charges.

The incorrectly spelt word is "commitee." i.e. option 3.

Key Points

• In the given options "commitee" is incorrectly spelt. Its correct spelling is "Committee."
• Let's look at its meaning.
• Committee: A small group of individuals appointed to speak on behalf of a bigger organisation and either make decisions or gather data for it.
• ​Example: He is a part of the school's development committee.
• ​Therefore, as per the points mentioned above, we find that the correct answer is Option 3.

​

Additional Information

Let's look at the meaning of other options also: 

The correct answer is As was customary, the palace staff brought the young prince to the podium in a diamond studded carriage for the coronation ceremony.

Key Points

• Given sentence "As was customary, the young prince was brought to the podium in a diamond studded carriage for the coronation ceremony" is in Simple Past Tense and in Passive voice.
• The rule for changing a Simple Past tense form into an Active voice is as follows:
• ​Interchange the subject with the object and vice-versa.
• Convert the verb into a past participle or 2^nd form of the verb.
• Structure: Subject + second form of verb + Object.
• Example: He drove the car. (Active Voice)

                           The car was driven by him. (Passive Voice)​​

Therefore, the correct answer is: As was customary, the palace staff brought the young prince to the podium in a diamond studded carriage for the coronation ceremony.

Additional Information

• The Active voice is used when the subject of the sentence does the action.
• The Passive voice is used when we focus on the object of the sentence.

The correct answer is Stay.

Key Points

• The sentence can be improved by replacing the "All stays" with "Stay."
• The given sentence is in Simple Present Tense.
• We use the "Simple Present Tense" when something is happening right now or frequently.
• Structure: Subject + first form of verb + s/es/ies (with singular verb) + Object.
• Example: She needs to do her homework.
• As "all" is a plural verb, "stay" should be used instead of "all stays."
• Thus, the correct option is Stay.

The correct sentence is: We should stay united.

The correct answer is Humiliated and started weeping.

Key Points

• The sentence can be improved by replacing the "Embarassed and started crying" with "Humiliated and started weeping."
• Embarrased: Feeling ashamed or timid.
• Example: She felt embarrassed when teacher scolded her.
• Humiliated: Someone who is made to feel foolish or ashamed.
• Example: She felt humiliated in front of the whole class.
• Thus, the correct option is Humiliated and started weeping.

The correct sentence is: Reena got humiliated and started weeping.

Additional Information

• Frightened refers to a person who is afraid or concerned.
• Frustrated refers to getting frustrated or feeling less confident as a result of not getting what you desire.
• Conscious refers to a person who is awake, reasoning, and aware of his surroundings.

The incorrectly spelt word is "Sabotege."

Key Points

• In the given options "Sabotege" is incorrectly spelt. Its correct spelling is "Sabotage."
• Let's look at its meaning.
• Sabotage: To tamper with or destroy machinery, tools, or structures in order to hinder the success of an adversary or rival.
• ​Example: Her image has been sabotaged by boys.
• ​Therefore, as per the points mentioned above, we find that the correct answer is Sabotege.

 

Additional Information

Let's look at the meaning of other options also: 

The correct answer is Gobble.

Key Points

• Nibble refers to eating something by taking several, little bits. 
• Example: A mouse has nibbled the cheese.
• Gobble means eating food too fast.
• Example: She gobbled her breakfast.
• Nibble​ and Gobble have different meanings.

Therefore, the correct option is Gobble. 

Key Points

• Muzzles refers to an animal's mouth and nose, especially one that is a dog.
• Elevated refers to something that is larger beyond what is reasonable or normal.
• Sprinted refers to go quickly across a small distance, either in a race or out of a strong desire to reach somewhere.

The correct answer is "gregarious"

Key Points

• In the given options, the most appropriate answer that can be used as a one-word substitute is 'gregarious.'
• Gregarious: like to be with other people.
• Example: Emma is a gregarious, outgoing sort of person.
• Therefore, the correct answer is option 2.​

Additional Information

• Let's look at the meanings of the other options.
• Misanthrope: someone who dislikes and avoids other people.
• Philanthropist: a person who helps the poor, especially by giving them money.
• Patent: the official legal right to make or sell an invention for a particular number of years.

The correct answer is "feeling ill"

Key Points

• The most appropriate meaning of the given idiom is 'feeling ill.'
• Under the weather: To feel sick.
• Example: I was still feeling a bit under the weather.
• Therefore, as per the points mentioned above, we find that the correct answer is option 4.

​ 

Additional Information

• Let's look at the usage and meaning of some idioms.

The correct answer is "alienating"

Key Points

• The highlighted word in the sentence is "divisive".
• The word "divisive" usually refers to something that causes disagreement or discord among people within a group, leading to the group being divided or split.
• ​Example: "His contentious stance on the new policy proved to be divisive among his colleagues, resulting in heated debates within the team."
• In this example, the word "divisive" is used to describe a situation where the individual's view on a policy causes disagreement or discord within the group (his colleagues), leading to the group being divided or conflicted.
• Alienating means to cause someone or a group of people to stop supporting and agreeing with you.
• • Example: All these changes to the newspaper have alienated its traditional readers.
• In the given options, the most appropriate answer is 'alienating.'
• Therefore, the correct answer is option 3.

