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Algebraic Functional Equations

XAT 2015 · Quantitative Ability & Data Interpretation · Medium · Quantitative Ability

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If f(x2−1)=x4−7x2+k1f(x^2 - 1) = x^4 - 7x^2 + k_1 and f(x3−2)=x6−9x3+k2f(x^3 - 2) = x^6 - 9x^3 + k_2, then the value of (k2−k1)(k_2 - k_1) is:

  1. A.

    6

  2. B.

    7

  3. C.

    8

  4. D.

    9

  5. E.

    None of the above

Answer

C

Explanation

Let x2−1=0  ⟹  x2=1x^2 - 1 = 0 \implies x^2 = 1. f(0)=(1)2−7(1)+k1=k1−6f(0) = (1)^2 - 7(1) + k_1 = k_1 - 6.

Let x3−2=0  ⟹  x3=2x^3 - 2 = 0 \implies x^3 = 2. f(0)=(2)2−9(2)+k2=k2−14f(0) = (2)^2 - 9(2) + k_2 = k_2 - 14.

Equating both expressions for f(0)f(0): k1−6=k2−14  ⟹  k2−k1=8k_1 - 6 = k_2 - 14 \implies k_2 - k_1 = 8.

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