SSC CHSL Previous Paper 4 (Held On: 18 March 2020 Shift 1) (Previous Year Paper)

The correct answer is 'A new version of Shakespeare's 'Macbeth' has been performed by the students'.

Key Points

• The given sentence is in Active Voice. As per the given question we have to change it into Passive Voice.
• The structure of the given sentence as follows: 
• Subject+has/have+V3+Object. (Active Voice)
• Subject (Objective Case)+has/have+V3+Object (Subjective Case). (Passive Voice)
• Example:
• We have finalized the site for our new factory. (Active Voice)
• The site for our new factory has been finalized by us. (Passive Voice)
• The subject of the given sentence is 'The students'. And the object of the given sentence is 'A new version'.
• The subject will be put in place of the object and the object will be put in place of the subject.
• And 'have performed' will be changed into 'has been performed'.
• Therefore, as per the points mentioned above, we find that the correct answer is Option 3. 
  

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Correct Sentence: A new version of Shakespeare's 'Macbeth' has been performed by the students.

Additional Information

• The transformation of imperative sentences:
• If someone gives a command, do request, express a desire or wish, in sentences then such sentences are called an imperative sentence.
• If the notion of a sentence is 'Command' then structure: 
• V1 + Object. (Active Voice)
• Let + subject + be + V3. (Passive Voice)
• Example: 
• Teach your students. (Active Voice)
• Let your students be taught. (Passive Voice)
• If the notion of a sentence is that of an 'Order', then structure: 
• V1 + Object. (Active Voice)
• You are ordered + V 1+ Object. (Passive Voice)
• Example: 
• Drive the car. (Active Voice)
• You are ordered to drive the car. (Passive Voice)

The correct answer is 'Shanti asked me why I had kept that smartphone in the bin'. 

Key Points

• The given sentence is in Direct Speech. As per the given question we have to change it into Indirect Speech.
• The process of transformation as follows:
• There will be no change in reporting verb. Because we know that the given sentence is interrogative with a question word.
• Comma and inverted commas will be removed.
• 'Why' will be used as a conjunction because we know that in an interrogative sentence with a question word the question word itself becomes the conjunction.
• 'You' will be changed into 'I' because in 'direct to indirect transformation' second-person pronoun (you) is changed according to the object of reporting verb that is 'me'.
• 'Did keep' will be changed into 'Had kept'.
• Therefore, as per the points mentioned above, we find that the correct answer is Option 3. 
  

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Correct Sentence: Shanti asked me why I had kept that smartphone in the bin. 

Additional Information

• Transformation of Imperative Sentence:
• 'Orders', 'Command', 'Request', 'Advice' are given in these kinds of sentences.
• Reporting Verb is changed into 'ordered, advised, requested, commanded, beg, forbade, suggested, proposed, assured, asked, reminded, warned, agreed, refused, promised', etc.
• Inverted Commas will be removed. And conjunction 'to/not to' will be used. 
• Example: 
• He said, "John, please bring me my towel." (Direct Speech)
• He requested John to bring him his towel. (Indirect Speech)

The correct answer is 'Completely disappear'. 

Key Points

• The most appropriate meaning of the given idiom is 'Completely disappear'.
• Vanish into thin air: to disappear suddenly and completely
• Example: The ship simply vanished into thin air.
• Therefore, as per the points mentioned above, we find that the correct answer is Option 3. 
  

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Correct Answer: Completely disappear. 

Additional Information

• Let's look at some idioms of the word 'Cast':

The correct answer is 'Defenition'. 

Key Points

• In the given question 'Defenition' is misspelt.
• The correct spelling of it is 'Definition'.
• It means a statement that explains the meaning of a word or phrase. 
• Example: The legal definition of what is and what is not pornography is very unsatisfactory.
• Therefore, as per the points mentioned above, we find that the correct answer is Option 2. 
  

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Correct Answer: Defenition.

Additional Information

• Let's look at the meaning of the other words given in the Options:

The correct answer is 'both'. 

Key Points

• The most appropriate fill in the blank for the given sentence is 'both'.
• We use both to refer to two things or people together. We use both … and to emphasize the link between two things. This puts more emphasis on the linkage than and alone. 
• Example: Both Britain and France agree on the treaty. (stronger link than Britain and France agree on the treaty.)
• Therefore, as per the points mentioned above, we find that the correct answer is Option 3. 
  

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Correct Sentence: "The internet is fast becoming trusted by both children and adults..."

The correct answer is 'source'. 

Key Points

• The most appropriate option for blank 2 is 'source'.
• Here, 'source of information' means 'someone or something that supplies information'. 
• Example: The journalist refused to reveal her sources (= say who had given the information to her).
• Therefore, as per the points mentioned above, we find that the correct answer is Option 3. 
  

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Correct Sentence: "The internet is fast becoming trusted by both children and adults as reliable and accurate source of information".

Additional Information

• Let's look at the meaning and examples of the other words given in the option:

The correct answer is 'access'. 

Key Points

• The most appropriate fill in the blank for the given sentence is 'access'.
• It means 'to be able to use or obtain something such as a service'. 
• Example: It needs to be easier for small businesses to access government resources and advice.
• Therefore, as per the points mentioned above, we find that the correct answer is Option 3. 
  

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Correct Sentence: "Through the internet children now have access to an almost..."

Additional Information

• Let's look at the meaning and examples of the other options:

The correct answer is 'interaction'. 

Key Points

• The most appropriate fill in the blank for the given sentence is 'interaction'.
• It means 'an occasion when two or more people or things communicate with or react to each other'. 
• Example: There's not enough interaction between the management and the workers.
• Therefore, as per the points mentioned above, we find that the correct answer is Option 1. 
  

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Correct Sentence: "Through the internet children now have access to an almost endless supply of information and opportunity for interaction".