​ 

Additional Information

• Let's understand the usage and meaning of the other options.

The correct answer is "base"

Key Points

• Nadir means the point at which something is at its lowest value or level.
• Example: At its nadir in the mid-1980s, the pound was almost at parity with the dollar.
• Base means the lowest part or edge of something.
• Example: She sat down at the base of a tree.
• Therefore, the correct answer is option 3.

 Additional Information

• ​Let's understand the usage and meaning of the other options.

The correct answer is "ACBD"

Key Points

• Sentence A should come first as it is the opening of the paragraph.
• Sentence C is the second sentence as it states about toxic pollutants caused by fossil fuels.
• Sentence B is the third sentence as it states that unchecking the burning of fossil fuel leads to getting dried.
• Sentence D comes at the end as it is the conclusion of the paragraph.

 

Additional Information

• ​The paragraph after the re-arrangement is as follows:
• The sharp increase in our consumption of fossil fuels has led to their depletion at an alarming rate. The toxic pollutants released from burning these fuels are also a cause for concern. Unchecked burning of fossil fuel is like an unchecked dripping tap which will eventually run dry. This has led to the tapping of various non-conventional sources of energy that are cleaner alternatives to fossil fuels.

The correct answer is "indecent"

Key Points

• Pure means not mixed with anything else.
• Example: A pure Arab horse.
• Marked option indecent means morally offensive.
• Example: She accused him of making indecent suggestions to her.
• Therefore, the most appropriate antonym of the given word 'pure' is 'indecent.'

​ 

Additional Information

• Let's understand the usage and meaning of the other options.

The correct answer is "am going"

Key Points

• The given sentence is grammatically incorrect.
• Here, the verb 'am' should be changed into 'are.'
• The verb 'are' is used in the present tense form with the second person singular and all plurals.
• The verb 'are' is used with the pronouns 'you' and 'they.'
• Example: They are both helpful.
• The above example indicates second person singular, all present tense.
• Therefore, as per the points mentioned above, the correct answer is option 3.

The correct answer is "definetly"

Key Points

• In the given options, the incorrectly spelled word is 'definetly.'
• The correct spelling of the word is 'definitely.'
• Definitely: without any doubt.
• Example: He definitely said he would be here.
• Therefore, the correct answer is option 4.

Additional Information

• ​Let's look at the usage and meaning of the other options.

The correct answer is "prohibited from receiving"

Key Points

• In the given options, the most appropriate answer is 'prohibited from receiving.'
• Forbidden: not allowed.
• Example: Smoking is forbidden in the cinema.
• Therefore, as per the points mentioned above, we find that the correct answer is option 2.

The correct answer is "very quickly"

Key Points

• The most appropriate meaning of the given idiom is 'very quickly.'
• By leaps and bounds: someone or something is increasing or improving quickly.
• Example: He is improved in leaps and bounds this season.
• Therefore, as per the points mentioned above, we find that the correct answer is option 4.

​ 

Additional Information

• Let's look at the usage and meaning of some idioms.

The correct answer is "in"

Key Points

• In the given options, the most appropriate answer is 'in'
• It is a preposition.
• A preposition is a word or group of words used before a noun or pronoun or noun phrase to show direction, time, place, location, etc.
• Example: He reads in the evening.
• Therefore, as per the points mentioned above, the correct answer is option 4.

The correct answer is "of"

Key Points 

• In the given options, the most appropriate answer is 'of'
• It is a preposition.
• A preposition is a word or group of words used before a noun or pronoun or noun phrase to show direction, time, place, location, etc.
• Example:  She stared at the floor of her room.
• Therefore, as per the points mentioned above, the correct answer is option 4.

The correct answer is "of"

Key Points 

• In the given options, the most appropriate answer is 'of'
• It is a preposition.
• A preposition is a word or group of words used before a noun or pronoun or noun phrase to show direction, time, place, location, etc.
• Example: He resigned from his position as a member of the school board.
• Therefore, as per the points mentioned above, the correct answer is option 2.

The correct answer is "into"

Key Points

• In the given options, the most appropriate answer is 'into'
• It is a preposition.
• A preposition is a word or group of words used before a noun or pronoun or noun phrase to show direction, time, place, location, etc.
• Example: She dived into the water.
• Therefore, as per the points mentioned above, the correct answer is option 1.

The correct answer is "dreamy"

Key Points

• In the given options, the most appropriate answer is 'dreamy.'
• Dreamy: seeming to be in a dream and not paying attention to what is happening around you.
• ​Example: She gets this dreamy expression on her face when she talks about food.
• ​ Therefore, as per the points mentioned above, the correct answer is option 2.

​ 

Additional Information

• ​Let's look at the usage and meaning of the other options.

The logic followed here is:

Logic: (First number - 7) × first number = second number

1. 13 : 78

⇒ (13 - 7) = 6 

⇒ 6 × 13 = 78

2. 19 :228

⇒ (19 - 7) = 12

⇒ 19 × 12 = 228

Similarly,

3. 23 : '?'

⇒ (23 - 7) = 16

⇒16 × 23 = 368 

Hence, the correct answer is "368".

