CAT 2023 Question Paper | Quant Slot 2

CAT 2023 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 21 : If $p^2+q^2-29=2 p q-20=52-2 p q$, then the difference between the maximum and minimum possible value of $\left(p^3-q^3\right)$ is

Video Explanation

Explanatory Answer

$p ^ { 2 } + q ^ { 2 } - 29 = 2 p q - 20 = 52 - 2 p q$
$2 p q - 20 = 52 - 2 p q$
$4 p q = 72$
$p q = 18$
$p ^ { 2 } + q ^ { 2 } - 29 = 2 p q - 20$
$p ^ { 2 } + q ^ { 2 } - 2 p q = 29 - 20$
$( p - q ) ^ { 2 } = 3 ^ { 2 }$
$p - q = \pm 3$
$( p - q ) ^ { 3 } = p ^ { 3 } - q ^ { 3 } - 3 p q ( p - q )$
$p ^ { 3 } - q ^ { 3 } = ( p - q ) ^ { 3 } + 3 p q ( p - q )$
The maximum value of $p ^ { 3 } - q ^ { 3 }$ occurs when $p - q = 3$
$p ^ { 3 } - q ^ { 3 } = 3 ^ { 3 } + 3 ( 18 ) 3 = 3 ^ { 3 } ( 1 + 6 ) = 27 \times 7 = 189$
The minimum value of $p ^ { 3 } - q ^ { 3 }$ occurs when $p - q = - 3$
$p ^ { 3 } - q ^ { 3 } = ( - 3 ) ^ { 3 } + 3 ( 18 ) ( - 3 ) = - 189$
The difference between the maximum and minimum values $= 189 - ( - 189 ) = 2 ( 189 ) = 378$

The question is " If $p^2+q^2-29=2 p q-20=52-2 p q$, then the difference between the maximum and minimum possible value of $\left(p^3-q^3\right)$ is "

Hence, the answer is '378'

Choice C is the correct answer.

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