Additional Information

• Let's look at the meaning and examples of the other options:

The correct answer is 'unsupervised'. 

Key Points

• The most appropriate fill in the blank for the given sentence is 'unsupervised'.
• It means 'without anyone watching to make sure that nothing dangerous or wrong is done or happening'. 
• Example: Young children should not be left alone with animals unsupervised.
• Therefore, as per the points mentioned above, we find that the correct answer is Option 1. 
  

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Correct Sentence: "However, there can be real risks and dangers for an unsupervised child". 

Additional Information

• Let's look at the meaning and examples of the other options:

The correct answer is 'As long as'. 

Key Points

• The most appropriate option to substitute the underlined segment in the given sentence is 'As long as'.
• As long as: during the whole time that
• Example: They have been there as long as anyone can remember.
• We use 'as long as' to refer to the intended duration of a plan or idea, most commonly referring to the future.
• We always use the present simple to refer to the future after 'as long as'.
• Therefore, the correct answer is Option 4. 
  

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Correct Sentence: As long as you are here with me, who cares about the outcome of the issue. 

Mistake Points

• We tend to go for Option 1 but it would be wrong because 'So long as' is the correct conjunction but here in the given option only 'So long' is used.
• Similarly, Option 2 is also grammatically incorrect because the 'As long to' kind of conjunction does not exist.
• Therefore, the correct answer is Option 4.

Additional Information

• The structure so…that
• It is used to show cause and effect. 
• Example: She was very tired. She could not walk.
• We can combine these two sentences using so…that.
• Example: She was so tired that she could not walk.

The correct answer is 'Joyous'. 

Key Points

• The most appropriate synonym of the given word 'Jovial' is 'Joyous'.
• Let's look at the meaning and examples of the given options:

The correct answer is 'Comemorate'. 

Key Points

• In the given question 'Comemorate' is incorrectly spelt.
• It's correct spelling is 'Commemorate'.
• Let's look at the meaning and examples of the given words:

The correct answer is 'Cardinal'. 

Key Points

• The most appropriate one-word for the given group of words is 'Cardinal'.

The correct answer is 'provided that'. 

Key Points

• The most appropriate option to substitute the underlined segment in the given sentence is 'provided that'.
• The conditional 'provided that' means 'if, or only if'. 
• Example: ILO conventions allow schoolchildren to work during their holidays and free time provided that strict safety measures are taken.
• Here, in the given question 'provided with' is incorrectly used because the correct usage is 'provide sb with sth'. It means 'to give something to a person to make it available for them to use'. 
• Example: Customers, suppliers, and contractors will be provided with the specialist training they need to work on the project.
• Therefore, as per the points mentioned above, we find that the correct answer is Option 4. 
  

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Correct Sentence: Before it was modified, the Law provided that the owner could take possession of the goods at any time. 

Additional Information

• Let's look at some different usage of 'provide' when it is used as a verb:
• When we use 'provide' as a verb it means 'to give something to a person, company, or organization, or to make it available for them to use'. 
• provide sth for sb/sth: 
• Example: The enhanced cash flow will also provide funds for expansion into television.
• provide sth to sb:
• Example: Our digital imaging center provides a 3D scanning service to a wide range of customers in Europe, the Middle East, and Africa.
• be provided by sb/sth:
• Example: According to Census data provided by the state Department of Labor and Industry, Hispanic workers constitute about 3% of the workforce.

The correct answer is 'CBDA'. 

Key Points

• The given sentence is an example of a Sentence Jumble.
• The first statement of a sentence jumble question is usually an independent general statement, a noun, a universal fact, the starting of an incident, or it starts with 'most' or 'once'.
• Part C will be the first sentence because it is talking about a past happening. It talks about a girl named 'Tejaswini' who is known in her village for her devotion to the Lord.
• Part B will be the second sentence because it is connected with Part B. And further explains 'How many times she visits the temple a day'.
• Part D will be used next because it tells us about her visit in the morning to the temple.
• And finally, Part A will be used because it tells us about her visit to the temple in the evening.
• Therefore, as per the points mentioned above, we find that the correct answer is Option 2. 
  

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Correct Sentence: "Tejaswini was known in the village for her devotion to the Lord. She would go to the Lord’s temple twice a day. In the morning, she would take with her a pot of milk and a bunch of flowers as an offering. In the evening, Tejaswini would sing songs praising the Lord."

The correct answer is 'BDCA'. 

Key Points

• The given sentence is an example of a Sentence Jumble.
• The first statement of a sentence jumbled question is usually an independent general statement, a noun, a universal fact, the starting of an incident, or it starts with 'most' or 'once'.
• Part B will be the first sentence because it is talking about a past happening. It is talking about a large mango tree in a forest.
• Part D will be the second sentence because it further explains that tree. How the tree looks etc.
• Part C will be the next sentence because it is connected with Part D. Part D talks about the physical appearance of that tree. And Part D further explains it.
• Finally, Part A will be used because Part C starts with 'The Branches' and Part A further explains an incident that 'One day' a crow from the west landed on one of the branches of that mango tree.
• Therefore, as per the points mentioned above, we find that the correct answer is Option 4. 
  

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Correct Sentence: "There was a large mango tree deep inside a thick forest. Its branches spread in all directions, lobbing a large shadow on the ground. The branches were full of leaves, which crackled when the storm blew. One day, a strange crow from the west, landed on one of the branches."

The correct answer is 'fracture'. 

Key Points

• The most appropriate word to fill in the blank is 'fracture'.
• It means 'If something hard, such as a bone, fractures, or is fractured, it breaks or cracks'.
• 'Bone' is fractured not 'contaminated or any other options' given in the question.  
• Example: She fractured her skull in the accident.
• Therefore, as per the points mentioned above, we find that the correct answer is Option 1.  
  