We have, 25 March 1995 is Saturday.

Since 1996 is leap year = 2 odd days. So, 25 March 1996 = Saturday + 2 = Monday

Again 1997 is ordinary year = 1 odd day. So 25 March 1997 = Monday +1 = Tuesday

Next, we will count the remaining days = 6 + 30 + 31 + 30 + 31 + 31 + 30 + 31 + 30 + 12 = 262 ÷ 7 = 3 odd days

Thus, on 12 Dec 1997 = Tuesday + 3 = Friday

Hence, the correct answer is "Friday".

The logic here is:

Logic: Each term is added by 18 to get the next term of the same row. 

First row → 54 + 18 = 72 ⇒ 72 + 18 = 90

And,

Second row → 79 + 18 = 97 ⇒ 97 + 18 = 115,

Similarly,

Third row → 83 + 18 = 101 ⇒ 101 + 18 = 119

Hence, the correct answer is "119".

The figure is given below:

 

The shaded portion is embedded in the figure.

Hence, 'option 1' is the correct answer.

Here, we can see that the given figure is moving in clockwise direction.

Given:  

In the 1^st figure the arrow is bottom left facing left, next it is at top left facing top.

Similarly seeing this, the missing image will be same as the 1^st figure given.

First, we will check all the options:

We see that, the next figure will have the arrow at the bottom left and so option 4 is eliminated.

Next, we see that the rectangle image will be in horizontal direction and so option 3 is eliminated.

Again, we see that, the black part is on top and so option 1 is eliminated.

Thus, we are left with option 2 

Hence, the correct answer is option 2.

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

‘PAIR’ is written as ‘ZKIR’ 

And,

‘SAIL’ is written as ‘ZHOR’

Similarly,

‘BULK’ be written as '?'

Hence, the correct answer is "FYPO".

Here, we need to find the missing blanks

The logic here is: Make of group of 1^st seven letters.

Given: K a _ _ N _ a K _ R a _ K a _a R _ N K a

1) R a K a N K a: K a R a N K a  - K a R a N K a  - K a R a N K a

2) K a K a N K a: K a K a N K a  - K a R a N K a  - K a R a N K a 

3) R a K a N R a: K a R a N K a  - K a R a N K a  - R a R a N K a

4) R a R a N K a: K a R a N R a  - K a R a N K a  - K a R a N K a

In the option 1, the "R a K a N K a" is repeating itself.

From this we can say the pattern: K a R a N K a - K a R a N K a - K a R a N K a 

Hence, the missing blanks is 'R a K a N K a'.

BODMAS Rule:

Here, we have a equation as

21 A 34 C 6 B (72 D 9) A 13 B 1 + (56 D 4)

Let us decode the given sign:

= 21 + 34 - 6 × (72 ÷ 9) + 13 × 1 + (56 ÷ 4)

Applying BODMAS rule, we will get,

= 21 + 34 - 6 × 8 + 13 × 1 + 14

= 21 + 34 - 48 + 13 + 14

= 82 - 48

= 34

Hence, the correct answer is "34".

Here, we need to arrange them in alphabetic order:

The order is:

2. Pronoun
3. Proof
4. Propel
5. Proper
6. Prophecy

Hence, the alphabetic order is '2, 3, 5, 1, 4'.

The logic followed here is:

Logic: (3^rd number - 2^nd number) × 2^nd number = 1^st number.

Now,

For, (50, 10, 15)

→ (15 - 10) × 10

→ 5 × 10 = 50.

For, (60, 10, 16)

→ (16 - 10) × 10

→ 6 × 10 = 60.

Similarly, for (60, 15, 19)

→ (19 - 15) × 15

→ 4 × 15 = 60.

Hence, the correct answer is "Option 1".

Alternate Method

The pattern followed here is: 

Logic: In each pair the 1^st number is divided by the 2^nd number and the answer is added to the 2^nd number to get the 3^rd number.

1. (50, 10, 15 ): 50 ÷ 10 = 5, 10 + 5 = 15

2. (60, 10, 16): 60 ÷ 10 = 6, 10 + 6 = 16

Next, we will check all the options one by one:

1) (60, 15, 19): 60 ÷ 15 = 4, 15 + 4 = 19

2) (60, 10, 6): 60 ÷ 10 = 6, 10 + 6 = 16 ≠ 6. This is not true. 

3) (80, 20, 16): 80 ÷ 20 = 4, 20 + 4 = 24 ≠16. This is not true. 

4) (80, 6, 15): 80 ÷ 6 = 13.33, 6 + 13.33 = 19.333 ≠ 15. This is not true.

Hence, the correct answer is "(60, 15, 19)".

The pattern followed here is:

The logic here is: Each odd term is subtracted by 12 to get the next odd term and each even terms are added by 12 to get the next even terms.

Hence, the correct answer is "74".

The least possible Venn Diagram is given below:

Conclusion:

I. Some avocados are bananas → Follow → (As all bananas are avocados. So some avocados are bananas is possible. This is true)

II. Some grapes are bananas → Does not follow → (As they have not mentioned the relationship between grapes and bananas. This is not true)

III. Some avocados are strawberries → Follow → (As some strawberries are bananas and all bananas are avocados. So some avocados can be strawberries. This is true)

Hence, the answer is "Only conclusion I and III follow".

