CAT 2023 Slot 3 | Past Year Question Paper

Question 1.

For a real number x , if $\frac{1}{2}, \frac{\log _3\left(2^x-9\right)}{\log _3 4}$ , and $\frac{\log _5\left(2^x+\frac{17}{2}\right)}{\log _5 4}$ are in an arithmetic progression, then the common difference is

A

\log _4 7

B

\log _4\left(\frac{3}{2}\right)

C

\log _4\left(\frac{7}{2}\right)

D

\log _4\left(\frac{23}{2}\right)

Question 2.

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

A

14

B

42

C

16

D

44

Question 3.

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

A

15

B

33

C

55

D

25

Question 4.

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

A

40 \sqrt{5}

B

36 \sqrt{5}

C

30 \sqrt{5}

D

34 \sqrt{5}

Question 5.

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

Question 6.

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

Question 7.

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

Question 8.

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

17

B

21

C

16

D

20

Question 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

24

C

16

D

17

Question 10.

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

75000

C

72000

D

73000

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cat 2023 Complete Paper Solution | Slot 3

Question 1.

For a real number x , if $\frac{1}{2}, \frac{\log _3\left(2^x-9\right)}{\log _3 4}$ , and $\frac{\log _5\left(2^x+\frac{17}{2}\right)}{\log _5 4}$ are in an arithmetic progression, then the common difference is

Question 2.

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

Question 3.

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

Question 4.

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

Question 5.

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

Question 6.

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

Question 7.

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

Question 8.

Question 9.

Question 10.

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cat 2023 Complete Paper Solution | Slot 3

Question 1.

Question 2.

Let n and m be two positive integers such that there are exactly 41 integers greater than 8m8^m8m and less then 8n8^n8n, which can be expressed as powers of 2. Then, the smallest possible value of n + m is

Question 3.

For some real numbers a and b, the system of equations x + y = 4 and (a+5)x+(b2−15)y=8b(a+5) x+\left(b^2-15\right) y=8 b(a+5)x+(b2−15)y=8b has infinitely many solutions for x and y. Then, the maximum possible value of ab is

Question 4.

If x is a positive real number such that x8+(1x)8=47x^8+\left(\frac{1}{x}\right)^8=47x8+(x1​)8=47, then the value of x9+(1x)9x^9+\left(\frac{1}{x}\right)^9x9+(x1​)9 is

Question 5.

A quadratic equation x2+bx+c=0x^2+b x+c=0x2+bx+c=0 has two real roots. If the difference between the reciprocals of the roots is 13\frac{1}{3}31​, and the sum of the reciprocals of the squares of the roots is 59\frac{5}{9}95​, then the largest possible value of (b + c) is

Question 6.

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

Question 7.

Let n be any natural number such that 5n−1<3n+15^{n-1} \lt 3^{n+1}5n−1<3n+1. Then, the least integer value of m that satisfies 3n+1<2n+m3^{n+1} \lt 2^{n+m}3n+1<2n+m for each such n, is

Question 8.

Question 9.

Question 10.

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

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

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

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

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