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 5 : If \( 9 ^ { x ^ { 2 } + 2 x - 3 } - 4 \left( 3 ^ { x ^ { 2 } + 2 x - 2 } \right) + 27 = 0 \), then the product of all possible values of \( x \) is

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Video Explanation

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Explanatory Answer

Let \(k=3^{x^2+2x-3}\).

Notice that \(3^{x^2+2x-2}=3^{x^2+2x-3+1}=3k\).
Also, \(3^{2x^2+4x-6}=k^2\).

Solve the quadratic equation in terms of \(k\)

\(k^2-4(3k)+27=0\)
\(k^2-12k+27=0\)
\((k-3)(k-9)=0\Rightarrow k=3\) or \(k=9\)

Case 1:
\(3^{x^2+2x-3}=3^1\)
\(x^2+2x-3=1\)
\(x^2+2x-4=0\)
Product of roots \(P_1=\frac{c}{a}=-4\)

Case 2:
\(3^{x^2+2x-3}=3^2\)
\(x^2+2x-3=2\)
\(x^2+2x-5=0\)
Product of roots \(P_2=\frac{c}{a}=-5\)

Compute the total product of all roots
Total Product \(=P_1\times P_2=(-4)\times(-5)=20\)

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The question is " If \( 9 ^ { x ^ { 2 } + 2 x - 3 } - 4 \left( 3 ^ { x ^ { 2 } + 2 x - 2 } \right) + 27 = 0 \), then the product of all possible values of \( x \) is "

Hence, the answer is '\( 20 \)'

Choice 2 is the correct answer.

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