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Question: Let \[\begin{bmatrix} x \\ x^{2} \\ 1 \end{bmatrix} = \begin{bmatrix} x \\ 2\alpha x + \beta x^{2} \...

Let [xx21]=[x2αx+βx25x+γx2+3]\begin{bmatrix} x \\ x^{2} \\ 1 \end{bmatrix} = \begin{bmatrix} x \\ 2\alpha x + \beta x^{2} \\ 5x + \gamma x^{2} + 3 \end{bmatrix} for at least 3 values of x and f(x) is a differentiable function satisfying f(xy) = f(x) + x^{2}(y^{2}–1)+y– 1, ∀ x, y \in \mathbb{R} and f(1) = 3, then ∫–11(αf(x)+βf′(x)+γ)\int_{–1}^{1}{(\alpha f(x) + \beta f'(x) + \gamma)}dx is equal to -

A

10

B

20

C

25

D

35

Answer

20

Explanation

Solution

[100620543]\begin{bmatrix} 1 & 0 & 0 \\ 6 & 2 & 0 \\ 5 & 4 & 3 \end{bmatrix} [xx21]\begin{bmatrix} x \\ x^{2} \\ 1 \end{bmatrix} =

= [x6x+2x25x+4x2+3]\begin{bmatrix} x \\ 6x + 2x^{2} \\ 5x + 4x^{2} + 3 \end{bmatrix} = [x2αx+βx25x+γx2+3]\begin{bmatrix} x \\ 2\alpha x + \beta x^{2} \\ 5x + \gamma x^{2} + 3 \end{bmatrix}

\ f(y) = f(1) + y2 – 1 + y – 1 = y2 + y + 1

{Q f(1) = 3}

\ f(x) = x2 + x + 1

f '(x) = 2x + 1

\ ∫−11(αf(x)\int_{- 1}^{1}{(\alpha f(x)}+ bf '(x) + g)dx = 20