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CAT 2022 Slot 2 Question Paper with Solutions

66 questions in 120 minutes, in three timed sections. Below: how the paper is built, the topics each section tested, and a solved sample from every section. Take the whole paper in the simulator to get your Score Waterfall, Topic ROI Map and Strategic IQ.

Paper structure and topics

VARC

Verbal Ability & Reading Comprehension

24 questions · 21 MCQ · 3 TITA

  • Reading Comprehension · 16
  • Verbal Ability · 8

DILR

Data Interpretation & Logical Reasoning

20 questions · 14 MCQ · 6 TITA

  • Logical Reasoning · 15
  • Data Interpretation · 5

QA

Quantitative Ability

22 questions · 14 MCQ · 8 TITA

  • Arithmetic · 10
  • Algebra · 7
  • Geometry · 3
  • Modern Math · 2

Sample questions with solutions

14 of the 66 questions: the first passage, the first DILR set and the first QA questions of the paper.

VARC · Verbal Ability & Reading Comprehension

Passage

The passage below is accompanied by a set of questions. Choose the best answer to each question.

Humans today make music. Think beyond all the qualifications that might trail after this bald statement: that only certain humans make music, that extensive training is involved, that many societies distinguish musical specialists from nonmusicians, that in today’s societies most listen to music rather than making it, and so forth. These qualifications, whatever their local merit, are moot in the face of the overarching truth that making music, considered from a cognitive and psychological vantage, is the province of all those who perceive and experience what is made. We are, almost all of us, musicians — everyone who can entrain (not necessarily dance) to a beat, who can recognize a repeated tune (not necessarily sing it), who can distinguish one instrument or one singing voice from another. I will often use an antique word, recently revived, to name this broader musical experience. Humans are musicking creatures. . . .

The set of capacities that enables musicking is a principal marker of modern humanity. There is nothing polemical in this assertion except a certain insistence, which will figure often in what follows, that musicking be included in our thinking about fundamental human commonalities. Capacities involved in musicking are many and take shape in complicated ways, arising from innate dispositions . . . Most of these capacities overlap with nonmusical ones, though a few may be distinct and dedicated to musical perception and production. In the area of overlap, linguistic capacities seem to be particularly important, and humans are (in principle) language-makers in addition to music-makers — speaking creatures as well as musicking ones.

Humans are symbol-makers too, a feature tightly bound up with language, not so tightly with music. The species Cassirer dubbed Homo symbolicus cannot help but tangle musicking in webs of symbolic thought and expression, habitually making it a component of behavioral complexes that form such expression. But in fundamental features musicking is neither language-like nor symbol-like, and from these differences come many clues to its ancient emergence.

If musicking is a primary, shared trait of modern humans, then to describe its emergence must be to detail the coalescing of that modernity. This took place, archaeologists are clear, over a very long durée: at least 50,000 years or so, more likely something closer to 200,000, depending in part on what that coalescence is taken to comprise. If we look back 20,000 years, a small portion of this long period, we reach the lives of humans whose musical capacities were probably little different from our own. As we look farther back we reach horizons where this similarity can no longer hold — perhaps 40,000 years ago, perhaps 70,000, perhaps 100,000. But we never cross a line before which all the cognitive capacities recruited in modern musicking abruptly disappear. Unless we embrace the incredible notion that music sprang forth in full-blown glory, its emergence will have to be tracked in gradualist terms across a long period.

This is one general feature of a history of music’s emergence . . . The history was at once sociocultural and biological . . . The capacities recruited in musicking are many, so describing its emergence involves following several or many separate strands.

Question 1 · MCQ

Which one of the following statements, if true, would weaken the author’s claim that humans are musicking creatures?
  1. A.
    As musicking is neither language-like nor symbol-like, it is a much older form of expression.
  2. B.
    Nonmusical capacities are of far greater consequence to human survival than the capacity for music.
  3. C.
    Musical capacities are primarily socio-cultural, which explains the wide diversity of musical forms.
    ✓ Correct
  4. D.
    From a cognitive and psychological vantage, musicking arises from unconscious dispositions, not conscious ones.
Solution
The author argues that musicking is a fundamental human commonality arising from both biological and socio-cultural evolution, with innate dispositions. If musical capacities were shown to be primarily socio-cultural (Option 3), it would undermine the biological/innate aspect of the author's claim that musicking is a universal, inherent trait of modern humanity.

