JEE Advance - Mathematics (2008 - Paper 1 Offline - No. 19)
STATEMENT - 1: $$\mathop {\lim }\limits_{x \to 0} \,\,\left[ {g\left( x \right)\cot x - g\left( 0 \right)\cos ec\,x} \right] = f''\left( 0 \right)$$ and
STATEMENT - 2: $$f'\left( 0 \right) = g\left( 0 \right)$$
STATEMENT - 2: $$f'\left( 0 \right) = g\left( 0 \right)$$
Statement - 1 is True, Statement - 2 is True; Statement - 2 is a correct explanation for Statement - 1
Statement - 1 is True, Statement - 2 is True; Statement - 2 is NOT a correct explanation for Statement - 1
Statement - 1 is True, Statement -2 is False
Statement - 1 is False, Statement -2 is True
Explanation
$$\mathop {\lim }\limits_{x \to 0} {{g(x)\cos x - g(0)} \over {\sin x}}$$
$$ = \mathop {\lim }\limits_{x \to 0} {{g'(x)\cos x - g(x)\sin x} \over {\cos x}}$$ (Applying L-Hospital rule)
$$g'(0) - 0 = 0 = f''(0)$$
Statement - 1 is True.
$$f'(x) = g(x)\cos x + g'(x)\sin x$$
$$f'(0) = g(0)$$
Statement - 2 is True.
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