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$f (x) = a \cos x + b x \sin x$ and $f^{1} (x) = 5 \sin x + 3 x \cos x$ then $(a, b)$ is

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$(2, 3)$

$(- 2, 3)$

$(- 2, - 3)$

$(2, - 3)$

answer is B.

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Detailed Solution

$\begin{array}{l} f (x) = a \cos x + b x \sin x \\ f^{1} (x) = - a \sin x + b x \cos x + b \sin x \\ 5 \sin x + 3 x \cos x = (b - a) \sin x + b x \cos x \\ Comparing on both sides \\ b - a = 5 \\ 3 - a = 5 \\ a = - 2 \end{array}$

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