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Correct Sentence: I had a broken bone in the hand which the doctor called a fracture and suggested immediate surgery. 

Additional Information

• Let's look at the meaning and examples of the other words given in the Options:

The correct answer is 'Amateur'. 

Key Points

• The most appropriate one-word for the given group of words is 'Amateur'.

The correct answer is 'Show admiration'.  

Key Points

• The most appropriate meaning of the given idiom 'Take one's hat off to someone' is 'Show admiration'.
• Take one's hat off to someone: To express admiration; a form of applause.
• Example: We should take off our hats to them and wish them godspeed.
• Therefore, as per the points mentioned above, we find that the correct answer is Option 2. 
  

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Correct Answer: Show admiration

Additional Information

• Let's look at the meaning and examples of some idioms of the word 'Play':

The correct answer is 'Clear'. 

Key Points

• The most appropriate antonym of the given word 'Obscure' is 'Clear'.
• Let's look at the meaning and examples of the given options:

The correct answer is 'Celestial'.

Key Points

• The most appropriate antonym of the given word 'Earthly' is 'Celestial'.
• Let's look at the meaning and examples of the given options:

The correct answer is 'explore'. 

Key Points

• The most appropriate word to fill in the blank is 'explore'.
• It means 'to search and discover (about something)'. 
• Example: The best way to explore the countryside is on foot.
• Therefore, as per the points mentioned above, we find that the correct answer is Option 1. 
  

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Correct Sentence: She has shown a great interest towards space science, since her early childhood and a passion to explore the outer space.  

Additional Information

• Let's look at the meaning and examples of the other words given in the option:

The correct answer is 'who have been paying back'. 

Key Points

• The given sentence is grammatically incorrect.
• Here, 'who has been paying back' should be used instead of 'who have been paying back'.
• The given sentence is an example of the relative pronoun.
• Since the antecedent of the relative pronoun is a person that is 'Thomas'.
• 'Thomas' is a singular noun. Hence, a singular helping verb that is 'has' should be used.
• Therefore, as per the points mentioned above, we find that the correct answer is Option 1. 
  

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Correct Sentence: Thomas is a man of word who has been paying back the borrowed money in installments. 

Additional Information

• Let's have a look at different usage of 'who':

The correct answer is 'out in fear'. 

Key Points

• The given sentence is grammatically incorrect.
• Here, 'out of fear' should be used instead of 'out in fear'.
• In the given question 'Out of fear' means 'a feeling of distress, apprehension, or alarm caused by impending danger, pain, etc. 
• Example: The surgeon comes to look at Paul, who refuses to be chloroformed out of fear that they will amputate his leg.
• Therefore, as per the points mentioned above we find that the correct answer is Option 3. 
  

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Correct Sentence: Though she was able to finish the work on time, she couldn't do that out of fear.

The correct answer is 'Selective'. 

Key Points

• The most appropriate synonym of the given word 'Choosy' is 'Selective'.
• Let's look at the meaning and examples of the given options:

According to given conditions,

Rahul and Sheela are brother and sister.

Hence, ‘Brother-Sister’ is the correct answer.

1. f, f, e, f, f

Efge / effgge / eefffggg

After filling the blanks we observe a particular pattern is formed.

When the series is divided into three parts, we observe that in 1^st part ‘f’ and ‘g’ appears once, in the 2^nd part ‘f’ and ‘g’ appears twice and in the 3^rd part ‘f’ and ‘g’ appears thrice.

2. f, g, f, g, g

Efge / efgggf / eegfgggg

No pattern is formed.

3. g, f, e, f, f

Egge / effgge / eefffggg

No pattern is formed.

4. f, f, f, f, f

Efge / effggf / eefffggg

No pattern is formed.

Hence, ‘f, f, e, f, f’ is the correct answer.

Age of Rishabh = 10 years

If his mother is 20 years older than him.

So, age of his mother = (10 + 20) years

= 30 years

If his mother is 6 years younger than his father.

Age of his father = (30 + 6) years

= 36 years

Hence, ’36 years’ is the correct answer.

The pattern followed here is:

According to the alphabetical positions of the letters,

Reverse letters of each letter is shown in the above table.

These reverse letters when written from right to left, we get GLXRIKZ which is the code for the word APRICOT.

Similarly,

These reverse letters when written from right to left, gives the code VTMZIL.

Hence, ‘VTMZIL’ is the correct answer.

The number of triangles in the given figure is shown below:

Hence, ‘8’ is the correct answer.

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

Hence, ‘option 2’ is the correct answer.

The pattern followed here is:

According to the alphabetical positions of the letters,

1. US → U – 2 = S

2. OM → O – 2 = M

3. AY → A – 2 = Y

4. BE → B + 3 = E

Hence, ‘BE’ is the odd one out.

The logic is:

9 : 121 → 9 : (9 + 2)²

Similarly,

7 : ? → 7 : (7 + 2)²

=> ? = 9²

=> ? = 81

Hence, ‘81’ is the correct answer.

The option which is the exact mirror image of the given alphanumeric figure when the mirror is held at the right side is shown below:

Hence, ‘option 3’ is the correct answer.

1. • and –

Given expression: 4 ÷ 6 + 9 - 48 × 8 = 27

After replacing the symbols, we get:

4 ÷ 6 – 9 + 48 × 8 = 27

=> 0.67 – 9 + 384 = 27

=> 384.67 – 9 = 27

=> 375.67 ≠ 27

2. • and ×

Given expression: 4 ÷ 6 + 9 - 48 × 8 = 27

After replacing the symbols, we get:

4 ÷ 6 × 9 - 48 + 8 = 27

=> 6 – 48 + 8 = 27

=> 14 – 48 = 27

=> -34 ≠ 27

3. ÷ and +

Given expression: 4 ÷ 6 + 9 - 48 × 8 = 27

After replacing the symbols, we get:

4 + 6 ÷ 9 - 48 × 8 = 27

=> 4 + 0.67 – 384 = 27

=> 4.67 – 384 = 27

=> -379.33 ≠ 27

4. ÷ and ×

Given expression: 4 ÷ 6 + 9 - 48 × 8 = 27

After replacing the symbols, we get:

4 × 6 + 9 - 48 ÷ 8 = 27

=> 24 + 9 – 6 = 27

=> 33 – 6 = 27

=> 27 = 27

Hence, ‘÷ and ×’ is the correct answer.

