CAT 2023 Question Paper Slot -3 | QA
CAT Quantitative Aptitude (QA) | CAT Previous Year Paper
Unlike previous years, the CAT 2023 Quant section was not dominated by Arithmetic, though it remained the most tested topic, followed by Algebra. Within Arithmetic, questions were primarily focused on Speed-Time-Distance, Mixture, and Alligations. The section included eight TITA (Type-in-the-answer) questions. Overall, the CAT 2023 Quantitative Aptitude Section was difficult paper compared to the previous years.
CAT 2023 Slot 1 - Quantitative Aptitude
CAT 2023 Slot 3 – QA
1. If x is a positive real number such that x³ + (1/3)8 = 47, then the value of x9 + (1/x)9
A. 40√5
B. 36√5
C. 36√4
D. 30√5
CAT 2023 Slot 3 – QA
2. For some real numbers a and b, the system of equations x + y = 4 and (a + 5)x + (b2 – 15)y = 8b has infinitely many solutions for x and y. Then, the maximum possible value of ab is
A. 25
B. 33
C. 55
D. 15
CAT 2023 Slot 3 – QA
3. Let n and m be two positive integers such that there are exactly 41 integers greater than 8m and less than 8n which can be expressed as powers of 2. Then, the smallest possible value of n + m is
A. 44
B. 42
C. 14
D. 16
4. For a real number x, 1/2, log3(2x – 9)/ log34 and log5 (2x + 17/2) / log54 are in an arithmetic progression, then the common difference is
A. log4 7/2
B. log4 7
C. log4 3/2
D. log4 23/2
CAT 2023 Slot 3 – QA
5. The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is
CAT 2023 Slot 3 – QA
6. A quadratic equation x² + 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
CAT 2023 Slot 3 – QA
7. Let n be any natural number such that 5n-1 <3n+1. Then, the least integer value of m that satisfies 3n+1 < 2n+m for each such n, isCAT 2023 Slot 3 – QA
8. A merchant purchases a cloth at a rate of Rs.100 per meter and receives 5 cm length of cloth free for every 100 cm length of cloth purchased by him. He sells the same cloth at a rate of Rs.110 per meter but cheats his customers by giving 95 cm length of cloth for every 100 cm length of cloth purchased by the customers. If the merchant provides a 5% discount, the resulting profit earned by him is
A. 15.5%
B. 9.7%
C. 16%
D. 4.2%
CAT 2023 Slot 3 – QA
9. Anil mixes cocoa with sugar in the ratio 3:2 to prepare mixture A, and coffee with sugar in the ratio 7:3 to prepare mixture B. He combines mixtures A and B in the ratio 2:3 to make a new mixture C. If he mixes C with an equal amount of milk to make a drink, then the percentage of sugar in this drink will be
A. 21%
B. 17%
C. 16%
D. 24%
CAT 2023 Slot 3 – QA
10. There are three persons A, B and C in a room. If a person D joins the room, the average weight of the persons in the room reduces by x kg. Instead of D, if person E joins the room, the average weight of the persons in the room increases by 2x kg. If the weight of E is 12 kg more than that of D, then the value of x is
A. 0.5
B. 1.5
C. 2
D. 1
CAT 2023 Slot 3 – QA
11. Rahul, Rakshita and Gurmeet, working together, would have taken more than 7 days to finish a job. On the other hand, Rahul and Gurmeet, working together would have taken less than 15 days to finish the job. However, they all worked together for 6 days, followed by Rakshita, who worked alone for 3 more days to finish the job. If Rakshita had worked alone on the job then the number of days she would have taken to finish the job, cannot be
A. 20
B. 21
C. 17
D. 16
CAT 2023 Slot 3 – QA
12. A boat takes 2 hours to travel downstream a river from port A to port B, and 3 hours to return to port A. Another boat takes a total of 6 hours to travel from port B to port A and return to port B. If the speeds of the boats and the river are constant, then the time, in hours, taken by the slower boat to travel from port A to port B is