Question 2 · MCQ

Which one of the following sets of terms best serves as keywords to the passage?
  1. A.
    Humans; Psychological vantage; Musicking; Cassirer; Emergence of music.
  2. B.
    Musicking; Cognitive psychology; Antique; Symbol-makers; Modernity.
  3. C.
    Humans; Capacities; Language; Symbols; Modernity.
  4. D.
    Humans; Musicking; Linguistic capacities; Symbol-making; Modern humanity.
    ✓ Correct
Solution
Option 4 captures the core themes of the passage: humans as musicking creatures, the role of linguistic capacities, symbol-making, and how these define modern humanity. Other options either focus on minor details (like Cassirer) or miss key elements like linguistic/symbolic capacities.

Question 3 · MCQ

Based on the passage, which one of the following statements is a valid argument about the emergence of music/musicking?
  1. A.
    Although musicking is not language-like, it shares the quality of being a form of expression.
  2. B.
    All musical work is located in the overlap between linguistic capacity and music production.
  3. C.
    Anyone who can perceive and experience music must be considered capable of musicking.
  4. D.
    20,000 years ago, human musical capacities were not very different from what they are today.
    ✓ Correct
Solution
The passage explicitly states: "If we look back 20,000 years, a small portion of this long period, we reach the lives of humans whose musical capacities were probably little different from our own." This directly supports Option 4.

Question 4 · MCQ

“Think beyond all the qualifications that might trail after this bald statement . . .” In the context of the passage, what is the author trying to communicate in this quoted extract?
  1. A.
    Thinking beyond qualifications allows us to give free reign to musical expressions.
  2. B.
    A bald statement is one that is trailed by a series of qualifying clarifications and caveats.
  3. C.
    Although there may be many caveats and other considerations, the statement is essentially true.
    ✓ Correct
  4. D.
    A bald statement is one that requires no qualifications to infer its meaning.
Solution
The author uses this phrase to introduce qualifications (such as training, specialization, and passive listening) but immediately dismisses them as "moot in the face of the overarching truth" that humans are inherently musicking creatures. Thus, despite caveats, the main statement remains essentially true.

DILR · Data Interpretation & Logical Reasoning

Set

A speciality supermarket sells 320 products. Each of these products was either a cosmetic product or a nutrition product. Each of these products was also either a foreign product or a domestic product. Each of these products had at least one of the two approvals – FDA or EU.
The following facts are also known:
  1. There were equal numbers of domestic and foreign products.
  2. Half of the domestic products were FDA approved cosmetic products.
  3. None of the foreign products had both the approvals, while 60 domestic products had both the approvals.
  4. There were 140 nutrition products, half of them were foreign products.
  5. There were 200 FDA approved products. 70 of them were foreign products and 120 of them were cosmetic products.

Question 1 · TITA (type in the answer)

How many foreign products were FDA approved cosmetic products?

Answer: 40

Solution
Step 1: Find Total Products by Category
  • Total products = 320. Equal Domestic (D) and Foreign (F) \(\Rightarrow\) D = 160, F = 160.
  • Nutrition (N) = 140 \(\Rightarrow\) Cosmetic (C) = 180.
  • Foreign Nutrition (\(F_N\)) = 140 / 2 = 70 \(\Rightarrow\) \(F_C\) = 90, \(D_N\) = 70, \(D_C\) = 90.
Step 2: Analyze Approvals
  • Total FDA+ = 200. F_FDA+ = 70 \(\Rightarrow\) D_FDA+ = 130.
  • Total C_FDA+ = 120. D_C_FDA+ = 160 / 2 = 80 \(\Rightarrow\) F_C_FDA+ = 120 - 80 = 40.
  • Since F_FDA+ = 70, F_N_FDA+ = 70 - 40 = 30.
  • Foreign products have 0 'Both' approvals, so FDA+ means FDA only.
  • F_C_EU_only = 90 - 40 = 50; F_N_EU_only = 70 - 30 = 40.
  • For Domestic: D_C_FDA+ = 80 \(\Rightarrow\) D_C_EU_only = 90 - 80 = 10.
  • D_N_FDA+ = D_FDA+ - D_C_FDA+ = 130 - 80 = 50 \(\Rightarrow\) D_N_EU_only = 70 - 50 = 20.
  • Let Domestic Cosmetic 'Both' = \(x\). Then D_C_FDA_only = \(80 - x\).
  • Total D_Both = 60 \(\Rightarrow\) D_N_Both = \(60 - x\).
  • D_N_FDA_only = 50 - (60 - x) = \(x - 10\).
  • To keep all values \(\ge 0\), we need \(10 \le x \le 60\).
The question asks for the number of foreign products that were FDA approved cosmetic products (F_C_FDA+), which is exactly 40.
CategoryFDA onlyEU onlyBothTotal
Foreign Cosmetic4050090
Foreign Nutrition3040070
Domestic Cosmetic\(80 - x\)10\(x\)90
Domestic Nutrition\(x - 10\)20\(60 - x\)70

