CAT 2023 Slot 2 QA Question 21: If $p^{2} + q^{2} - 29 = 2pq - 20 = 52 - 2pq$, then the difference between the maximum and minimum possible value of $p^{3} - q^{3}$ is

Question

If $p^{2} + q^{2} - 29 = 2pq - 20 = 52 - 2pq$, then the difference between the maximum and minimum possible value of $p^{3} - q^{3}$ is

Accepted Answer

Since all three expressions are equal, equate the second and third expressions first.

$2pq - 20 = 52 - 2pq$
$4pq = 72$
$pq = 18$

Now equate the first and second expressions.
$p^{2} + q^{2} - 29 = 2pq - 20$
$p^{2} + q^{2} - 2pq = 29 - 20$
$(p - q)^{2} = 9$
$p - q = \pm 3$

Also,
$p^{2} + q^{2} = 2p

If p2+q2−29=2pq−20=52−2pqp^{2} + q^{2} - 29 = 2pq - 20 = 52 - 2pqp2+q2−29=2pq−20=52−2pq, then the difference between the maximum and minimum possible value of p3−q3p^{3} - q^{3}p3−q3 is