The pattern followed here is:

According to the alphabetical positions of the letters,

INTEX : EINTX → Letters are arranged in alphabetical order.

Similarly,

SMALL : ? → Letters when arranged in alphabetical order will give ALLMS.

Hence, ‘ALLMS’ is the correct answer.

1. acbdecdeaacecdd

abacbcba / decdddea / abcdacba / aecddcba

No pattern is formed.

2. abeaacdecdeecdd

ababecba / aacdddec / abcddeba / aecddcba

No pattern is formed.

3. abbaacdeecdecdd

ababbcba / aacdddee / abcdcdba / aecddcba

No pattern is formed.

4. cddabccbadcbcdd

abcddcba / abcddcba / abcddcba / abcddcba

The series when divided into four equal parts, we observe that a particular pattern is repeated every single time.

Hence, ‘cddabccbadcbcdd’ is the correct answer.

LOGIC: Sum of position values of all the terms – Number of terms = Code

The pattern followed here is:

According to the alphabetical positions of the letters,

Similarly,

Hence, ‘62’ is the correct answer.

The pattern followed here is:

According to the alphabetical positions of the letters,

Note: Vowel ‘A’ is moved eight steps forward and vowel ‘E’ is moved ten steps forward, while consonants are moved three steps backward.

Similarly,

Hence, ‘IKDOI’ is the correct answer.

The logic is:

(2nd number - 1st number) × 8 = 3rd number

(2, 6, 32) → (6 – 2) × 8 = 32

Similarly,

(8, 18, 80) → (18 – 8) × 8 = 80

Hence, ‘(8, 18, 80)’ is the correct answer.

Words are arranged in the order of their technological advantages.

1. Letter - By letter, we can send information to someone.

3. Telephone - We can call to someone. 

4. Smartphone - We can voice call, video call, and text messages to one person/more than one person.

2. Satellite - It is used in a wide variety of applications such as global mobile communication, private business networks, Long distance telephone transmission, weather forecasting, radio/TV signal broadcasting, connecting remote areas, television distribution etc.

Hence, ‘1, 3, 4, 2’ is the correct answer.

The logic is:

Hence, ‘73’ is the correct answer.

The logic is:

Chair : Furniture → Chair is categorized as a furniture.

Similarly,

Letter paper : stationery → Letter paper is categorized as a stationary item.

Hence, ‘Letter paper :  Stationary ’ is the correct answer.

People who like both tea and coffee, but do NOT like juice is shown below:

Hence, ‘12’ is the correct answer.

The following figure when folded to form a cube, then the faces opposite to each other is shown below:

1 dot will be on the face opposite to the face having 2 dots.

Hence, ‘1’ is the correct answer.

The option figure that will come next in the following series is shown below:

Dividing the series into two parts we observe that:

Images at odd position: Rotating in anti-clockwise direction.

Images at even position: Rotating in anti-clockwise direction.

So, the image that will complete the series will be similar to ‘option 3’ as shown below.

Hence, ‘option 3’ is the correct answer.

The paper when unfolded will appear as shown below:

Hence, ‘option 2’ is the correct answer.

Total work is 90 units.

A worked for 6 days and efficiency of A is 3.

So, total work done by A in 6 days = (6 × 3)

= 18 units

Remaining work = (90 – 18) units

= 72 units

Efficiency of A and B is 3 and 5 respectively.

So, together they will complete the remaining work in [72 ÷ (3 + 5)] days

= [72 ÷ 8] days

= 9 days

 

 Number of days in which the whole work was completed = (6 + 9) days

= 15 days

Hence, ‘15’ is the correct answer.

1. 14 and 18, + and –

Given expression: 16 × 18 + 2 - 14 ÷ 3 = 38

After replacing the numbers and symbols, we get:

16 × 14 – 2 + 18 ÷ 3 = 38

=> 224 – 2 + 6 = 38

=> 230 – 2 = 38

=> 228 ≠ 38

2. 14 and 18, + and ×

Given expression: 16 × 18 + 2 - 14 ÷ 3 = 38

After replacing the numbers and symbols, we get:

16 + 14 × 2 - 18 ÷ 3 = 38

=> 16 + 28 – 6 = 38

=> 44 – 6 = 38

=> 38 = 38

3. 16 and 3, - and ÷

Given expression: 16 × 18 + 2 - 14 ÷ 3 = 38

After replacing the numbers and symbols, we get:

3 × 18 + 2 ÷ 14 - 16 = 38

=> 54 + 0.14 – 16 = 38

=> 54.15 – 16 = 38

=> 38.15 ≠ 38

4. 16 and 14, - and ×

Given expression: 16 × 18 + 2 - 14 ÷ 3 = 38

After replacing the numbers and symbols, we get:

14 - 18 + 2 × 16 ÷ 3 = 38

=> 14 – 18 + 10.67 = 38

=> 24.67 – 18 = 38

=> 6.67 ≠ 38

 Hence, ‘14 and 18, + and ×’ is the correct answer.

Concept:

1. We have to check the conclusions are 100% match with the statement. If the conclusion is 100% match with the given statement then the conclusion is true. If there is a 1% chance of not following the statement then the conclusion is considered to be false.

