CAT 2022 Slot 3 QA Q11 Logarithms & Exponents

Question 11 of the Quantitative Ability (QA) section, CAT 2022 Slot 3, with CAT's official answer and a worked solution.

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

TITA: type in the answer. There are no options and no negative marking.

Answer: 14 Logarithms & Exponents TITA

Solution

  1. \(\frac{875}{2401}=\frac{5^3\cdot7}{7^4}=\left(\frac75\right)^{-3}\), and the left side is \(\left(\frac75\right)^{\frac{3x-y}2}\), so \(3x-y=-6\).
  2. \(\left(\frac{4a}b\right)^{6x-y}=\left(\frac{2a}b\right)^{-(6x-y)}\) for all non-zero \(a,b\) means \(\left(\frac{8a^2}{b^2}\right)^{6x-y}=1\) always, forcing \(6x-y=0\).
  3. So \(y=6x\), \(3x-6x=-6\Rightarrow x=2,\ y=12\), and \(x+y=14\).

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