cat 2023 Question 21

Question

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

Accepted Answer

Given that $ 2pq - 20 = 52 - 2pq \Rightarrow 4pq = 72 \Rightarrow pq = 18 $ ---(1)

Now, $ p^2 + q^2 - 29 = 2pq - 20 \Rightarrow p^2 + q^2 - 2pq = 9 \Rightarrow (p-q)^2 = 9 \Rightarrow p - q = \pm 3 $

Also, $ p^2 + q^2 - 29 = 2pq - 20 \Rightarrow p^2 + q^2 = 2pq + 9 = 2(18) + 9 = 45 $

Now, $ p^3 - q^3 = (p-q)(p^2 + pq + q^2) = (p-q)(45 + 18) = (p-q)(63) $

=> When $ p-q = -3 $ => The value is $ 63(-3) = -189 $ and when $ p-q = 3 $ => The value is $ 63(3) = 189 $.

=> The difference = $ 189 - (-189) = 378 $.

Question 21.