2. The possibility case can be considered to be "false" in "definite case".

3. In Possibility case, either "can" or "possible" word must be present.

Explanation

The least possible Venn diagram is:

I. All cups are doors. → False (It can be possible but it is not definite)

II. All doors are cups. → True (It is definite)

III. All teachers are doors. → False (It can be possible but it is not definite)

IV. Some cups are teachers. → True (It is definite)

Hence, only conclusions II and IV follow.

Additional Information

GIven:

$\frac{{1 - \tan A}}{{1 + \tan A}} = \frac{{\tan 3^\circ \tan 15^\circ \tan 30^\circ \tan 75^\circ \tan 87^\circ }}{{\tan 27^\circ \tan 39^\circ \tan 51^\circ \tan 60^\circ \tan 63^\circ }}$

Formula used:

If  A + B = 90° 

Then tan A × tan B = 1

Tanθ = 1/cotθ 

Tan 30° = 1/√3

Tan 60° = √3

Calculation:

Using the above formula 

⇒ tan 3° × tan 87° = 1

⇒ tan 15° × tan 75° = 1 

⇒ tan 27° × tan 63° = 1

⇒ tan 39° × tan 51° = 1

Put values in the given equation we get

⇒ (1 - tan A)/(1 + tan A) = (1/√3)/√3

⇒ (1 - tan A)/(1 + tan A) = 1/3

⇒ 3 - 3tan A = 1 + tan A

⇒ 4 tan A = 2

⇒ tan A = 2/4

⇒ cot A = 2

∴ The value of cot A is 2.

Given:

The number is divivible by 2, 5 and 10

Concept used:

Number must be divisible by LCM of the numbers

Calculation:

Lcm of 2,5 and 10

⇒ LCM = 10

Number must have a zero at the end to be divisble by 10

So, Option 4 is divisible by 2,5 and 10

∴ 19400 is divisible by 2, 5 and 10.

Shortcut Trick Number which is divided by 10 should have 0 at its unit digit. And out of all the question we have one such option available.

GIven:

a SinA + b cosA = c

Formula used:

If a Sinθ + b Cosθ = c

Then, a Cosθ – b Sinθ = $√ {{a^2} + {b^2} - {c^2}}$

Calculation:

Apply above formula on a SinA + b cosA = c 

We get

⇒ a CosA – b SinA = $√ {{a^2} + {b^2} - {c^2}}$

∴ a CosA – b SinA is equla to  $√ {{a^2} + {b^2} - {c^2}}$

 Alternate Method

Given, a sinA + b cosA = c             -----(1)

Let, x = a cosA - b sinA                  -----(2)

On Square equation (1)

⇒ a² sin²A + b²cos²A + 2a sinA × b cosA = c²      -----(3)

On Square equation (2)

⇒ a² cos²A + b² sin²A - 2a cosA × b sinA = x²      -----(4)

On adding (3) and (4)

⇒ x² + c² = a² + b²

⇒ x = √(a² + b² - c²)

∴ a CosA – b SinA is equla to  √(a2 + b2 - c2).

Given:

Formula used:

Percentage = (part value/ whole value) × 100

Calculation:

Percentage of students qualified in college B

⇒ (420/560) × 100 

⇒ 75%

Percentage of students qualified in college D

⇒ (432/480) × 100

⇒ 90%

Difference between there percentage is 

⇒ 90% – 75% = 15%

∴ The difference between the percentage of students that qualified, from college B and D is 15%.

Given:

Formla used:

Percentage = (part value/whole value) × 100

Calculation:

Total number of student = 9800

Students in college B is 

⇒ (9800/100) × 16 = 1568

Ratio of Boys and Girls is 5 : 9

⇒ 14x = 1568

⇒ x = 1568/14 = 112

⇒ 9x = 9 × 112 = 1008

∴ Number of girls in college B is 1008.

GIven:

Ravi starts from his house to school at 8:20 a.m with speed of 10 km/h and reaches 8 minutess late

If he ride the cycle at speed of 16 km/h he reaches his school 10 minutes early

Formula used:

Calculation:

Let the distance between home and school be x km

And time take to reach school at the time be t hrs

With the speed of 10 km/h, Ravi reaches 8 minutes late

⇒ x/10 = t + 8/60

⇒ x/10 = t + 2/15      ----(1)

With the speed of 16 km/h, Ravi reaches 10 minutes early

⇒ x/16 = t – 10/60

⇒ x/16 = t – 1/6      ----(2)

Now, Subtract (2) from (1)

⇒ x/10 – x/16 = t + 2/15 – t + 1/6

⇒ (5x – 8x)/80 = (4 + 5)/30

⇒ 3x × 30 = 9 × 80

⇒ x = 720/90

⇒ x = 8 km

Now, put the value of x in (1)

⇒ 8/10 = t + 2/15

⇒ t = 8/10 – 2/15

⇒ t = (24 – 4)/30 

⇒ t = 2/3

To convert it into minutes by 60

⇒ (2/3) × 60 = 40 minutes

College starts at 8:20 + 40 mins = 9 A.M

∴ College starts at 9 A.M

 

⇒ 8x = 48 minutes

Ravi is 8 min late, the correct time will be

⇒ 48 - 8 = 40 minutes

College starts at 8:20 + 40 mins = 9 A.M

∴ College starts at 9 A.M

Given: 

Value of x is square of $(\frac{2}{5}of6\frac{1}{4} \div \frac{3}{7}) of 1\frac{2}{7}$/ $11\frac{1}{4}$

Formula used:

Follow BODMAS rule to solve this question, as per the order given below, 

Calculation:

Value of x is square of $(\frac{2}{5}of6\frac{1}{4} \div \frac{3}{7}) of 1\frac{2}{7}$/ $11\frac{1}{4}$

⇒ $(\frac{2}{5}of6\frac{1}{4} \div \frac{3}{7}) of 1\frac{2}{7}$/ $11\frac{1}{4}$

