CAT 2023 Slot 3 QA Question Paper is available here for free download. CAT 2023 Slot 3 paper has been conducted on November 26 from 4.30 PM to 6.30 PM. CAT 2023 Slot 3 question paper QA comprises 22 questions to be attempted in 40 minutes. According to initial students reaction, CAT 2023 Slot 3 QA was moderate to difficult.

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CAT 2023 Slot 3 QA Question Paper with Solutions PDF

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CAT 2023 Slot 3 QA Questions with Solution

Q. No. Question (with options) Correct Answer Solution
1 For a real number x, if 1/2, log4((x^2 - 9)/x), and log4((x^2 + 17)/x) are in an arithmetic progression, then the common difference is:
1. log4(3/2)
2. log4(23/2)
3. log4(7/2)
4. log4(7)		 3. log4(7/2) Since the terms form an arithmetic progression, we can equate the differences between them. Solving for x, we determine that the common difference is log4(7/2).
2 Let n and m be two positive integers such that there are exactly 41 integers greater than 8^n and less than 8^m that can be expressed as powers of 2. Then, the smallest possible value of n + m is:
1. 44
2. 14
3. 16
4. 42		 3. 16 Using the constraints on the powers of 2 between 8^n and 8^m, we calculate the range and find that n + m = 16 is the smallest possible sum.
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. 55
3. 15
4. 25		 1. 33 For the system to have infinitely many solutions, the two equations must be proportional. Setting up proportional conditions and solving for a and b, we find that the maximum possible value of ab is 33.
4 If x is a positive real number such that x^8 + (1/x)^8 = 47, then the value of x^9 + (1/x)^9 is:
1. 34 * sqrt(5)
2. 40 * sqrt(5)
3. 30 * sqrt(5)
4. 36 * sqrt(5)		 1. 34 * sqrt(5) Breaking down x^8 + (1/x)^8 into simpler terms, we use identities to find that x^9 + (1/x)^9 equals 34 * sqrt(5).
5 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:
Options omitted for brevity		 9 Given conditions on the reciprocals, we apply Vieta’s formulas: sum of roots equals -b and product of roots equals c. By translating the reciprocal conditions, we solve for the maximum value of b + c, which is 9.
6 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:
1. 5
2. 4
3. 3
4. 6		 5 To solve, we take logarithms and simplify both inequalities. Testing small values of n, we find that m must be at least 5 to satisfy all cases.
7 The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is:
1. 468
2. 372
3. 600
4. 504		 468 Using the divisor formula, we identify the smallest numbers with exactly 15 divisors. The two smallest such numbers are 144 and 324, and their sum is 468.
8 A merchant purchases cloth at a rate of Rs.100 per meter and receives 5 cm free for every 100 cm purchased. He sells it at Rs.110 per meter but gives 95 cm for every 100 cm. If he provides a 5% discount, the profit percentage is:
1. 9.7%
2. 15.5%
3. 4.2%
4. 16%		 15.5% Calculating the effective cost and selling price per cm, and factoring in the free and reduced lengths, we find the profit margin to be 15.5%.
9 A boat takes 2 hours to travel downstream from port A to B, and 3 hours to return. Another boat takes 6 hours for a round trip from B to A. Find the time taken by the slower boat to travel from A to B:
1. 3(3 + sqrt(5))
2. 3(3 - sqrt(5))
3. 3(sqrt(5) - 1)
4. 12(sqrt(5) - 2)		 3(3 - sqrt(5)) Using relative speeds for downstream and upstream movement, we derive an equation for the slower boat's travel time from A to B, resulting in 3(3 - sqrt(5)).
10 There are three persons A, B, and C in a room. If a fourth person D joins, the average weight reduces by x kg. If E joins instead, the average increases by 2x kg. If E weighs 12 kg more than D, find x:
1. 2
2. 1
3. 1.5
4. 0.5		 Option 2. 1 Setting up equations based on the change in average weight, we find that the difference in weights provides x = 1 as the solution.
11 The population of a town was 100,000 in 2020. It decreased by y% in 2021 and increased by x% in 2022. If the 2022 population is greater than 2020’s and x - y = 10, the minimum population in 2021 was:
