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 1 : If \( m \) and \( n \) are integers such that \( ( m + 2 n ) ( 2 m + n ) = 27 \), then the maximum possible value of \( 2 m - 3 n \) is

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Video Explanation

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Explanatory Answer

Identify all integer factor pairs of 27

Since \(m\) and \(n\) are integers, \(m+2n\) and \(2m+n\) must be integer factor pairs of 27:
\((9,3),\ (3,9),\ (27,1),\ (1,27),\)
\((-9,-3),\ (-3,-9),\ (-27,-1),\ (-1,-27)\)

Test factor pair \((3,9)\)
\(m+2n=3\)
\(2m+n=9\)

Adding both equations gives \(3m+3n=12\Rightarrow m+n=4\).

Subtracting \(m+n=4\) from \(m+2n=3\) yields \(n=-1\) and \(m=5\).
\(2m-3n=2(5)-3(-1)=10+3=13\)

Test factor pair \((-9,-3)\)
\(m+2n=-9\)
\(2m+n=-3\)

Adding both equations gives \(3m+3n=-12\Rightarrow m+n=-4\).

Subtracting \(m+n=-4\) from \(m+2n=-9\) yields \(n=-5\) and \(m=1\).
\(2m-3n=2(1)-3(-5)=2+15=17\)

(Note: Factor pairs containing 27 and 1 yield non-integer values for \(m\) and \(n\).)

The maximum possible value of \(2m-3n\) is 17.

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The question is " If \( m \) and \( n \) are integers such that \( ( m + 2 n ) ( 2 m + n ) = 27 \), then the maximum possible value of \( 2 m - 3 n \) is "

Hence, the answer is '17'

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