# CAT 2021 Question Paper | Quant Slot 3

###### CAT Previous Year Paper | CAT Quant Questions | Question 22

CAT 2021 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 22 : If $f(x)=x^{2}-7 x$ and $g(x)=x+3$, then the minimum value of $f(g(x))-3 x$ is

1. - 20
2. - 15
3. - 12
4. - 16

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g(x) = x + 3
f(x) = x2 - 7x
f(g(x)) - 3x
= f(x + 3) - 3x
= (x + 3)2 - 7(x + 3) - 3x
= x2 + 9 + 6x - 7x - 21 - 3x
= x2 - 4x - 12
= x2 - 4x + 4 - 4 - 12
= x2 - 4x + 4 - 16
= (x - 2)2 - 16
f(g(x)) - 3x is minimum when (x - 2)2 - 16 is minimum.
(x - 2)2 - 16 is minimum when (x - 2)2 is minimum.
Since (x - 2)2 is non-negative, the minimum value it can take is 0.
Hence the minimum value of (x - 2)2 - 16 = 0 - 16 = -16
Therefore, the minimum value of f(g(x)) - 3x is -16

The question is " If $f(x)=x^{2}-7 x$ and $g(x)=x+3$, then the minimum value of $f(g(x))-3 x$ is "

##### Hence, the answer is '- 16'

Choice D is the correct answer.

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