⇒ $(\frac{2}{5}of\frac{25}{4} \div \frac{3}{7}) of \frac{9}{7}$/ $\frac{45}{4}$

⇒ $(\frac{5}{2} \div \frac{3}{7}) × \frac{9}{7}$/ $\frac{45}{4}$

⇒ $\frac{35}{6} × \frac{9}{7} \div \frac{45}{4}$

⇒ $\frac{35}{6} × \frac{9}{7} × \frac{4}{45}$

⇒ 2/3

Value of x is square of 2/3 = 4/9

Value of 81x is 

⇒ 81 × (4/9) = 36

∴ value of 81x is 36

Given:

In 2013 % increase = 26.88%

In 2017 % increase = 12.64%

In 2015 % increase = 22.96%

In 2018 % increase = 17.46%

Calculation:

According to the question,

In 2013 profit % is the greatest.

∴ The greatest year in profit percent is 2013.

Given:

${[{{ {(\frac{2}{3})^3}} ^{(2x + 3)}}]^{\frac{{ - 3}}{4}}} = {[{{ {(\frac{2}{3})^{\frac{2}{3}}}} ^{(3x + 7)}}]^{\frac{{ - 6}}{5}}}$

Formula Used:

(a^b)^c = a^{b × c}

Calculation:

(2/3)^{3 × (2x + 3) × (–3/4)} = (2/3)^{(2/3) × (3x + 7) × (–6/5)}

On comparing the powers,

⇒ 3 × (2x + 3) × (–3/4) = (2/3) × (3x + 7) × (–6/5)

⇒ 45 × (2x + 3) = 16 × (3x + 7)

⇒ 90x + 135 = 48x + 112

⇒ 90x – 48x = 112 – 135

⇒ 42x = –23

$\sqrt {2 - 42x}$

⇒ √{2 – (–23)}

⇒ √(2 + 23)

⇒ √25

⇒ 5

∴ The value of $\sqrt {2 - 42x}$ is 5.

Given:

AB = 10 units, BC = 8 units, and CA = 12 units

Concept Used:

The length of two tangents drawn on the same circle, from the same external point is same.

Calculation:

⇒ AD = AE, CD = CF, BE = BF

Let AD = AE be x, CD = CF be y, and BE = BF be z.

AB = AE + EB

⇒ 10 = x + z      ----(1)

BC = BF + FC 

⇒ 8 = z + y      ----(2)

AC = AD + DC

⇒ 12 = x + y      ----(3)

Adding equation 1, 2 and 3, we get

⇒ 30 = x + z + z + y + x + y

⇒ 30 = 2 (x + y + z)

⇒ x + y + z = 15

From equation 1, x + z = 10

⇒ y = 15 – (x + z)

⇒ y = 15 – (10)

⇒ y = 5

From equation 2, z + y = 8

⇒ x = 15 – (y + z)

⇒ x = 15 – (8)

⇒ x = 7

From equation 3, x + y = 12

⇒ z = 15 – (x + y)

⇒ z = 15 – 12

⇒ z = 3

Now AD × BF,

⇒ x × z

⇒ 7 × 3

⇒ 21

∴ The value of AD × BF is 21.

Given:

tanα = 2/√13

Concept Used:

tanθ = Opposite side/Adjacent side

secθ = Hypotenuse/Adjacent side

cosecθ = Hypotenuse/Opposite side

 

 

Calculation:

Applying Pythagoras theorem;

⇒ H² = 2² + (√13)²

⇒ H2 = 4 + 13

⇒ H = √17   

Cosecθ = √17/2 and Secθ = √17/√13

⇒ Cosec²θ = 17/4 and Sec²θ = 17/13

Putting values in the given equation,

⇒ {(17/4) + 2(17/13)}/{(17/4) – 3(17/13)} 

⇒ (17/4 + 34/13)/(17/4 – 51/13) 

⇒ {(221 + 136)/52}/{(221 – 204)/52}

⇒ 357/17

⇒ 21

∴ The value of $\frac{{cosec^2a\ +\ 2{{\sec }^2}a}}{{cosec^2a\ -\ 3{{\sec }^2}a}}$is 21.

Given:

Total items purchased = 40 items

Sold some items at 30% profit 

Remaining items sold at 18% profit 

Formula Used:

SP = (100 + Profit%)/CP × 100

Calculation:

Let the cost price be 40x

And number of items sold at 30% profit be y

⇒ Cost price of y items = xy

⇒ Selling price of y items = 130/100 × (xy)

According to the question,

Selling price of y items = Cost price of 26 items

⇒ 130/100 × (xy) = 26x

⇒ 13/10 × y = 26

⇒ y = 20

⇒ Number of items sold at 30% profit = 20

Remaining items sold at profit 18%,

⇒ 40 – 20 = 20

Selling price of items sold at 30%,

⇒ 130/100 × 20x = 26x

Selling price of items sold at 18%, 

⇒ 118/100 × 20x = 23.6x

Total profit percent,

⇒ (26x + 23.6x – 40x)/40x × 100

⇒ 9.6/40 × 100 = 24%

∴ Total profit percentage is 24%

 

Let the CP of 40 items be Rs.100

⇒ Total CP = 40 × 100 =  4,000

Let say he sold x items at 30% profit

⇒ SP of x items = x × 130/100 × 100

⇒ SP of x items = x × 130

Which is equal to the CP of 26 items.

