CAT 2024 Question Paper | Quant Slot 2

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 9 If \(a, b\) and \(c\) are positive real numbers such that \(a>10 \geq b \geq c\) and \(\frac{\log _8(a+b)}{\log _2 c}+\frac{\log _{27}(a-b)}{\log _3 c}=\frac{2}{3}\), then the greatest possible integer value of \(a\) is

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Video Explanation

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Explanatory Answer

\[ \begin{array}{l} \log _8(a+b) / \log _2 c=\{\log (a+b) / \log 8\} /\{\log c / \log 2\}=(1 / 3) \log _c(a+b) \\ \log _{27}(a-b) / \log _3 c=\{\log (a-b) / \log 27\} /\{\log c / \log 3\}=(1 / 3) \log _c(a-b) \\ (1 / 3) \log _c(a+b)+(1 / 3) \log { }_c(a-b)=(1 / 3) \log _c\{(a+b)(a-b)\}=2 / 3 \\ \log _c\{(a+b)(a-b)\}=2 \\ c^2=a^2-b^2 \\ a^2=b^2+c^2 \end{array} \] a should be an integer. The greatest possible integer value of \(a\) is possible if \(b=c=10\) \[ a^2 \leq 200 \] The maximum value of \(a=14\)

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The question is "If \(a, b\) and \(c\) are positive real numbers such that \(a>10 \geq b \geq c\) and \(\frac{\log _8(a+b)}{\log _2 c}+\frac{\log _{27}(a-b)}{\log _3 c}=\frac{2}{3}\), then the greatest possible integer value of \(a\) is"

Hence, the answer is '14'

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