Let f(x)= 7 x^ 10 +9 x^ 8 (1+x^ 2 +2 x^ 9 )^ 2 d x, x0, _ x 0 f(x)=0 and f(1)= 1 4 . If A = [ array ccc 0 & 0 & 1 \ 1 4 & f^ (1) & 1 \ ^ 2 & 4 & 1 array ] and B = adj ( adj A ) be such that | B |=81, then ^ 2 is equal to

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Let f(x)= 7 x^ 10 +9 x^ 8 (1+x^ 2 +2 x^ 9 )^ 2 d x, x>0, _ x 0 f(x)=0 and f(1)= 1 4 . If A = [ array ccc 0 & 0 & 1 \ 1 4 & f^ (1) & 1 \ ^ 2 & 4 & 1 array ] and B = adj ( adj A ) be such that | B |=81, then ^ 2 is equal to

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Correct answer: D. 4

Divide numerator and denominator by x^ 18 , let t = x^ -9 + x^ -7 + 2, then dt = -(9x^ -10 +7x^ -8 )dx. f(x) = 1 t + C = x^9 1+x^2+2x^9 + C. _ x 0 f(x) = 0 C = 0. f(1) = 1 4 (verified). f'(x) = 9x^8(1+x^2+2x^9) - x^9(2x+18x^8) (1+x^2+2x^9)^2 . f'(1) = 36-20 16 = 1. |A| = vmatrix 0 & 0 & 1 \ 1/4 & 1 & 1 \ ^2 & 4 & 1 vmatrix = 1 - ^2. |B| = | adj ( adj A)| = |A|^ (3-1)^2 = (1- ^2)^4 = 81 = 3^4. |1- ^2| = 3 ^2 = 4.

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