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Question

If p2+q2+r2=2 for p,q,rC and g(x)=∣ ∣ ∣1+p2x(1+q2)x(1+r2x)(1+p2)x1+q2x(1+r2)x(1+p2)x(1+q2)x1+r2x∣ ∣ ∣ for all xR, then the value of 41g(x)dx is

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Solution

g(x)=∣ ∣ ∣1+p2x(1+q2)x(1+r2)x(1+p2)x1+q2x(1+r2)x(1+p2)x(1+q2)x1+r2x∣ ∣ ∣

Applying C1C1+C2+C3
g(x)=∣ ∣ ∣1(1+q2)x(1+r2)x11+q2x(1+r2)x1(1+q2)x1+r2x∣ ∣ ∣ (p2+q2+r2=2)

Applying R1R1R3, R2R2R3
g(x)=∣ ∣ ∣00x10(x1)(x1)1(1+q2)x1+r2x∣ ∣ ∣

g(x)=(x1)2 xR
So, 41 (x1)2 dx=13[(x1)3]41=9

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