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CAT Practice Test: Week 21 - CAT MCQ


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30 Questions MCQ Test - CAT Practice Test: Week 21

CAT Practice Test: Week 21 for CAT 2024 is part of CAT preparation. The CAT Practice Test: Week 21 questions and answers have been prepared according to the CAT exam syllabus.The CAT Practice Test: Week 21 MCQs are made for CAT 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for CAT Practice Test: Week 21 below.
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CAT Practice Test: Week 21 - Question 1
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If cot (x + 27°) = tan y, then which of the following equations is true?  

Detailed Solution for CAT Practice Test: Week 21 - Question 1

cot (90 - θ) = tan θ
So, tan y = cot (90° - y)
Hence, cot (x + 27°) = cot (90° - y)
x + 27° = 90° - y
x + y = 90° - 27° = 63°

CAT Practice Test: Week 21 - Question 2
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Solve for x: 2 cos (3x) - 1 = 0; where, x ∈ (0, 2π)

Detailed Solution for CAT Practice Test: Week 21 - Question 2

2 cos 3x - 1 = 0
⇒ cos 3x = 1/2 = cos 60°
The general solution of above equation is 

The solution in the interval (0, 2π) are

Hence, correct answer that match the given options are 20°, 100°

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CAT Practice Test: Week 21 - Question 3
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The value of cos 24° + cos 55° + cos 125° + cos 204° + cos 300° is

Detailed Solution for CAT Practice Test: Week 21 - Question 3

The value of cos 24° + cos 55° + cos 125° + cos 204° + cos 300° ?
We know that, cos(180° ± θ) = - cosθ
⇒ cos 24° + cos 55° + cos(180° - 55°) + cos(180° + 24°) + cos(360° - 60°)
= cos 24° + cos 55° - cos 55° - cos 24° + cos 60°
= cos 60° = 1/2

CAT Practice Test: Week 21 - Question 4
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Solve the equation 2 sin2 x + 3 cos x = 0.
Detailed Solution for CAT Practice Test: Week 21 - Question 4

2 sin2 x + 3 cos x = 0
⇒ 2(1 - cos2 x) + 3 cos x = 0
⇒ 2 - 2 cos2 x + 3 cos x = 0
⇒ 2 cos2 x - 3 cos x - 2 = 0
⇒ 2 cos2 x - 4 cos x + cos x - 2 = 0
⇒ 2 cos x(cos x - 2) + 1(cos x - 2) = 0
⇒ (cos x - 2)(2 cos x + 1) = 0
Either cos x - 2 = 0 or 2 cos x + 1 = 0
But cos x - 2 = 0
i.e. cos x = 2, which is not possible.
Now, from 2 cos x + 1 = 0, we get:
cos x = -1/2

Therefore, general solution of the equation is:

CAT Practice Test: Week 21 - Question 5
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Find the value of cos2 15° - cos2 30° + cos2 45° - cos2 60° + cos2 75°.
Detailed Solution for CAT Practice Test: Week 21 - Question 5

cos2 15° - cos2 30° + cos2 45° - cos2 60° + cos2 75° = (cos2 15° + cos2 75°) - (cos2 30° + cos2 60°) + cos2 45°
= [cos2 15° + cos2 (90° - 15°)] - [cos2 30° + cos2 (90° - 30°)] + cos2 45°
= [cos2 15° + sin2 15°] - [cos2 30° + sin2 30°] + cos2 45° [∵ cos (90° - θ) = sin θ]

CAT Practice Test: Week 21 - Question 6
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Two pillars of equal height stand on either side of a road, which is 150 m wide. At a point on the road between the pillars, the angles of elevation of their tops are 60° and 30°. Find the height of the pillars.
Detailed Solution for CAT Practice Test: Week 21 - Question 6

Let AB and CD be the two pillars of equal height 'h' m.
If P is the point between them, then
∠APB = 60° and ∠CPD = 30°

Let BP = x m
⇒ PD = (150 - x) m
In ΔABP, tan 60° = 
⇒ h = √3x … (1)

⇒ 3x = 150 - x or 4x = 150
∴ x = 37.5 and h = √3x = (1.732)37.5 = 64.95 m

CAT Practice Test: Week 21 - Question 7
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If cosθ = 2cos2θ , then 2sec2θ – cos2θ + tan2θ =
Detailed Solution for CAT Practice Test: Week 21 - Question 7

cosθ = 2 cos2θ

secθ = 2
secθ = sec 60°
θ = 60°
2 sec2θ – cos2θ + tan2θ = 2 sec2 60° - cos2 60° + tan2 60° = 

CAT Practice Test: Week 21 - Question 8
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Find the value of
Detailed Solution for CAT Practice Test: Week 21 - Question 8



(∵ sin (90° - θ) = cos θ and cos (90° - θ) = sin θ)