1. 73,000
2. 75,000
3. 74,000
4. 72,000		 73,000 Using percentage decreases and increases, we calculate that for the population to exceed 100,000 in 2022 with x - y = 10, the minimum population in 2021 must be 73,000.
12 Anil mixes cocoa and sugar in the ratio 3:2 for mixture A, and coffee and sugar in 7:3 for mixture B. He combines A and B in a 2:3 ratio to make mixture C, then adds equal milk to make a drink. The sugar percentage in the drink is:
1. 24%
2. 16%
3. 17%
4. 21%		 17% Calculating the sugar content in each mixture, then adjusting for the dilution by milk, we find that the sugar percentage in the final drink is 17%.
13 Rahul, Rakshita, and Gurmeet can finish a job in over 7 days. Rahul and Gurmeet together would take under 15 days. They all worked together for 6 days, and Rakshita worked alone for 3 more days to complete the job. Rakshita’s time to finish alone would be:
1. 20
2. 21
3. 16
4. 17		 21 Using inequalities to estimate each person's work rate, we determine that Rakshita alone would need at least 21 days to complete the job.
14 The number of coins collected per week by two collectors A and B are in the ratio 3:4. If A collects coins in multiples of 7 over 5 weeks and B collects multiples of 24 over 3 weeks, the minimum coins A collects in one week is:
1. 28
2. 42
3. 21
4. 56		 42 Using the ratios and divisibility conditions, we find the smallest value for weekly collection that satisfies both conditions. This results in A collecting 42 coins weekly.
15 Gautam and Suhani can finish a job in 20 days. If Gautam works at 60% of his usual rate, Suhani must do 150% of her rate to compensate. The faster worker’s solo time to complete the job is:
1. 36 days
2. 30 days
3. 25 days
4. 40 days		 36 Setting up equations for adjusted work rates, we determine each worker's individual rate. Gautam, the faster worker, would take 36 days to complete the job alone.
16 A fruit seller starts with mangoes, bananas, and apples, with mangoes making up 40% of stock. Selling half the mangoes, 96 bananas, and 40% of apples, he sells 50% of the total. Minimum stock at start is:
1. 320
2. 340
3. 360
4. 380		 340 Using the initial proportions and sold quantities, we set equations to find the smallest initial stock that allows for these proportions, resulting in a minimum stock of 340.
17 In an isosceles triangle ABC, AB = AC. Altitudes AD and BE intersect at O such that ∠AOB = 105°. The length of AD is:
1. 2 cos 15°
2. sin 15°
3. 2 sin 15°
4. cos 15°		 2 cos 15° Using trigonometric identities in the isosceles triangle and given angle, we find AD’s length as 2 cos 15°.
18 A rectangle with maximum area is inscribed in a semicircle of radius 2 cm. The ratio of the rectangle's largest to smallest side is:
1. 1:1
2. 2:1
3. sqrt(5):1
4. sqrt(2):1		 2:1 Applying geometry and Pythagorean theorem in the semicircle’s constraints, we find that the ratio of the rectangle's sides is 2:1 for maximum area.
19 In a regular polygon, any interior angle exceeds the exterior angle by 120°. The number of diagonals is:
1. 36
2. 54
3. 72
4. 60		 54 Using the angle relationship in a regular polygon, we calculate that the polygon has 12 sides, resulting in 54 diagonals.
20 Calculate the value of the series:
(1 + 1/4 + 1/16 + ...) + (1/3)(1 + 1/4 + 1/16 + ...) + (1/9)(1 + 1/4 + ...) + ...
1. 15/8
2. 15/13
3. 16/11
4. 27/12		 16/11 Recognizing the nested sums as geometric progressions, we simplify to find that the series sums to 16/11.
21 Let an = 46 + 8n and bn = 98 + 4n be sequences for n ≤ 100. Find the sum of terms common to both sequences:
1. 14602
2. 14798
3. 15000
4. 14900		 14900 Identifying the common terms and using the sum of arithmetic progression formula, we find the total sum of common terms is 14900.
22 If f(x, y) = 19x for all real numbers x and y, find x when f(x, 2x) = 27:
1. 4
2. 2
3. 3
4. 5		 3 Setting f(x, 2x) equal to 27 and solving for x, we determine that x = 3 satisfies the equation.