text: # 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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