cat 2023 Complete Paper Solution | Slot 3

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

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

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

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

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

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