⇒ x × 130 = 26 × 100

⇒ x = 20

He sold 20 items at 30% profit,

⇒ Total SP of 20 items = Rs. 2,600 

⇒ Remaining items = 40 – 20 ⇒ 20

He sold remaining items at 18% profit 

⇒ Remaining 20 articles SP = 20 × 118/100 × 100

⇒ Remaining 20 articles SP = Rs.2,360

Total SP of 40 articles = 2,600 + 2,360 ⇒ Rs.4,960

Total SP of 40 articles = Rs.4,960

Profit % = (SP – CP)/CP × 100

⇒ Profit % = (4,960 – 4,000)/4,000 × 100

Profit % = 960/4000 × 100 ⇒ 24% 

∴ Total profit percentage is 24%

Given:

For 48 students, average marks reduced from 78 to 66

The remaining students increased by 3.5 marks.

Total average reduced by 4.5 marks.

Concept Used:

Using mixture and allegation

Calculation:

Let the total number of students be x

Number of students whose average is reduced is 48

Number of students whose average is increased is x – 48.

⇒ 48 : (x – 48)

⇒ The marks of 48 students reduced by, 

⇒ 78 – 66

⇒ The marks of 48 students reduced by 12,

And the marks of remaining students increased by 3.5.

⇒ 

⇒ 48 : (x – 48) = 8 : 7.5

⇒ 48 : (x – 48) = 16 : 15

⇒ 48/x – 48 = 16/15 

⇒ 3/x – 48 = 1/15

⇒ x – 48 = 45

⇒ x = 45 + 48

⇒ x = 93

∴ Total number of students is 93.

Given:

CP of item = Rs.500

Tax payed on purchasing = 12% 

Profit % while selling = 20%

Discount % given while selling = 16%

Concept Used:

Profit value = {(100 + % Profit)/100} × CP

Discounted price = {100/(100 – % Discount)} × CP

Calculation:

CP = Rs.500

After-tax paid CP,

⇒ 112/100 × 500 = 560

He wants a profit of 20%

The selling price should be,

⇒ 120/100 × 560 = Rs. 672

The discount he wants to give is 16%,

⇒ 100/84 × 672 = 800

∴ The marked price of the article is Rs. 800.

Given:

In 56 liters, 

The ratio of milk to water is 5 : 2

Calculation:

The ratio of milk to water is 5 : 2

⇒ Milk in the total mixture = 5/7 of the total mixture

⇒ Quantity of Milk = 5/7 × 56

⇒ Quantity of Milk = 40 liters

Quantity of Water = Total mixture – Quantity of Milk

⇒ Quantity of Water = 56 – 40

⇒ Quantity of Water = 16 liters

Now, in the final mixture,

the ratio of milk to water 7 : 2

⇒ Water in the final mixture = 2/9 of the total mixture

⇒ 16 = 2/9 × Total mixture

⇒ Total mixture = (16 × 9)/2

⇒ Total mixture = 72 liters

Quantity of Milk = Total mixture – Quantity of Water

⇒ Quantity of Milk = 72 –  16

⇒ Quantity of Milk = 56 liters

∴ The quantity of milk in the new mixture is 56 liters.

Given:

 a% of 240 = c

c% of a = 117.6

Concept used:

m% of n = (m × n)/100

Calculation:

(a × 240)/100 = c

⇒ a/c = 5/12

Let the number a and c be 5x and 12x

(12x × 5x)/100 = 117.6

⇒ 60x² = 11760

⇒ x² = 11760/60

⇒ x² = 196

⇒ x = 14

(a + c) = (5x + 12x) = 17x

After putting the value of x

17x = 17 × 14

⇒ (a + c) = 238

∴ The value of (a + c) is 238

Given:

[(3x√y + 2y√x)/(3x√y - 2y√x)] - [(3x√y - 2y√x)/(3x√y + 2y√x)] = √x × √y

Concept Used:

We need to take LCM and simplify the equation.

Formula Used:

(a + b)² - (a - b)² = 4ab

(a² - b²) = (a + b)(a - b)

Calculation:

Take a = 3x√y and b = 2y√x

Given expression will be transformed to [(a + b)/(a - b)] - [(a - b)/(a + b)]

⇒ [(a + b)² - (a - b)²]/[(a + b)(a - b)] = √(xy)

⇒ 4ab/(a² - b²) = √(xy)

⇒ (4 ×  3x√y × 2y√x)/[(3x√y)² - (2y√x)²] = √(xy)

⇒ (24x√x × y√y)/(9x²y - 4y²x) = √(xy)

⇒ [24 × xy × √(xy)]/[(xy)(9x - 4y) = √(xy)

⇒ 24/(9x - 4y) = 1

⇒ 9x - 4y = 24

∴ The correct relation between x and y is 9x - 4y = 24

Given:

Increase in Length = 12%

Decrease in Breadth = 8%

Formula Used:

Increase in Area = +(Length %) - (Breadth %) - [(Length × Breadth)/100]

Calculation:

Increase in Area = (12%) - (8%) - [(12 × 8)/100]

⇒ Increase in Area = (4 - 0.96)%

⇒ Increase in Area = 3.04%

∴ The increase in area will be 3.04%

Given:

One side of triangle = 7 cm

Perimeter of triangle = 18 cm

Area of triangle = √108 cm²

Formula Used:

If sides of triangle are a, b, and c

then,

Semi perimeter (S) = (a + b + c)/2

Area of triangle = √[s(s - a)(s - b)(s - c)

Calculation:

18 = (7 + b + c)

⇒ (b + c) = 11

⇒ c = 11 - b 

S = 18/2 = 9

√108 = √[(9)(9 - 7)(9 - b)(9 - c)]

Squaring both sides,

108 = 18 × (9 - b) × (9 - c)

Substituting Value of c as 11 - b

6 = (9 - b)[9 - (11 - b)]

⇒ 6 = (9 - b)(b - 2)

⇒ 6 = 9b + 2b - b² - 18

⇒ b² - 11b + 24 = 0

⇒  b2 - 8b - 3b + 24 = 0

⇒ b(b - 8) - 3(b - 8) = 0

⇒ (b - 8)(b - 3) = 0

⇒ b = 8, 3

If we Take side b as 8

Then,

c = 11 - 8 = 3

∴ The other two sides of triangle are 8 cm and 3 cm.

