CAT 2023 · Slot 3 — Quantitative Ability

All 22 Quant questions from the CAT 2023 · Slot 3 paper, with the correct answer for each.

Question 1

If $x$ is a positive real number such that $x^8 + \frac{1}{x^8} = 47$, then the value of $x^9 + \frac{1}{x^9}$ is

  1. $40\sqrt{5}$
  2. $30\sqrt{5}$
  3. $36\sqrt{5}$
  4. $34\sqrt{5}$

Correct answer: D · $34\sqrt{5}$

Question 2

Let $n$ and $m$ be two positive integers such that there are exactly 41 integers greater than $8^m$ and less than $8^n$ which can be expressed as powers of 2. Then, the smallest possible value of $n + m$ is

  1. 42
  2. 44
  3. 14
  4. 16

Correct answer: D · 16

Question 3

For some real numbers $a$ and $b$, the system of equations $x + y = 4$ and $(a+5)x + (b^2-15)y = 8b$ has infinitely many solutions for $x$ and $y$. Then, the maximum possible value of $ab$ is

  1. 33
  2. 25
  3. 15
  4. 55

Correct answer: A · 33

Question 4

For a real number $x$, if 1/2, $\frac{\log_3(2^x - 9)}{\log_3(4)}$, and $\frac{\log_5(2^x + 17/2)}{\log_5(4)}$ are in an arithmetic progression, then the common difference is

  1. $\log_4(3/2)$
  2. $\log_4 7$
  3. $\log_4(23/2)$
  4. $\log_4(7/2)$

Correct answer: D · $\log_4(7/2)$

Question 5

Let $n$ be any natural number such that $5^{n-1} < 3^{n+1}$. Then, the least integer value of $m$ that satisfies $3^{n+1} < 2^{n+m}$ for each such $n$, is

Correct answer: 5

Question 6

The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is

Correct answer: 468

Question 7

A quadratic equation $x^2 + bx + c = 0$ has two real roots. If the difference between the reciprocals of the roots is 1/3, and the sum of the reciprocals of the squares of the roots is 5/9, then the largest possible value of $(b + c)$ is

Correct answer: 9

Question 8

A merchant purchases cloth at Rs.100 per meter and receives 5 cm free for every 100 cm purchased. He sells at Rs.110 per meter but gives 95 cm for every 100 cm the customer pays for. If the merchant provides a 5% discount, the resulting profit earned by him is

  1. 4.2%
  2. 9.7%
  3. 15.5%
  4. 16%

Correct answer: C · 15.5%

Question 9

Rahul, Rakshita and Gurmeet together would take more than 7 days to finish a job. Rahul and Gurmeet together would take less than 15 days. They all work together for 6 days, then Rakshita works alone for 3 more days to finish. If Rakshita had worked alone, the number of days she would take CANNOT be

  1. 20
  2. 17
  3. 16
  4. 21

Correct answer: D · 21

Question 10

The population of a town in 2020 was 100000. It decreased by y% from 2020 to 2021, then increased by x% from 2021 to 2022, where x and y are natural numbers. If the 2022 population exceeded the 2020 population and x - y = 10, the lowest possible population in 2021 was

  1. 72000
  2. 74000
  3. 73000
  4. 75000

Correct answer: C · 73000

Question 11

Anil mixes cocoa with sugar in ratio 3:2 (mixture A) and coffee with sugar in ratio 7:3 (mixture B). He combines A and B in ratio 2:3 to make mixture C. If he mixes C with an equal amount of milk to make a drink, the percentage of sugar in the drink is

  1. 17
  2. 16
  3. 21
  4. 24

Correct answer: A · 17

Question 12

There are three persons A, B and C in a room. If D joins, the average weight reduces by x kg. Instead of D, if E joins, the average increases by 2x kg. If E is 12 kg heavier than D, then the value of x is

  1. 2
  2. 0.5
  3. 1
  4. 1.5

Correct answer: C · 1

Question 13

A boat takes 2 hours downstream from A to B and 3 hours to return. Another boat takes a total of 6 hours to travel from B to A and back to B. If speeds are constant, the time (in hours) taken by the slower boat to travel from A to B is

  1. $12(\sqrt{5}-2)$
  2. $3(3+\sqrt{5})$
  3. $3(\sqrt{5}-1)$
  4. $3(3-\sqrt{5})$

Correct answer: D · $3(3-\sqrt{5})$

Question 14

The number of coins collected per week by two collectors A and B are in ratio 3:4. If A's total over 5 weeks is a multiple of 7 and B's total over 3 weeks is a multiple of 24, then the minimum possible number of coins collected by A in one week is

Correct answer: 42

Question 15

A fruit seller has mangoes, bananas and apples (at least one of each). Mangoes are 40% of the stock. He sells half the mangoes, 96 bananas, and 40% of the apples, ending up selling 50% of all fruits. The smallest possible total number of fruits at the start is

Correct answer: 340

Question 16

Gautam and Suhani together finish a job in 20 days. If Gautam does only 60% of his usual work on a day, Suhani must do 150% of her usual work that day to exactly make up for it. The number of days required by the faster worker alone is

Correct answer: 36

Question 17

Let triangle ABC be isosceles with AB = AC. AD is the altitude from A on BC and BE the altitude from B on AC. If AD and BE intersect at O such that angle AOB = 105 degrees, then AD/BE equals

  1. sin15°
  2. cos15°
  3. 2cos15°
  4. 2sin15°

Correct answer: C · 2cos15°

Question 18

A rectangle with the largest possible area is drawn inside a semicircle of radius 2 cm. Then, the ratio of the lengths of the largest to the smallest side of this rectangle is

  1. 2:1
  2. 1:1
  3. $\sqrt{5}:1$
  4. $\sqrt{2}:1$

Correct answer: A · 2:1

Question 19

In a regular polygon, any interior angle exceeds the exterior angle by 120 degrees. Then, the number of diagonals of this polygon is

Correct answer: 54

Question 20

The value of 1 + (1 + 1/3)(1/4) + (1 + 1/3 + 1/9)(1/16) + (1 + 1/3 + 1/9 + 1/27)(1/64) + ... is

  1. 15/13
  2. 27/12
  3. 15/8
  4. 16/11

Correct answer: D · 16/11

Question 21

Let a_n = 46 + 8n and b_n = 98 + 4n be two sequences for natural numbers n <= 100. Then, the sum of all terms common to both sequences is

  1. 14900
  2. 14798
  3. 15000
  4. 14602

Correct answer: A · 14900

Question 22

Suppose f(x, y) is a real-valued function such that f(3x + 2y, 2x - 5y) = 19x for all real x and y. The value of x for which f(x, 2x) = 27, is

Correct answer: 3