Here, we have two dices given and have face 2 has common in both.

Since, we can see all the numbers in both the dice except number 6. 

We will start writing the numbers starting from the common number 2, in clockwise direction as:

For 1^st dice: 2 - 5 - 1

For 2^nd dice: 2 - 4 - 3

So, we see that the number 6 has the opposite of number 2.

Hence, the number 6 will be on the face opposite to the one showing '2'.

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

'BROWSER' is written as 'DTQYUGT'

And

‘MONITOR’ is written as ‘OQPKVQT’

Similarly,

'PRINTER' be written as '?'

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

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

Logic: The 1^st two letters interchange their positions in code language and similarly the last two letters. Also, the middle two letters are added by 2 to get the code language.

BURGER : UBTIRE

And

EDITOR : DEKVRO

PLASMA : ? 

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

Logic: First number × 8 + 16 =  Second number

Explanation :

13 : 120 = 13 × 8 + 16 = 120

17 : 152 = 17 × 8 + 16 = 152

Similarly,

21 : 184 = 21 × 8 + 16 = 184

Hence, 184 is the correct answer.

The logic followed here is:

• The second digit is completely divisible by the first digit.

12 : 1764 = 1764 ÷ 12

And,

8 : 528 = 528 ÷ 8

Similarly,

16 : 4160 = 4160 ÷ 16

Only, Option 4 is totally divisible by 16

Hence, "4160" is the correct answer

Explanation :

• The line MN is the mirror
• The word "ma26w" ends with number "3". so, the mirror image will start with 

. because in mirror image left changes to right and the right changes to left vice - versa.

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

Hence, "Option 4" is the correct answer.

Explanation :

• Pari is taller than only veer.
• Krishi is taller than only three people.

       ..... >.......>.......> Krishi .....> Pari > veer

• Misha is taller than Minshu.
• Misha is shorter than Mohi.
• Priya is taller than Mohi.

       Priya > Mohi > Misha > Krishi > Minshu > Pari > Veer

Hence, Mohi is taller than Krishi.

The logic followed here is :

Hence, 44 is the correct answer.

The pattern followed here is : 

 

Hence, 404 is the correct answer.

The mirror image of XY will form like the following given image: 

Detailed Explanation:

• The line XY is the mirror.
• The word "123abc" ends with the Letter "c". So, the mirror image will start with "c"  ⇒ Option 1 and 3 are eliminated.
• The word "123abc" has the last second letter "b". So, the mirror image should be " ⇒ Option 4 is eliminated.

Hence, the correct answer is "Option 2".

Hence, "Option 2" is the correct answer.

Explanation :  

  The option figure that is embedded in the given figure when rotation is not allowed is shown below : 

 Hence, Option 2"  is the correct answer.

BODMAS rule

Option 1) ÷, +, =, −

Given expression : 91 * 13 * 26 * 51 * 18

After replacing  *  with ÷, +, =, − respectively, we get

⇒ 91 ÷ 13 + 26  = 51 − 18

⇒ 7 + 26  = 51 - 18

⇒ 33 = 51 - 18

⇒ 33 = 33

Option 2)  ×, −, =, +

Given expression : 91 * 13 * 26 * 51 * 18

After replacing  *  with ×, −, =, + respectively, we get

⇒ 91 × 13 - 26 = 51 + 18

⇒ 1183 - 26 = 51 + 18

⇒ 1183 - 26 = 69

⇒ 1157 ≠ 69

Option 3)  ÷, =, +, −

Given expression : 91 * 13 * 26 * 51 * 18

After replacing  *  with ÷, =, +, −  respectively, we get

⇒ 91 ÷ 13 = 26 + 51 - 18

⇒ 7 =  77 - 18

⇒ 7  ≠ 59

Option 4) ×, =, −, −

Given expression : 91 * 13 * 26 * 51 * 18

After replacing  *  with ×, =, −, −   respectively, we get

⇒ 91 × 13 = 26 − 51 − 18     

⇒ 1183 = - 43

⇒ 1183  ≠ - 43

Hence, the  ÷, +, =, − is the correct answer.

Explanation: Checking options one by one

Option 1)  E V E F I B V I F I C V E F V  D  I E F A  Not following the pattern.

Option 2)  A V E F I B V I F I C V E F E D V E F I Not following the pattern.

Option 3)  A V E F I B V E F I C V E F I  D V E F I  Pattern is followed here.

Option 4)  E V E F I B V I F I C V E F V D A E F I   Not following the pattern

Hence, A E I V I  is the correct answer.

GIVEN:

Price of an article reduced by 30%

Sale increased by 35%

CALCULATION:

Let the price of the article be Rs100 and the daily sale be 100 units

Daily revenue = 100 × 100 = Rs10,000

Now, after change in price and daily sale we have

New receipts = 70 × 135 = Rs9,450.

Decrease = Rs. (10,000 - 9,450 ) = Rs550.

⇒ %Decrease = $550\over 10000$ × 100 =  5.5% or $5 \frac{1}{2}$ %.

Hence, net percentage effect is $5 \frac{1}{2}$ % decrease. 