Question 2 · MCQ

How many cosmetic products did not have FDA approval?
  1. A.
    10
  2. B.
    60
    ✓ Correct
  3. C.
    50
  4. D.
    Cannot be determined
Solution
Based on the set analysis (see table):
  • Total products = 320. Domestic (D) = 160, Foreign (F) = 160.
  • Nutrition (N) = 140 \(\Rightarrow\) Cosmetic (C) = 180.
  • \(F_N\) = 70 \(\Rightarrow\) \(F_C\) = 90, \(D_N\) = 70, \(D_C\) = 90.
  • Total FDA+ = 200. F_FDA+ = 70 \(\Rightarrow\) D_FDA+ = 130.
  • Total C_FDA+ = 120. D_C_FDA+ = 160 / 2 = 80 \(\Rightarrow\) F_C_FDA+ = 40.
  • Foreign products have 0 'Both' approvals, so F_C_EU_only = 90 - 40 = 50.
  • For Domestic: D_C_FDA+ = 80 \(\Rightarrow\) D_C_EU_only = 90 - 80 = 10.
Products that 'did not have FDA approval' are those with 'EU only' approval.
Total Cosmetic EU only = D_C_EU_only + F_C_EU_only = 10 + 50 = 60.
CategoryFDA onlyEU onlyBothTotal
Foreign Cosmetic4050090
Foreign Nutrition3040070
Domestic Cosmetic\(80 - x\)10\(x\)90
Domestic Nutrition\(x - 10\)20\(60 - x\)70

Question 3 · MCQ

Which among the following options best represents the number of domestic cosmetic products that had both the approvals?
  1. A.
    At least 10 and at most 80
  2. B.
    At least 20 and at most 70
  3. C.
    At least 20 and at most 50
  4. D.
    At least 10 and at most 60
    ✓ Correct
Solution
Based on the set analysis (see table):
  • Domestic FDA+ Cosmetic (D_C_FDA+) = 160 / 2 = 80.
  • Total Domestic FDA+ = 130 \(\Rightarrow\) Domestic FDA+ Nutrition (D_N_FDA+) = 130 - 80 = 50.
  • Let Domestic Cosmetic with 'Both' approvals = \(x\).
  • Then Domestic Cosmetic 'FDA only' = \(80 - x\).
  • Total Domestic 'Both' = 60 \(\Rightarrow\) Domestic Nutrition 'Both' = \(60 - x\).
  • Domestic Nutrition 'FDA only' = 50 - (60 - x) = \(x - 10\).
Since the number of products in any category cannot be negative, we must satisfy:
  1. \(80 - x \ge 0 \Rightarrow x \le 80\)
  2. \(60 - x \ge 0 \Rightarrow x \le 60\)
  3. \(x - 10 \ge 0 \Rightarrow x \ge 10\)
Combining these, we get \(10 \le x \le 60\). Thus, the number is at least 10 and at most 60.
CategoryFDA onlyEU onlyBothTotal
Foreign Cosmetic4050090
Foreign Nutrition3040070
Domestic Cosmetic\(80 - x\)10\(x\)90
Domestic Nutrition\(x - 10\)20\(60 - x\)70

Question 4 · MCQ

If 70 cosmetic products did not have EU approval, then how many nutrition products had both the approvals?
  1. A.
    10
    ✓ Correct
  2. B.
    20
  3. C.
    30
  4. D.
    50
Solution
Based on the set analysis (see table):
  • Let Domestic Cosmetic with 'Both' approvals = \(x\).
  • Domestic Cosmetic 'FDA only' = \(80 - x\).
  • Foreign Cosmetic 'FDA only' = 40.
  • Domestic Nutrition 'Both' = \(60 - x\).
  • Foreign Nutrition 'Both' = 0.
'Did not have EU approval' means the product had 'FDA only' approval.
We are given Total Cosmetic 'FDA only' = 70.
\((80 - x) + 40 = 70 \Rightarrow 120 - x = 70 \Rightarrow x = 50\).

