cat 2022 Question 11

Question

If $\left(\sqrt{\frac{7}{5}}\right)^{3 x-y}=\frac{875}{2401}$ and $\left(\frac{4 a}{b}\right)^{6 x-y}=\left(\frac{2 a}{b}\right)^{y-6 x}$, for all non-zero real values of a and b, then the value of x + y is

Accepted Answer

$\left(\sqrt{\frac{7}{5}}\right)^{3x - y} = \frac{875}{2401}$

$\left(\frac{7}{5}\right)^{\frac{3x - y}{2}} = \frac{125}{343}$

$\left(\frac{7}{5}\right)^{\frac{3x - y}{2}} = \left(\frac{7}{5}\right)^{-3}$

3x − y = −6

$\left(\frac{4a}{b}\right)^{6x - y} = \left(\frac{2a}{b}\right)^{y - 6x}$

Therefore, y = 6x as the bases are different so the power should be zero for the results to be equal.

3x − y = −6

or, 3x − 6x = −6

or x = 2

y = 6x = 12

Question 11.

Question Explanation