CAT Practice Test: Week 21 - Question 9
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The angle of elevation of a cloud from a point 60 m above a lake is 30° and the angle of depression of its image in lake is 60°. The height of the cloud is
Detailed Solution for CAT Practice Test: Week 21 - Question 9

Let A be the point of observation and C be the position of the cloud. Let DE be the surface of the lake and C' be the reflection of C in the lake.
∠CAB = 30°
And ∠C'AB = 60°
Let BC = h m

In right-angled ΔABC, BC/AB = tan 30°


In right-angled ΔABC',

h = 60 m
Height of cloud = 120 m

CAT Practice Test: Week 21 - Question 10
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Solve for the number of solution of the equation: sin(x) + cos(x) = 1 on the interval 0° ≤ x < 360°.  
Detailed Solution for CAT Practice Test: Week 21 - Question 10

sin(x) + cos(x) = 1
Squaring both side,
(sin(x) + cos(x))2 = 12
[sin2(x) + cos2(x)] + 2sin(x)cos(x) = 1 (Using identity: (a + b)2 = a2 + b2 + 2ab)
1 + 2sin(x)cos(x) = 1
2sin(x)cos(x) = 0
sin(x)cos(x) = 0
From the last line above, either sine is zero or cosine is zero, so solutions are:
x = 0°, 90°, 180°, 270°
Checking the value of 'x' in the above equation,
x = 0°, 90° satisfies.
x = 180° and 270° does not satisfies.
Hence, total number of solution = 2

CAT Practice Test: Week 21 - Question 11
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Direction: In the following question, two statements are given followed by three or four conclusions numbered I, II, III and IV. You have to take the given statements to be true even if they seem to be at variance from the commonly known facts and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.

Statements: All branches are flowers. All flowers are leaves.

Conclusions:
I. All branches are leaves.
II. All leaves are branches.
III. All flowers are branches.
IV. Some leaves are branches.
Detailed Solution for CAT Practice Test: Week 21 - Question 11

Since both the premises are universal and affirmative, the conclusion must be universal affirmative and should not contain the middle term. So, it follows that 'All branches are leaves'. Thus, I follows. IV is the converse of this conclusion and so it also holds.

CAT Practice Test: Week 21 - Question 12
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Direction: In the following question, two statements are given followed by three or four conclusions numbered I, II, III and IV. You have to take the given statements to be true even if they seem to be at variance from the commonly known facts and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.

Statements: All politicians are honest. All honest are fair.

Conclusions:
I. Some honest are politicians.
II. No honest is politician.
III. Some fair are politicians.
IV. All fair are politicians.
Detailed Solution for CAT Practice Test: Week 21 - Question 12

Clearly, it follows that 'All politicians are fair'. I is the converse of the first premise, while III is the converse of the above conclusion. So, both I and III hold.

CAT Practice Test: Week 21 - Question 13
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Direction: In the following question, two statements are given followed by three or four conclusions numbered I, II, III and IV. You have to take the given statements to be true even if they seem to be at variance from the commonly known facts and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.

Statements: Some clothes are marbles. Some marbles are bags.

Conclusions:
I. No cloth is a bag.
II. All marbles are bags.
III. Some bags are clothes.
IV. No marble is a cloth.
Detailed Solution for CAT Practice Test: Week 21 - Question 13

Since both the premises are particular, no definite conclusion follows. However, I and III involve only the extreme terms and form a complementary pair. Thus, either I or III follows.

CAT Practice Test: Week 21 - Question 14
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Direction: In the following question, two statements are given followed by three or four conclusions numbered I, II, III and IV. You have to take the given statements to be true even if they seem to be at variance from the commonly known facts and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.

Statements: Some tables are TVs. Some TVs are radios.

Conclusions:
I. Some tables are radios.
II. Some radios are tables.
III. All radios are TVs.
IV. All TVs are tables.
Detailed Solution for CAT Practice Test: Week 21 - Question 14

Since both the premises are particular, no definite conclusion follows.

CAT Practice Test: Week 21 - Question 15
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Direction: In the following question, two statements are given followed by three or four conclusions numbered I, II, III and IV. You have to take the given statements to be true even if they seem to be at variance from the commonly known facts and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.

Statements: All terrorists are guilty. All terrorists are criminals.

Conclusions:
I. Either all criminals are guilty or all guilty are criminals.
II. Some guilty persons are criminals.
III. Generally criminals are guilty.
IV. Crime and guilt go together.
Detailed Solution for CAT Practice Test: Week 21 - Question 15

Since the middle term 'terrorists' is distributed twice in the premises, the conclusion cannot be universal. So, it follows that 'Some guilty persons are criminals'. Thus, II holds.

CAT Practice Test: Week 21 - Question 16
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Direction: In the following question, two statements are given followed by three or four conclusions numbered I, II, III and IV. You have to take the given statements to be true even if they seem to be at variance from the commonly known facts and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.

