CAT 2022 Question Paper | Quant Slot 2

CAT 2022 Quant was dominated by Arithmetic followed by Algebra. 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 20 : Let \(f(x)\) be a quadratic polynomial in \(x\) such that \(f(x) \geq 0\) for all real numbers \(x\). If \(f(2)=0\) and \(f(4)=6\), then \(f(-2)\) is equal to

Video Explanation

Explanatory Answer

Let, \( f ( x ) = a x ^ { 2 } + b x + c \)
Since, \( f ( x ) \geq 0 \),
This means that graph of f(x) is U shaped and above the x-axis.
Since f(x) = 0 at x = 2, the graph is centered at x = 2.
This means f(2 - k) = f(2 + k).
f(2 - 2) = f(2 + 2)
f(0) = f(4)
f(0) = 6
f(0) = 6
c = 6
f(4) = 6
16 a + 4 b + 6 = 6
f(2) = 0
4 a + 2 b + 6 = 0
Solving the two equations…
a = \( \frac { 3 } { 2 } \); b = -6
f(x) = \( \frac { 3 } { 2 } x ^ { 2 } - 6 x + 6 \)
f(-2) = \( \frac { 3 } { 2 } 4 + 12 + 6 = 24 \)

The question is " Let \(f(x)\) be a quadratic polynomial in \(x\) such that \(f(x) \geq 0\) for all real numbers \(x\). If \(f(2)=0\) and \(f(4)=6\), then \(f(-2)\) is equal to "

Hence, the answer is '24'

Choice C is the correct answer.

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