CAT 2025 Question Paper | Quant Slot 2

CAT Questions

CAT 2025 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 2 : If \( \log _ { 64 } x ^ { 2 } + \log _ { 8 } \sqrt { y } + 3 \log _ { 512 } ( \sqrt { y } z ) = 4 \), where \( x , y \) and \( z \) are positive real numbers, then the minimum possible value of \( ( x + y + z ) \) is

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Video Explanation

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Explanatory Answer

Simplify all logarithms to base 2

\(\log_{64}x^2=\frac{2}{6}\log_2x=\frac{1}{3}\log_2x\)
\(\log_8\sqrt{y}=\frac{1}{2}\cdot\frac{1}{3}\log_2y=\frac{1}{6}\log_2y\)
\(3\log_{512}(\sqrt{y}z)=3\cdot\frac{1}{9}\log_2(\sqrt{y}z)\)
\(=\frac{1}{3}\left(\frac{1}{2}\log_2y+\log_2z\right)\)
\(=\frac{1}{6}\log_2y+\frac{1}{3}\log_2z\)

Combine all logarithmic terms

\(\frac{1}{3}\log_2x+\frac{1}{6}\log_2y+\frac{1}{6}\log_2y+\frac{1}{3}\log_2z=4\)
\(\frac{1}{3}\log_2x+\frac{1}{3}\log_2y+\frac{1}{3}\log_2z=4\)
\(\frac{1}{3}\log_2xyz=4\)
\(\log_2xyz=12\Rightarrow xyz=2^{12}=4096\)

Apply the AM-GM Inequality

\(\frac{x+y+z}{3}\geq\sqrt[3]{xyz}\)
\(\frac{x+y+z}{3}\geq\sqrt[3]{4096}=16\)
\(x+y+z\geq3\times16=48\)

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The answer is '\( 48 \)'

Choice 2 is the correct answer.

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