Statements: Some books are pens. No pen is pencil.

Conclusions:
I. Some pens are books.
II. Some pencils are books.
III. Some books are not pencils.
IV. All pencils are books.
Detailed Solution for CAT Practice Test: Week 21 - Question 16

Since one premise is particular and the other negative, the conclusion must be particular negative and should not contain the middle term. Thus, III follows. I is the converse of the first premise and so it also holds.

CAT Practice Test: Week 21 - Question 17
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Direction: In the following question, two statements are given followed by three or four conclusions numbered I, II, III and IV. You have to take the given statements to be true even if they seem to be at variance from the commonly known facts and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.

Statements: Some houses are offices. Some offices are schools.

Conclusions:
I. Some schools are houses.
II. Some offices are houses.
III. No house is school.
IV. Some schools are offices.
Detailed Solution for CAT Practice Test: Week 21 - Question 17

Since both the premises are particular, no definite conclusion follows. However, I and III involve only the extreme terms and form a complementary pair. So, either I or III follows. II is the converse of the first premise while IV is the converse of the second premise. Thus, both of them hold.

CAT Practice Test: Week 21 - Question 18
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Direction: In the following question, three statements are given followed by four conclusions numbered I, II, III and TV. You have to take the given statements to be true even if they seem to be at variance with commonly known facts and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.

Statements: All buildings are windows. No toys is building. Some tigers are toys.

Conclusions:
I. Some tigers are buildings.
II. Some windows are tigers.
III. All toys are tigers.
IV. Some windows are toys.
Detailed Solution for CAT Practice Test: Week 21 - Question 18

No toy is building. All buildings are windows.

Since the middle term 'buildings' is distributed twice and one premise is negative, the conclusion must be particular negative and should not contain the middle term.

So, it follows that 'Some windows are not toys'.

Some tigers are toys. No toy is building.

Since one premise is particular and the other premise is negative, the conclusion must be particular negative and should not contain the middle term. So, it follows that 'Some tigers are not buildings'.

CAT Practice Test: Week 21 - Question 19
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Direction: In the following question, three statements are given followed by four conclusions numbered I, II, III and TV. You have to take the given statements to be true even if they seem to be at variance with commonly known facts and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.

Statements: Some papers are cats. All cats are bats. No bat is horse.

Conclusions:
I. Some papers are horses.
II. No horse is cat.
III. Some bats are papers.
IV. All papers are bats.
Detailed Solution for CAT Practice Test: Week 21 - Question 19

Some papers are cats. All cats are bats.

Since one premise is particular, the conclusion must be particular and should not contain the middle term. So, it follows that 'Some papers are bats'. III is the converse of this conclusion and so it holds.

All cats are bats. No bat is horse.

Since both the premises are universal and one premise is negative, the conclusion must be universal negative and should not contain the middle term. So, it follows that 'No cat is horse'. II is the converse of this conclusion and so it holds.

Some papers are bats. No bat is horse.

Since one premise is particular and the other negative, the conclusion must be particular negative and should not contain the middle term. So, it follows that 'Some papers are not horses'.

CAT Practice Test: Week 21 - Question 20
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In each of the following questions two statements followed by four conclusions numbered I, II, III and IV are given, You have to take the two given statements to take true even if they seem to be at variance from commonly known facts. Read all the conclusions and then decide which of the given conclusions logically follows from the statements.

Statements
All branches are flowers.
All flowers are leaves.

Conclusions
I. All branches are leaves.
II. All leaves are branches.
III. All flowers are branches.
IV. Some leaves are branches.
Detailed Solution for CAT Practice Test: Week 21 - Question 20

Solution

The correct option is B Both I and IV follow
Mediate term of the following statement is
All branches are flowers (A - type)
All flowers are leaves (A - type)
A+A→A type
All branches are leaves
(Conclusion I follows)
All branches are leaves
Conversion
Some leaves are branches
(Conclusion IV follows)

CAT Practice Test: Week 21 - Question 21
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Directions: The sentence given below may have an error. The sentence has been rewritten under the answer options (1) to (5). Select the option that is the most coherent and precise representation of the sentence.

A balding, smooth faced man, he could have been anywhere between forty and sixty.
Detailed Solution for CAT Practice Test: Week 21 - Question 21

The phrase 'a balding smooth faced man' is an appositive and can be used as a sentence opener. It is placed beside 'he' to explain or describe the pronoun.

CAT Practice Test: Week 21 - Question 22
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Directions: The sentence below has been divided into four parts. Identify the part having an error. If no part has an error, mark (5).

(1) When reminded of his obligation under the rules, (2) he complied with the requirement that all graduate students (3) in education (4) should write a thesis. (5) No error
Detailed Solution for CAT Practice Test: Week 21 - Question 22

Error of importance. Delete ‘should’. Here, 'should' implies a choice where there is no choice. 