The question asks for the number of nutrition products with both approvals:
Total Nutrition 'Both' = Domestic Nutrition 'Both' + Foreign Nutrition 'Both' = \((60 - 50) + 0 = 10\).
CategoryFDA onlyEU onlyBothTotal
Foreign Cosmetic4050090
Foreign Nutrition3040070
Domestic Cosmetic\(80 - x\)10\(x\)90
Domestic Nutrition\(x - 10\)20\(60 - x\)70

Question 5 · TITA (type in the answer)

If 50 nutrition products did not have EU approval, then how many domestic cosmetic products did not have EU approval?

Answer: 50

Solution
Based on the set analysis (see table):
  • Let Domestic Cosmetic with 'Both' approvals = \(x\).
  • Domestic Nutrition 'FDA only' = \(x - 10\).
  • Foreign Nutrition 'FDA only' = 30.
  • Domestic Cosmetic 'FDA only' = \(80 - x\).
'Did not have EU approval' means the product had 'FDA only' approval.
We are given Total Nutrition 'FDA only' = 50.
\((x - 10) + 30 = 50 \Rightarrow x + 20 = 50 \Rightarrow x = 30\).

The question asks for the number of domestic cosmetic products that did not have EU approval (i.e., 'FDA only'):
Domestic Cosmetic 'FDA only' = \(80 - x = 80 - 30 = 50\).
CategoryFDA onlyEU onlyBothTotal
Foreign Cosmetic4050090
Foreign Nutrition3040070
Domestic Cosmetic\(80 - x\)10\(x\)90
Domestic Nutrition\(x - 10\)20\(60 - x\)70

QA · Quantitative Ability

Question 1 · MCQ

In triangle ABC, altitudes AD and BE are drawn to the corresponding bases. If \(\angle BAC = 45^\circ\) and \(\angle ABC = \theta\), then \(\frac{AD}{BE}\) equals
  1. A.
    1
  2. B.
    \(\sqrt{2} \cos \theta\)
  3. C.
    \(\frac{\sin \theta + \cos \theta}{\sqrt{2}}\)
  4. D.
    \(\sqrt{2} \sin \theta\)
    ✓ Correct
Solution
The area of \(\triangle ABC\) can be expressed in two ways using the altitudes \(AD\) and \(BE\):
\(\text{Area} = \frac{1}{2} \times BC \times AD = \frac{1}{2} \times AC \times BE\)
This gives the ratio \(\frac{AD}{BE} = \frac{AC}{BC}\).
Using the Law of Sines in \(\triangle ABC\):
\(\frac{AC}{\sin \angle ABC} = \frac{BC}{\sin \angle BAC}\)
Substitute the given angles \(\angle ABC = \theta\) and \(\angle BAC = 45^\circ\):
\(\frac{AC}{\sin \theta} = \frac{BC}{\sin 45^\circ}\)
\(\frac{AC}{BC} = \frac{\sin \theta}{\sin 45^\circ} = \frac{\sin \theta}{1/\sqrt{2}} = \sqrt{2} \sin \theta\)
Therefore, \(\frac{AD}{BE} = \sqrt{2} \sin \theta\).

Question 2 · MCQ

Consider the arithmetic progression 3, 7, 11, ... and let \(A_n\) denote the sum of the first n terms of this progression. Then the value of \(\frac{1}{25} \sum_{n=1}^{25} A_n\) is
  1. A.
    404
  2. B.
    415
  3. C.
    455
    ✓ Correct
  4. D.
    442
Solution
The given arithmetic progression has first term \(a = 3\) and common difference \(d = 4\).
The sum of the first \(n\) terms is:
\(A_n = \frac{n}{2} [2a + (n-1)d] = \frac{n}{2} [6 + (n-1)4] = 2n^2 + n\)
We need to evaluate \(\frac{1}{25} \sum_{n=1}^{25} A_n\):
\(\sum_{n=1}^{25} A_n = \sum_{n=1}^{25} (2n^2 + n) = 2 \sum_{n=1}^{25} n^2 + \sum_{n=1}^{25} n\)
Using the standard summation formulas \(\sum n^2 = \frac{n(n+1)(2n+1)}{6}\) and \(\sum n = \frac{n(n+1)}{2}\) for \(n = 25\):
\(\sum_{n=1}^{25} n^2 = \frac{25 \times 26 \times 51}{6} = 25 \times 13 \times 17\)
\(\sum_{n=1}^{25} n = \frac{25 \times 26}{2} = 25 \times 13\)
Substitute these back into the required expression:
\(\frac{1}{25} \sum_{n=1}^{25} A_n = \frac{1}{25} [2(25 \times 13 \times 17) + 25 \times 13]\)
\(= 2(13 \times 17) + 13 = 442 + 13 = 455\)