Shortcut Trick 

⇒ Decrease  = 200 - 189 = 11

⇒ Decrease % = (11/200) × 100 = 5.5 %

GIVEN:

The winning candidate gets the majority of votes = 572

Winning candidate got % of votes = 72 

CALCULATION:

Let us assume that 100% of votes are polled

Since, the winning candidate gets = 72% votes of polled votes

Hence, the other candidate gets = 28% votes of the polled votes.

⇒ Difference = (72 - 28) = 44% of polled  votes.

Now, According to the question winning candidate got the majority of votes = 572.

⇒ 44% of polled votes  = 572

⇒  100% polled votes = $572\over 44$ × 100  = 1300.

Hence, the number of votes polled = 1300.

GIVEN:

Amount at the end of 3 years  = Rs8,640.

Rate = 20%

Time = 3 years.

FORMULA USED: 

Amount =  P × $( 1 +R/100)^t$      Where P = Principal, R = Rate of interest, t = time.

CALCULATION:

Amount = Rs8,640  , R = 20% Time = 3 years

 Amount = P × $( 1+ R/100)^t$ 

⇒ 8640 = P × $( 1+20/100)^3$ 

⇒ 8640 = P × 6/5 × 6/5 × 6/5 

⇒ P = Rs5,000.

Hence, at present investment should be Rs5,000.

GIVEN:

Selling price = Rs4,740.

FORMULA USED:

S.P. = ( 1-d₁/100) × ( 1-d₂/100) × M.P.     

Where S.P. = Selling price,

M.P. = Marked price,

d₁and d₂ = successive discounts respectively.

CALCULATION:

Here, S.P.  = Rs4,740.,d₁ = 21% , d₂ = 40%

⇒ S.P. = ( 1-d1/100) × ( 1-d2/100) × M.P.   

⇒ 4,740 = ( 1- 21/100) × ( 1-40/100) × M.P.

⇒4,740 = 79/100 × 60/100 × M.P.

⇒ M.P. = Rs10,000.

Hence, the marked price of an article is Rs10,000.

GIVEN:

$1 + tan\theta = \sqrt3$

FORMULA USED:

$tan\theta = 1/cot\theta$

CALCULATION:

Here, we have $1+ tan\theta = \sqrt3$ 

⇒ 1+$1/cot\theta$  = $\sqrt3$ 

⇒ $1/cot\theta = \sqrt3 -1$ 

⇒ $cot\theta = 1/(\sqrt3 -1)$ 

Now, $\sqrt3 cot\theta - 1 =$ $\sqrt3$ × $1\over (\sqrt3 -1)$ -1 = $(\sqrt3 - \sqrt3 +1)\over (\sqrt3 -1)$ = $1\over (\sqrt3 -1)$

⇒ On rationalization, i.e. multiply $( \sqrt3 +1)$  in both numerator as well as in denominator we get

⇒$1\over (\sqrt3 -1)$ × $( \sqrt3 +1)\over (\sqrt3 +1)$ = $(\sqrt3 +1)\over 2$.

Hence, the value is $(\sqrt3 +1)\over 2$.

Given:-

CP = 10,000, Discount = 20%, profit = 40%

To find:-

Listed price

Formula used:- 

Profit %= $\frac{SP-CP}{CP}\times 100$

Calculation:- 

By the formula of profit percentage,

Profit %= $\frac{SP-CP}{CP}\times 100$ 

⇒ 40 = $\frac{SP-10,000}{10,000}\times 100$

⇒ SP = 4,000 + 10,000

⇒ SP = Rs.14,000

Let the marked or listed price be x,

By the relation of the listed price, discount, and selling price,

⇒ Listed price(Marked price) - discount = Selling price

⇒ x - 20% of x = 14,000

⇒ $x=\frac{14000}{1-\frac{20}{100}}$

⇒ x = $\frac{14000}{\frac{4}{5}}$

⇒ x = 17,500

The listed price of the processor is Rs.17500

Given:-

A beats B by 100 m , and B beats C by 150 m 

To find:-

Distance between A and C when A finish the race

Formula used:-

Distance  = speed × time

Calculation**:-**

It is given that A beats B by 100 m which means that when A finishes the race B was 100 m behind him,

⇒Distance travelled by A is 1500 m and B is 1400 m respectively

⇒A : B = 1500 : 1400 

Similarly, the Distance travelled by B and C when B beats C is 1500 m and 1350 m respectively

⇒B : C = 1500 : 1350 

We have to find the relation between A and C so

$\frac{A}{C}=\frac{A}{B}\times\frac{B}{C}$

⇒$\frac{A}{C}=\frac{1500}{1400}\times\frac{1500}{1350}$

⇒$\frac{A}{C}=\frac{15}{14}\times\frac{10}{9}$

⇒A : C = 150 : 126

The above ratio signifies that if A travelled 150 m then C would be travelling 126 m 

So, If A travels 1500 m then C would be travelling 1260 m

⇒When A finishes the race C will be at 1260 m from the

start, the distance from end-

⇒1500 - 1260 = 240

⇒Distance by which A beats C is 240 m

A beats C by 240 m.