CAT Practice Test: Week 21 - Question 23
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Directions: The sentence below has been divided into four parts. Identify the part having an error. If no part has an error, mark (5)

(1) Although it might take a laboratory analysis to determine the (2) exact proportions of sand, silt, clay, and organic matter in your garden soil, (3) feel the soil`s texture to get a good idea of (4) what they are. (5) No error
Detailed Solution for CAT Practice Test: Week 21 - Question 23

Error of parallel structure. (3) incorrectly turns an assertive sentence into an imperative one.

CAT Practice Test: Week 21 - Question 24
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Directions: A part of the sentence is underlined. Five different ways of phrasing the underlined part are given. Choose the best alternative among the five.

Observing the changes in economy sweeping through the world it can be seen that they are spontaneous.
Detailed Solution for CAT Practice Test: Week 21 - Question 24

(4) corrects the error of dangling participle by introducing the subject 'we'. Option 4 most accurately replaces the underlined part of the sentence. It specifies that 'we' can infer the changes happening in the world to be spontaneous. Option 2 is incorrect as it uses a semi-colon despite the former sentence being a complete sentence. No subject is present in option 3; hence, the meaning of the sentence cannot be inferred. Option 5 completely alters the construction of the sentence.

CAT Practice Test: Week 21 - Question 25
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Directions: The sentence below may have an error of grammar or syntax. The following options seek to correct the error (including an option that simply repeats the sentence). Select the option that best corrects the sentence.

New zeal and vigour can be generated by the government among youths by providing employment to youths.
Detailed Solution for CAT Practice Test: Week 21 - Question 25

(2) corrects the structure of the sentence.

CAT Practice Test: Week 21 - Question 26
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Directions: The underlined part the following sentence may have an error of grammar or syntax. The part has been rewritten in options (2) to (5) in four different ways; (1) being an exact restatement of the underlined part. Choose the option that corrects the error in the underlined part, if any. If there is no error, mark (1).

Seldom have the mathematical theories of games been of practical use in playing real games, although our comprehension of the selection process from among variables may have enhanced.
Detailed Solution for CAT Practice Test: Week 21 - Question 26

Error of parallel structure; `Seldom have…they may have`

CAT Practice Test: Week 21 - Question 27
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Directions: The sentence below has an underlined part. The underlined part may have an error of grammar or syntax. There are five options that seek to correct the error (including an option that simply repeats the underlined part). Select the option that best replaces the underlined part.

During his youth, he used to work out ten times as much as he jogged, but today in his middle age he is doing less of it.
Detailed Solution for CAT Practice Test: Week 21 - Question 27

What does ‘it’ refer to?; either 'work out' or 'jogging'. It is clarified only in (2).

CAT Practice Test: Week 21 - Question 28
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Directions: The sentence below has an underlined part. The underlined part may have an error of grammar or syntax. There are five options that seek to correct the error (including an option that simply repeats the underlined part). Select the option that best replaces the underlined part.

After gaining maturity, a child ought to be allowed to make decisions on his own and in pursuing his goals because by that time he is mature sufficiently to decide on his own.
Detailed Solution for CAT Practice Test: Week 21 - Question 28

Only option 5 maintains parallel structure. With 'to make decisions' we need '… and pursue' to avoid repetition of 'to'. Moreover the adverb 'sufficiently' should be placed before the adjective 'mature'. 

CAT Practice Test: Week 21 - Question 29
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Directions: Look at the underlined part of the given sentence. Below the sentence, are three possible substitutions for the underlined part. Choose the one that is better than the underlined part. In case no improvement is needed, mark (4) as the answer.

Having passed the examination, the college awarded him the best student's medal.
Detailed Solution for CAT Practice Test: Week 21 - Question 29

Error of misplaced modifier. Who passed the exam? The subject has to be placed with the modifying clause in the beginning.

CAT Practice Test: Week 21 - Question 30
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Directions: A part of the sentence is underlined. Five different ways of phrasing the underlined part are given. Choose the best alternative among the five.

Depending on factual evidences, argument is seldom used in pleading.
Detailed Solution for CAT Practice Test: Week 21 - Question 30

(2) corrects the error of structure. The subject is missing in the given sentence. Option 2 has a subject 'pleaders' which is correctly placed. Option 1 is incorrect as it makes 'argument' the subject of the sentence. It is not the 'argument' which is dependent on factual evidences. Option 3, although contextual, unnecessarily introduces the phrase 'which is'. It could simply have been 'factual evidences'. Thus, it is not the 'best' expression. Option 5 alters the meaning and implies the opposite of the original sentence. Option 4 is also incorrect as 'dependent on by pleaders' is ungrammatical.

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