CAT 2024 Question Paper | Quant Slot 1

CAT Questions

CAT 2024 Quant was dominated by Algebra followed by Arithmetic. 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 3 :The sum of all real values of \(k\) for which \(\left(\frac{1}{8}\right)^k \times\left(\frac{1}{32768}\right)^{\frac{1}{3}}=\frac{1}{8} \times\left(\frac{1}{32768}\right)^{\frac{1}{k}}\), is

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Video Explanation

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Explanatory Answer

$$ \begin{array}{l} 1 / 8=2^{-3} \\ 1 / 32767=2^{-15} \\ \left(2^{-3}\right)^k *\left(2^{-15}\right)^{1 / 3}=\left(2^{-3}\right)^*\left(2^{-15}\right)^{1 / k} \\ 2^{-3 k-5}=2^{-3-15 / k} \end{array} $$ Equating the power on both sides, we get $$ \begin{array}{l} -3 k-5=-3-15 / k \\ 3 k+2-15 / k=0 \end{array} $$ \(3 k^2+2 k-15=0\) is an quadratic equation of the form \(a x^{\wedge} 2+b x+c=0\)
Sum of all real values of \(k=-b / a=-2 / 3\)

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The question is "The sum of all real values of \(k\) for which \(\left(\frac{1}{8}\right)^k \times\left(\frac{1}{32768}\right)^{\frac{1}{3}}=\frac{1}{8} \times\left(\frac{1}{32768}\right)^{\frac{1}{k}}\), is "

Hence, the answer is '\(-\frac{2}{3}\)'

Choice B is the correct answer.

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