If f(x) = Asin ((π x)/2 ) + B, f' (1/2 ) = √(2) and int_0^1 f(x) dx = 2A/(π), then constants A and B are

If f(x) = Asin ((π x)/2 ) + B, f' (1/2 ) = √(2) and int_0^1… — JEE Main 1995 Mathematics PYQ

1995

f(x) = A\sin\left(\frac{\pi x}{2}\right) + B

f'\left(\frac{1}{2}\right) = \sqrt{2}

and

\int_0^1 f(x)\,dx = \frac{2A}{\pi}

, then constants

A

and

B

are

A.
$\frac{\pi}{2}$ and $\frac{\pi}{2}$
B.
$\frac{2}{\pi}$ and $\frac{3}{\pi}$
C.
$0$ and $-\frac{4}{\pi}$
D.
$\frac{4}{\pi}$ and $0$

Show answer and solution

Correct option: D

Solution

f'(x) = A·(π/2)·cos(πx/2), so f'(1/2) = A·(π/2)·cos(π/4) = A·(π/2)·(1/√2) = Aπ/(2√2) = √2, giving A = 4/π.

∫₀¹ f(x)dx = [-2A/π·cos(πx/2)]₀¹ + B = (2A/π)(1 - 0) + B... let me redo: ∫₀¹ A·sin(πx/2)dx = A·[-2/π·cos(πx/2)]₀¹ = A·(2/π)(1 - 0) = 2A/π, and ∫₀¹ B dx = B. So total = 2A/π + B = 2A/π, which gives B = 0.

Thus A = 4/π and B = 0.

[1995]

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