CAT 2022 Question Paper | Quant Slot 3

CAT 2022 Quant was dominated by Arithmetic followed by Algebra. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 11 : 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

Video Explanation

Explanatory Answer

\( \left( \sqrt { \frac { 7 } { 5 } } \right) ^ { 3 x - y } = \frac { 875 } { 2401 } \)
\( \left( \sqrt { \frac { 7 } { 5 } } \right) ^ { 3 x - y } = \left( \frac { 7 } { 5 } \right) ^ { \frac { 3 x - y } { 2 } } = \frac { 125 } { 343 } = \left( \frac { 5 } { 7 } \right) ^ { 3 } = \left( \frac { 7 } { 5 } \right) ^ { - 3 } \)
3x - y = -6
\( \left( \frac { 4 a } { b } \right) ^ { 6 x - y } = \left( \frac { 2 a } { b } \right) ^ { y - 6 x } \)
\( ( 2 ) ^ { 6 x - y } = \left( \frac { 2 a } { b } \right) ^ { 2 y - 12 x } \)
Let R = \( \frac { 2 a } { b } \), Since a and b are real numbers, R is also real.
\( ( 2 ) ^ { 6 x - y } = \left( R ^ { - 2 } \right) ^ { 6 x - y } \)
R = \( \frac { 1 } { \sqrt { 2 } } \), But \( \frac { 1 } { \sqrt { 2 } } \) is not real.
Hence, the given equation can be satisfied only when the powers zero.
That is 6x - y = 0
y = 6x
3x - y = -6
x = 2
y = 12
x + y = 2 + 12 = 14

The answer is '14'