CAT 2025 Question Paper | Quant Slot 3

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 11 : If \( f ( x ) = \left( x ^ { 2 } + 3 x \right) \left( x ^ { 2 } + 3 x + 2 \right) \), then the sum of all real roots of the equation \( \sqrt { f ( x ) + 1 } = 9701 \), is

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Video Explanation

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Explanatory Answer

Given:
\[ f(x)=(x^2+3x)(x^2+3x+2) \]

Let:
\[ n=x^2+3x+1 \]
Then:
\[ x^2+3x=n-1 \]
\[ x^2+3x+2=n+1 \]
Therefore:
\[ f(x)=(n-1)(n+1)=n^2-1 \]

Given:
\[ \sqrt{f(x)+1}=9701 \]
\[ \sqrt{n^2}=9701\Rightarrow |n|=9701 \]

Case 1: \(n=9701\)
\[ x^2+3x+1=9701 \Rightarrow x^2+3x-9700=0 \]
Its discriminant is positive, so it has two real roots. Their sum is:
\[ -\frac{3}{1}=-3 \]

Case 2: \(n=-9701\)
\[ x^2+3x+1=-9701 \Rightarrow x^2+3x+9702=0 \]
Its discriminant is negative, so there are no real roots.

Sum of all real roots \(=-3\).

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The question is " If \( f ( x ) = \left( x ^ { 2 } + 3 x \right) \left( x ^ { 2 } + 3 x + 2 \right) \), then the sum of all real roots of the equation \( \sqrt { f ( x ) + 1 } = 9701 \), is "

Hence, the answer is '\( - 3 \)'

Choice 2 is the correct answer.

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