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 15 : For real values of \( x \), the range of the function \( f ( x ) = \frac { 2 x - 3 } { 2 x ^ { 2 } + 4 x - 6 } \) is

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Video Explanation

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Explanatory Answer

Let: \[ y=\frac{2x-3}{2x^2+4x-6} \]

Rearranging:
\[ y(2x^2+4x-6)=2x-3 \]
\[ 2yx^2+(4y-2)x+(3-6y)=0 \]

For real \(x\), the discriminant must be non-negative:
\[ (4y-2)^2-4(2y)(3-6y)\geq0 \]
\[ 16y^2-16y+4-24y+48y^2\geq0 \]
\[ (2y-1)(8y-1)\geq0 \]

Therefore:
\[ y\leq\frac18\quad\text{or}\quad y\geq\frac12 \]

Hence the range is:
\[ \left(-\infty,\frac18\right]\cup\left[\frac12,\infty\right) \]

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The question is " For real values of \( x \), the range of the function \( f ( x ) = \frac { 2 x - 3 } { 2 x ^ { 2 } + 4 x - 6 } \) is "

Hence, the answer is '\( \left( - \infty , \frac { 1 } { 8 } \right] \cup \left[ \frac { 1 } { 2 } , \infty \right) \)'

Choice 2 is the correct answer.

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