A. 3 (√5 – 1)
B. 12(√5 – 2)
C. 3 (3+√5)
D. 3 (3-√5)
CAT 2023 Slot 3 – QA
13. The population of a town in 2020 was 100000. The population decreased by y% from the year 2020 to 2021, and increased by x% from the year 2021 to 2022, where x and y are two natural numbers. If population in 2022 was greater than the population in 2020 and the difference between x and y is 10, then the lowest possible population of the town in 2021 was
A. 74000
B. 73000
C. 75000
D. 72000
CAT 2023 Slot 3 – QA
14. A fruit seller has a stock of mangoes, bananas and apples with at least one fruit of each type. At the beginning of a day, the number of mangoes make up 40% of his stock. That day, he sells half of the mangoes, 96 bananas and 40% of the apples. At the end of the day, he ends up selling 50% of the fruits. The smallest possible total number of fruits in the stock at the beginning of the day is
CAT 2023 Slot 3 – QA
15. Gautam and Suhani, working together, can 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 on that day to exactly make up for it. Then, the number of days required by the faster worker to complete the job working alone is
CAT 2023 Slot 3 – QA
16. The number of coins collected per week by two coin-collectors A and B are in the ratio 3: 4. If the total number of coins collected by A in 5 weeks is a multiple of 7, and the total number of coins collected by B in 3 weeks is a multiple of 24, then the minimum possible number of coins collected by A in one week is
CAT 2023 Slot 3 – QA
17. 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
A.√2 : 1
B.√5 : 1
C. 2 : 1
D. 1 : 1
18. Let ΔABC be an isosceles triangle such that AB and AC are of equal length. AD is the altitude from A on BC and BE is the altitude from B on AC. If AD and BE intersect at O such that ∠AOB = 105°, then AD/BE equals
A. 2cos15°
B. sin15°
C. cos15°
D. 2sin15°
CAT 2023 Slot 3 – QA
19.
20. Let an = 46+8n and bn = 98+4n be two sequences for natural numbers n ≤ 100. Then, the sum of all terms common to both the sequences is
A. 15000
B. 14798
C. 14900
D. 14602
CAT 2023 Slot 3 – QA
21. 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
A. 16/11
B. 15/13
C. 27/12
D. 15/8
The answer option is A.
If 1/(√y+√z) is the arithmetic mean of 1/√x +1/√z & 1/(√x+√y)
Then = 2/(√y+√x) = 1/(√x+√z) + 1/(√x+√y)
Rationalizing,
(2 √y-√z)/(y-z) = (√x-√y)/(x-z) + (√x-√y)/(x-y) ———- (1)
So, starting with options
If y, x & z are in AP,
⇒x-y=z-x=d & z-y=2d
⇒from (1),
RHS,
= (√x-√z)/(-d) + (√x-√y)/d=(√x-√y+√z-√x)/d
= (√z-√y)/d
LHS =(2(√x-√(y)))/(-2d)=(√z-√y)/d= RHS.
Hence, LHS = RHS
So, ⇒(b)is true.
Using the examples of the circle L
x^2+y^2+4x-6y-3=0, we can find the point of center of the circle and the radius (x^2+4x+4)+(y^2-6y+9)-4-9-3=0
(〖x+2)]^2+(〖y-3)〗^2=16
So, radius (r) = 4, Centre = (-2, 3)
DIAGRAM
OP being the radius of bigger circle (L) with center at O.
We can find the measurement of OP using trigonometry.
Sin 30^0 =OQ/OP=1/2=4/OP
⇒OP=8=radius of bigger circle (L)
Therefore, Example of the bigger circle (L) with centre (-2, 3)
and radius 8 is 〖(x+2)〗^2+〖(y-3)〗^2=8^2
Substituting x=6,in the above circle equation we can find.
〖(6+2)〗^2+〖(y-3)〗^2=64
8^2+〖(y-3)〗^2=64
(y-3)^2=0
⇒ y = 3.
So, the solution of this question is (a) (6, 3).
CAT 2023 Slot 3 – QA
22. Suppose f(x, y) is a real-valued function such that f(3x+2y, 2x-5y) = 19x, for all real numbers x and y. The value of x for which f(x, 2x) = 27 is