CAT 2025 Question Paper | Quant Slot 1

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 19 : For any natural number \(k\), let \(a_k=3^k\). The smallest natural number \(m\) for which \(\left\{\left(a_1\right)^1 \times\left(a_2\right)^2 \times \ldots \times\left(a_{20}\right)^{20}\right\}<\left\{a_{21} \times a_{22} \times \ldots \times a_{(20+m)}\right\}\), is

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Video Explanation

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Explanatory Answer

Using the sum of squares formula for \(n=20\):

\(LHS=a_1^1\times a_2^2\times\ldots\times a_{20}^{20}\)
\(LHS=3^{1^2}\times3^{2^2}\times\ldots\times3^{20^2}=3^{1^2+2^2+\ldots+20^2}\)

\[ \sum_{k=1}^{n}k^2=\frac{n(n+1)(2n+1)}{6} \]
\(n=20\)
Exponent of LHS \(=\frac{20\times21\times41}{6}=2870\)

\(RHS=a_{21}\times a_{22}\times\ldots\times a_{20+m}\)
\(RHS=3^{21}\times3^{22}\times\ldots\times3^{20+m}=3^{21+22+\ldots+20+m}\)

Exponent of RHS \(=\frac{m}{2}(21+20+m)=\frac{m(41+m)}{2}\)

We require \(LHS<RHS\):
\(2870<\frac{m(41+m)}{2}\)
\(5740<m(41+m)=m^2+41m\)
\(m^2+41m-5740>0\)

Testing integer values for \(m\):

For \(m=57\):
\(m(41+m)=57\times98=5586<5740\) (False)

For \(m=58\):
\(m(41+m)=58\times99=5742>5740\) (True)

The smallest natural number \(m\) is \(58\).

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The answer is '58'

Choice 3 is the correct answer.

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