Class 12

30.Definite Integral question bank (Maths)

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Q1.

Evaluate \( \int_{0}^{\pi/4} \frac{\sin x + \cos x}{3 + \sin 2x} \, dx = \)

(A)

\( \frac{1}{4} \log 3 \)

(B)

\( \log 2 \)

(C)

\( \log 3 \)

(D)

\( \frac{1}{4} \log \left( \frac{5}{3} \right) \)

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Q2.

Let \( F : \mathbb{R} \rightarrow \mathbb{R} \) be a thrice differentiable function. Suppose that \( F(1)=0 \), \( F(3) = -4 \) and \( F'(x) < 0 \) for all \( x \in ( \frac{1}{2}, 3 ) \). Let \( f(x) = xF(x) \) for all \( x \in \mathbb{R} \).
The correct statement(s) is(are)

(A)

\( f'(1) < 0 \)

(B)

\( f(2) < 0 \)

(C)

\( f'(x) \neq 0 \) for any \( x \in (1,3) \)

(D)

\( f'(x) = 0 \) for some \( x \in (1,3) \)

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