Given:-

Two proportional numbers are 27, 18

To find:-

Third proportional number

Concept used:-

Proportionality theorem i.e, a : b :: c : d =  $\frac{a}{b}=\frac{c}{d}$

Calculation:-

Let the third proportional be x

then by proportionality theorem,

27 : 18 :: 18 : x

⇒$\frac{27}{18}=\frac{18}{x}$

⇒$x=\frac{18\times18}{27}$

⇒x = 12

The value of the third proportional is 12.

**Given:-**Average increased by 8 when 100 runs were added in the 10^th inning.

**To find:-**New average

Concept used:- Average = $\frac{sum\ of\ observation}{Total\ number\ observation}$

Calculation**:-**Let the initial average be 'x'

Total run scored till 9^th inning = 9 × x = 9x

After the 10^th inning, the total score will be,

⇒New Total = 9x + 100

⇒New Average = $\frac{9x+100}{10}$---------(i)

As the question suggests that the average is increased by 8,

⇒New average = x + 8-----------(ii)

By equation (i) and (ii)

⇒$\frac{9x+100}{10}$ = x + 8

 ⇒9x + 100 = 10x + 80

⇒x = 20

New average = 20 + 8 = 28

The new average of Raman is 28.

Alternate Method

 when innings were 9 the average was x

After the 10^th inning the average increased by 8, this means '8' is added to each observation (10 innings)

Basically, 80 runs are extra added to the 9x(initial sum of runs)

This simply means 20 was the old average because if in the 10^th inning he had scored 20 runs then the average would've been the same.

⇒20 was the old average and after the 10^th inning, it is increased by 8, 

⇒New average = 28

Given:-

 a + b + c = 0

To find:-

 $\frac{{{a^2} + {b^2} + {c^2}}}{{{a^2} - bc}}$

Formula used:-

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

Calculation**:-** 

As mentioned, a + b + c = 0

⇒ a = - b - c

Let the value of $\frac{{{a^2} + {b^2} + {c^2}}}{{{a^2} - bc}}$ = k

⇒ a2 + b2 + c2 = k(a² - bc)

It is given that a + b + c = 0, On squaring both sides

⇒ a2 + b2 + c2 + 2ab + 2bc + 2ac = 0

Putting, a2 + b2 + c2 = k(a2 - bc)

⇒ k(a2 - bc) + 2ab + 2bc + 2ac = 0

⇒ k(a(- b - c) - bc) + 2ab + 2bc + 2ac =0

⇒ k(- ab - ac - bc) = -2(ab + bc + ac)

⇒ -k(ab + ac + bc) = -2(ab + bc + ac)

By comparing, k = 2

The value of $\frac{{{a^2} + {b^2} + {c^2}}}{{{a^2} - bc}}$ is 2  

Shortcut Trick

a + b + c = 0 Put the value a = 1, b = 0, c = (-1)

Let the value of $\frac{{{a^2} + {b^2} + {c^2}}}{{{a^2} - bc}}$ = k  

⇒ [(1)² + 0 + (-1)²]/[(1)2-0 × (-1)] = 2 /1 = 2

Given:-

radius = 6.3 cm

To find:-

The volume of a solid hemisphere

Formula used:-

 $Volume = \frac{2}{3}\pi r^3$

Calculation**:-**

Radius (r) = 6.3 cm

By using the formula of volume of the solid hemisphere,

⇒ $Volume = \frac{2}{3}\pi r^3$

⇒ $Volume = \frac{2}{3}\pi \left(6.3\right)^3$

⇒ $V=\frac{2}{3}\times\frac{22}{7}\times(6.3)\times(6.3)\times(6.3)$

⇒ V = 523.908 cm3 

The volume of solid hemisphere is 523.908 cm3

**Given:-**Table which represents the data of production of different car type in different year.

To find:- The year in which production of D2 and D3 is equal to the total production of D4 and D5 when taken together

Calculation**:-** The total production of cars of type D2 and D3 taken together in each year are-

In 2012,

(D2)15 + (D3)20 = 35

In 2013,

(D2)11 + (D3)15 = 26

In 2014,

(D2)15 + (D3)12 = 27

In 2015, 

(D2)14 + (D3)13 = 27

The total production of cars of type D2 and D3 taken together in each year are-

In 2012,

(D4)5 + (D5)19 = 24

In 2013,

(D4)8 + (D5)15 = 23

In 2014,

(D4)18 + (D5)9 = 27

In 2015,

(D4)15 + (D5)25 = 40

We can see that in year 2014 the production of car type D2 and D3 is equal to the production of car type D4 and D5 when taken together.

