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Difference between max and min value of algebraic expression

CAT 2023 Slot 2 · Quantitative Ability · Medium · Algebra

This is a medium Quantitative Ability question from the CAT 2023 Slot 2 paper. It tests Algebra. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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

  1. A.

    486

  2. B.

    189

  3. C.

    378

  4. D.

    243

Answer

C

Explanation

From 2pq−20=52−2pq2pq - 20 = 52 - 2pq: 4pq=72  ⟹  pq=184pq = 72 \implies pq = 18

Now, p2+q2−29=2(18)−20=16  ⟹  p2+q2=45p^2 + q^2 - 29 = 2(18) - 20 = 16 \implies p^2 + q^2 = 45.

We know (p−q)2=p2+q2−2pq=45−36=9  ⟹  p−q=±3(p - q)^2 = p^2 + q^2 - 2pq = 45 - 36 = 9 \implies p - q = \pm 3.

Now, p3−q3=(p−q)(p2+pq+q2)=(p−q)(45+18)=63(p−q)p^3 - q^3 = (p - q)(p^2 + pq + q^2) = (p - q)(45 + 18) = 63(p - q).

  • Maximum value of p3−q3=63×3=189p^3 - q^3 = 63 \times 3 = 189
  • Minimum value of p3−q3=63×(−3)=−189p^3 - q^3 = 63 \times (-3) = -189

Difference =189−(−189)=378= 189 - (-189) = 378.

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