1
Let f(x) = 7x10 + 9x8(1 + x2 + 2x9)2 dx, x > 0, x 0 f(x) = 0 and f(1) = 14. If A = bmatrix 0 & 0 & 1 \ 14 & f'(1) & 1 \ 2 & 4 & 1 bmatrix and B = adj(adj A) be such that |B| = 81, then 2 is equal to
- 2
- 4
- 3
- 1
Answer
(2) 4
Solution
Since f(x) is an antiderivative of the integrand,
f'(x) = 7x10 + 9x8(1 + x2 + 2x9)2
f'(1) = 7 + 9(1 + 1 + 2)2 = 1616 = 1
(Check: dividing numerator and denominator by x18 and putting t = x-9 + x-7 + 2 gives f(x) = x92x9 + x2 + 1, which satisfies x 0 f(x) = 0 and f(1) = 14.)
A = bmatrix 0 & 0 & 1 \ 14 & 1 & 1 \ 2 & 4 & 1 bmatrix
Expanding along R1:
|A| = 1 ( 14 4 - 1 2 ) = 1 - 2
For a matrix of order n: |adj(adj A)| = |A|(n-1)2
|B| = |A|4 = 81
|A| = 3
1 - 2 = 3 or 1 - 2 = -3
2 = -2 (rejected, 2 0) or 2 = 4
2 = 4
Answer: (2)
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