Given:

[4/(1 + √2 + √3)] = a + b√2 + c√3 - d√6

Concept Used:

We need to rationalize the given equation.

Calculation:

[4/(1 + √2 + √3)] × [(1 + √2 - √3)/(1 + √2 - √3)] = a + b√2 + c√3 - d√6

⇒ [4 × (1 + √2 - √3)]/[(3 + 2√2) - (√3)²] = a + b√2 + c√3 - d√6

⇒ [4 × (1 + √2 - √3)]/2√2 = a + b√2 + c√3 - d√6

⇒ [√2 × (1 + √2 - √3)] = a + b√2 + c√3 - d√6

⇒ 2 + √2 - √ 6 = a + b√2 + c√3 - d√6

On comparing equations,

a = 2, b = 1, c = 0, and d = 1

Now,

a + b + c + d = 2 + 1 + 0 + 1

⇒ a + b + c + d = 4

∴ The Value of a + b + c + d is 4

Given:

Principal = Rs 10,000

Some part of money lent at 15%

Some part of it lent at 10%

Total interest earned = Rs. 6,500

Time = 5 years

Formula Used:

Interest = (Principal × Rate × Time)/100

Calculation:

Let the money lent at 15% be x

Then money lent at 10% be (10,000 - x)

6,500 = [(x × 15 × 5)/100] + [(10,000 - x) × 10 × 5]/100

⇒ 6,500 = (3x/4) + [(10,000 - x)/2]

⇒ 6,500 = (3x + 20,000 - 2x)/4

⇒ 26,000 = 20,000 + x

⇒ x = Rs 6,000

Then,

(10,000 - x) = (10,000 - 6,000) = Rs 4,000

Difference between the two sums lent = Rs 6,000 - Rs 4,000

⇒ Difference between the two sums lent = Rs 2,000

∴ The difference between the sums lent is Rs 2,000.

Shortcut TrickTotal S.I. on 10,000 = (10000 × 5 × R)/ 100 = 6500

R = 13%

 

Now the difference = (1/ 5) × 10000

⇒ 2,000 Rs.

Given:

People below average salary is 16% which is 80 in numbers.

Calculation:

16% of total people from Company D = 80

Total number of people in Company D (100%) = (80/16) × 100

⇒ Total number of people in Company D (100%) = 500

Number of people below average salary = 500 - 80

⇒ Number of people below average salary = 420

∴ The number of people below average salary in Company D is 420.

Given:

PA and PB are tangents to circle with center O

Radius of circle = 5 cm and PO = 13 cm

Formula Used:

1. In right angle triangle, by Pythagoras theorem

(Base)² + (Perpendicular)² = (Hypotenuse)²

2. Area of Triangle = (1/2) × Base × Height

Calculation:

∠OAP = 90° (Angle made my radius and tangent)

In ΔOAP, by Pythagoras theorem

AP² = 13² - 5²

⇒ AP² = 144

⇒ AP = 12 cm

ΔOAP and ΔOBP are congruent triangles

⇒ Area of quadrilateral PAOB = 2 × (Area of ΔOAP)

⇒ Area of quadrilateral PAOB = 2 × [(1/2) × 5 × 12]

⇒ Area of quadrilateral PAOB = 60 cm²         ----(01)

Also

Area of quadrilateral PAOB = Area of ΔOAB + Area of ΔPAB

⇒ Area of quadrilateral PAOB = [(1/2)AB × (x)] + [(1/2)AB × (13 - x)]

⇒ Area of quadrilateral PAOB = (1/2)AB [x + 13 - x]

⇒ Area of quadrilateral PAOB = (13/2)AB    ----(02)

From (01) and (02)

(13/2)AB = 60

⇒ AB = 120/13

Since ON ⊥ AB and O is the center, then ON will bisect AB

AN = AB/2 = 60/13

In ΔANO, by Pythagoras theorem

ON² = 5² - (60/13)²

⇒ ON² = 625/169

⇒ ON = 25/13

From the figure, OP = ON + NP

⇒ NP = 13 - (25/13)

⇒ NP = 144/13

According to the question,

M is the area of ΔPAB

Area of ΔPAB = M = (1/2) × AB × NP

⇒ M = (1/2) × (120/13) × (144/13)

⇒ M = 8640/169

Now,

$\sqrt{\frac{M}{15}}$ = $\sqrt{\frac{8640}{169\times15}}$

⇒ $\sqrt{\frac{M}{15}}$ = $\sqrt{\frac{576}{169}}$ = 24/13

∴ The value of √(M/15) is 24/13.

Alternate Method

In ΔOAP,

OP2 = OA2 + AP2

⇒132 = 52 + AP2

⇒ AP2 = 169 - 25

⇒ AP2 = 144

⇒ AP = 12 cm

sinx = OA/OP

⇒ Sinx = 5/13

cosx = AP/OP

⇒ cosx = 12/13

In ΔACP,

sinx = AC/AP

⇒ 5/13 = AC/12

⇒ AC = 60/13 cm

Similarly, BC = 60/13 cm

cosx = CP/AP

⇒ 12/13 = CP/12

⇒ CP = 144/13

Area of ΔPAB = M

⇒ (1/2) × AB × CP = M

⇒ (1/2) × (AC + BC) × CP = M

⇒ (1/2) × {(60/13) + (60/13)} × (144/13) = M

⇒ (1/2) × (120/13) × (144/13) = M

⇒ M = (60/13) × (144/13)

So, √(M/15) = √{(60/13) × (144/13) × (1/15)}

⇒ √(M/15) = 24/13

∴ The Value of √(M/15) is 24/13.