Question 3 · TITA (type in the answer)

In an examination, there were 75 questions. 3 marks were awarded for each correct answer, 1 mark was deducted for each wrong answer and 1 mark was awarded for each unattempted question. Rayan scored a total of 97 marks in the examination. If the number of unattempted questions was higher than the number of attempted questions, then the maximum number of correct answers that Rayan could have given in the examination is

Answer: 24

Solution
Let \( C \), \( W \), and \( U \) be the number of correct, wrong, and unattempted questions respectively.

Total questions: \( C + W + U = 75 \)

Total score: \( 3C - W + U = 97 \)

Subtracting the first equation from the second gives \( 2C - 2W = 22 \), which simplifies to \( W = C - 11 \).

Substituting \( W \) into the first equation: \( C + (C - 11) + U = 75 \implies 2C + U = 86 \).

We are given that unattempted questions are more than attempted questions: \( U > C + W \).

Since \( C + W + U = 75 \), this means \( U > 75 - U \implies 2U > 75 \implies U \ge 38 \).

To maximize \( C \), we must minimize \( U \). From \( 2C + U = 86 \), \( U \) must be an even integer.

The smallest even integer \( U \ge 38 \) is \( 38 \).

Substituting \( U = 38 \) gives \( 2C + 38 = 86 \implies 2C = 48 \implies C = 24 \).

Thus, the maximum number of correct answers is 24.

Question 4 · TITA (type in the answer)

The number of integer solutions of the equation \((x^2 - 10)^{(x^2 - 3x - 10)} = 1\) is

Answer: 4

Solution
The equation \( A^B = 1 \) holds in three distinct cases:

Case 1: \( A = 1 \). Here, \( x^2 - 10 = 1 \implies x^2 = 11 \), which yields no integer solutions.

Case 2: \( B = 0 \) and \( A \neq 0 \). Here, \( x^2 - 3x - 10 = 0 \implies (x - 5)(x + 2) = 0 \). This gives \( x = 5 \) or \( x = -2 \). For both values, \( A = x^2 - 10 \neq 0 \), so these are 2 valid integer solutions.

Case 3: \( A = -1 \) and \( B \) is an even integer. Here, \( x^2 - 10 = -1 \implies x^2 = 9 \implies x = 3 \) or \( x = -3 \).

For \( x = 3 \), \( B = 3^2 - 3(3) - 10 = -10 \) (which is even).

For \( x = -3 \), \( B = (-3)^2 - 3(-3) - 10 = 8 \) (which is even).

Both \( x = 3 \) and \( x = -3 \) are valid integer solutions.

The total number of integer solutions is \( 2 + 2 = 4 \).

Question 5 · TITA (type in the answer)

Suppose for all integers x, there are two functions f and g such that \(f(x) + f(x - 1) - 1 = 0\) and \(g(x) = x^2\). If \(f(x^2 - x) = 5\), then the value of the sum \(f(g(5)) + g(f(5))\) is

Answer: 12

Solution
Given \(f(x) + f(x - 1) = 1\), we have \(f(x) = 1 - f(x - 1)\).

Substituting \(x\) with \(x+1\), we get \(f(x+1) = 1 - f(x) = 1 - (1 - f(x-1)) = f(x-1)\).

Thus, \(f(x)\) is periodic with a period of 2, meaning \(f(\text{even})\) is constant and \(f(\text{odd})\) is constant.

For any integer \(x\), the product \(x^2 - x = x(x - 1)\) is the product of two consecutive integers, which is always an even integer.

We are given \(f(x^2 - x) = 5\), which means \(f(\text{even}) = 5\).

Since \(f(\text{odd}) + f(\text{even}) = 1\), we find \(f(\text{odd}) = 1 - 5 = -4\).

We need to evaluate \(f(g(5)) + g(f(5))\).

First, \(g(5) = 5^2 = 25\). Since 25 is odd, \(f(g(5)) = f(25) = -4\).

Next, since 5 is odd, \(f(5) = -4\). Then \(g(f(5)) = g(-4) = (-4)^2 = 16\).

The sum is \(-4 + 16 = 12\).

The other 52 questions are in the simulator

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CAT 2022 Slot 2 Question Paper with Solutions | Catalyst