To find:- Which of the following is divisible by 36

Concept used:- Divisibility of 6

Calculation**:-** Ther number divisible by 6 must be divisible by 36 as 6 is the factor of 36,

Divisibility of 6 :- The number divisible by both 2 and 3 must be divisible by 6 subsequently by 36

⇒The option whose unit digit is even and sum of digit is divisible by 3 must be divisible by 6

Checking option 1

136216

unit digit = 6, divisible by 2

Sum of digits = 1 + 3 + 6 + 2 + 1 + 6 = 19

⇒Sum of digit is not divisible by 3 hence it is not divisible by 6

Checking option 2

124344

unit digit = 4, divisible by 2

Sum of digits = 1 + 2 + 4 + 3 + 4 + 4 = 18

⇒Sum of digit is divisible by 3 hence it is divisible by 6

Checking option 3

259884

unit digit = 4, divisible by 2

Sum of digits = 2 + 5 + 9 + 8 + 8 + 4 = 36

⇒Sum of digit is divisible by 3 hence it is divisible by 6

Checking option 4

120816

unit digit = 6, divisible by 2

Sum of digits = 1 + 2 + 0 + 8 + 1 + 6 = 18

⇒Sum of digit is divisible by 3 hence it is divisible by 6

⇒Only option 1 is not divisible by 6 which means it will not be divisible by 36 either.

 

.·.The number which is not divisible by 36 is 136216.

Given: 

Two chords of length of 10 cm and 24 cm  on the same side of center

distance between chords is 7 cm

To find: 

Radius of circle

Concept used:

Pythagorean theorem

Calculation**:**

 

Let A be the point on RS and B be the point on PQ,

⇒ RA  =1/2(RS) = 1/2(24) = 12

⇒ PB = 1/2(PQ) = 1/2(10) = 5 

In Δ OAR, ∠ OAR = 90° 

Using the Pythagoras theorem, 

⇒ OR² = OA² + AR²

⇒ r² = OA² + 12²

⇒ r² = OA² + 144                        (1)

In Δ OBP, ∠ OBP = 90° 

Using the Pythagoras theorem, 

⇒ OP² = PB² + OB²

⇒ r² = 5² + (OA + 7)²

⇒ r² = 25 + OA² + 49 + 14OA 

⇒ r² = OA² + 14OA + 74            (2)

Equating equations 1 & 2, we get, 

⇒  OA² + 144  = ​OA² + 14OA + 74  

⇒ 70 = 14OA

⇒ OA = 70/14 = 5 

Now, in Δ OAR, 

⇒ OR² = OA²  + AR² 

⇒ ${r}^2={5}^2+144$

⇒ r² = √169 

⇒Radius = 13 cm

The radius of the circle is 13 cm

Given:- 26 kg of rice Rs.20/kg is mixed with 30kg of rice Rs.36/kg

To find:- Profit percentage

Formula used:- $Profit=\frac{Gain}{CP}$

Calculation**:-** Let the quantity of rice-1 and rice-2 be n₁ and n₂ respectively,

⇒n₁ = 26 kg and n₂ = 30 kg

Let the price of rice-1 and rice-2 be w₁ and w₂ respectively,

⇒w₁ = Rs. 20/kg and w₂ = Rs. 36/kg

⇒CP = 26 × 20 + 30 × 36

⇒CP =Rs. 1600

The selling price of 56Kg will be, 56 × 30

⇒SP =Rs. 1680

Profit % = $\frac{gain}{CP}\times 100$

gain = SP - CP = 1680 - 1600

Gain = Rs.80

Profit % = $\frac{80}{1600}\times 100$

Profit % = 5

The percentage of profit sellar makes is of 5%

To find:- 52 – 5[2 + 3{2 – 2(5 – 3) + 5} – 10] ÷ 8 

Concept used:- BODMAS[Brackets, Order, Division, Multiplication, Addition, Subtraction]

Calculation**:-** Calculating by using the rule of BODMAS- 

52 – 5[2 + 3{2 – 2(5 – 3) + 5} – 10] ÷ 8 

⇒52 – 5[2 + 3{2 – 2(2) + 5} – 10] ÷ 8 

⇒52 – 5[2 + 3{2 – 4 + 5} – 10] ÷ 8 

⇒52 – 5[2 + 3{3} – 10] ÷ 8 

⇒52 – 5[2 + 9 – 10] ÷ 8 

⇒52 – 5 ÷ 8 

⇒52 – 0.625

⇒51.375

The value of the given expression is 51.375

**Given:-**Total number of students = 52

To find:- Percentage of students who took the test and secured grade D

Formula used:- percentage = $\frac{parts}{whole}\times100$

Calculation**:-** Total number of students who secured grade D is 4

The total number of students who took the test is 52

⇒In terms of percentage(Student secured D with respect to the total number of students) =  $\frac{Number\ of\ students}{Total\ number\ of\ student}\times 100$

⇒percentage of students who took the test and secured a D grade = $\frac{4}{52}\times 100$ = 7$9 \over 13$%

There are 7$9 \over 13$% of the total students who took the test and secured grade D

Mistake Points Here though the data is given in Percent/Ratio form, 

We will consider it as a whole value.

As it will get canceled out, while calculating the % value.

Given:- average speed of train = 60km/h

To find:- Distance covered by train

Formula used:- Distance = speed × time

Calculation:- Let the time taken by train to reach destination be T ,

Let the distance travelled by train be D

By using the formula, distance = speed × Time

when train travels at an average speed of 60 km/h

⇒D = 60 × T ------(i)

when the train travels at an average speed of 45 km/h, takes 20 minutes late,

⇒D = 45 × (T + $\frac{20}{60}$)

⇒D = 5 × (3T + 1) --------(ii)

Equating (i) and (ii)

⇒ 60 × T = 15 × (3T + 1)

⇒60T = 45T + 15

⇒T = 1 hour

Substituting the value of T in the equation-(i)

⇒D = 60 × T = 60 × 1

⇒D = 60 

.·.Distance covered by the train is 60km

Shortcut Trick

Surface Area of Sphere = 616 cm² ⇒ 4πr² = 616 ⇒ r = 7 cm.

