CAT 2022 Slot 2 QA Question & Solution

AlgebraHard

Question

The number of distinct integer values of n satisfying $\frac{4-\log_{2}n}{3-\log_{4}n} < 0$, is

Solution

Let $\ \log_2n=y$

$\ \ \frac{\ 4-y}{3-\frac{y}{2}}<0$

$\ \ \left(4-y\right)\left(3-\frac{y}{2}\right)<0$

$\ \ \left(4-y\right)\left(6-y\right)<0$

$\ \ \left(y-4\right)\left(y-6\right)<0$

$4 < y < 6$

$4<\log_2n<6$

$2^4

$16

n can take values from 17 to 63(inclusive).

The number of n values possible = 47