Volume of Sphere = Volume of Cone.

(4/3)πr³ = (1/3)πR²h ⇒ 4 × 7³ = R² × 28.

4 × 343 = R² × 28 ⇒ R² = 1372 / 28 = 49 ⇒ R = 7 cm.

Diameter of Cone = 2 × R = 2 × 7 = 14 cm.

∴ The correct answer is 14 cm.

Alternate Method

Given: Surface area of sphere = 616 cm², Height of cone (h) = 28 cm.

Formula Used: Surface Area (Sphere) = 4πr²; Volume (Sphere) = (4/3)πr³; Volume (Cone) = (1/3)πR²h.

Step 1: Find radius of the sphere (r)

⇒ 4 × (22/7) × r² = 616

⇒ r² = (616 × 7) / (4 × 22) = 616 × 7 / 88

⇒ r² = 7 × 7 = 49 ⇒ r = 7 cm

Step 2: Equate volumes to find radius of cone (R)

⇒ (4/3) × π × (7)³ = (1/3) × π × R² × 28

⇒ 4 × 343 = R² × 28

⇒ R² = (4 × 343) / 28 = 343 / 7 = 49

⇒ R = 7 cm

Step 3: Calculate diameter of the cone

⇒ Diameter = 2 × R = 2 × 7 = 14 cm

∴ The correct answer is 14 cm.

Additional Information

Recasting Rule

When one solid is melted and recast into another, the Volume remains constant regardless of shape change.

Sphere Properties

Volume = (4/3)πr³ and Total Surface Area = 4πr². For a hemisphere, TSA = 3πr².

Cone Properties

Volume = (1/3)πR²h and Slant Height (l) = √(R² + h²).

Given:- Equation of square, 4x2 - 12x + 9

To find:- Side of the square

Calculation:- The given equation is, 4x2 - 12x + 9

Using mid-term split method to find the factors of the equation,

Since, the above equation is of the square then it must have two same factors and the factor will represent its side

⇒4x2 - 12x + 9 

⇒4x2 - (6 + 6)x + 9 

⇒4x2 - 6x - 6x + 9

⇒2x(2x - 3) - 3(2x -3)

⇒(2x - 3)(2x - 3)

⇒(2x - 3)²

The side of square is (2x-3)

GIVEN:

Volume = 4851$cm^3$ 

FORMULA USED:

Volume = $\frac{4}{3}$ $\pi r^3$  where r = radius

CALCULATION:

Volume of the sphere = $\frac{4}{3} \pi r^3$ 

⇒ 4851 = $\frac{4}{3}$ × $\frac{22}{7}$ × $r^3$ 

⇒  $441\times 21\over8$ = $r^3$ 

⇒ r =  21/2 = 10.5 cm.

Hence, radius is10.5cm.

GIVEN:

Total number of books = 2500.

CONCEPT USED:

100% represents Total number of books

CALCULATION:

   State. B = 24%  and State C = 18%

⇒ Since, 100% = 2500 

⇒ 24% = (2500/100)× 24 = 600 

⇒ 18% = (2500/100) × 18 = 450.

Hence, Total number of books in B and C together = ( 600 +450) = 1050.

State D = 25% and State E= 20%

⇒ Now, 25% = (2500/100)× 25 = 625.

⇒ 20% = (2500/100)× 20 = 500.

Hence, Total number of books in  D and E together = ( 625 +500) = 1125.

Now, Difference = ( books of D and E ) - (books of B and C) = (1125 - 1050) = 75.

Hence, the difference is 75.

GIVEN:

(a -b) = 3

$( a^3 - b^3)$ = 999.

FORMULA USED:

$( a^3 - b^3 ) = ( a -b) ( a^2 + b^2 +ab)$

$( a + b)^2 = a^2 + b^2 +2ab.$

$( a -b)^2 = a^2 + b^2 -2ab.$

CALCULATION:

Now, ( a -b) = 3 on squaring both the sides we get,

⇒ $( a-b)^2 = 3^2$ 

⇒ $a^2 + b^2 - 2.a.b$ = 9.

⇒ $( a^2 + b^2) = 9 +2 ab.$ ...........................(1)

⇒ $( a^3 - b^3) = ( a-b)( a^2 +b^2 +ab)$                      .....  put the value of equation 1 in the place of$( a^2 +b^2)$

⇒ 999 = 3 × ( 9+2ab +a b )

⇒ 333 = 9 +3 a b  ⇔ ab = 108.

⇒ Now, as,$( a+b )^2 = a^2 +b^2 + 2ab.$ = ( 9 +2ab +2ab) = ( 9 +4ab) =( 9 + 4 × 108) = 441.

⇒ Hence, ( a+ b) =  21.

Now, $( a^2 - b^2 ) = ( a+b ) ( a - b)$ = 21 × 3 = 63.

Hence, the value is 63.