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Q)

The maximum value of $3\cos\theta+4\sin\theta$ is

$(a)\;3\qquad(b)\;4\qquad(c)\;5\qquad(d)\;7$

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A)
We can have the maximum value at
$-\sqrt{3^2+4^2}\leq 3\cos\theta+4\sin\theta\leq \sqrt{3^2+4^2}$
$\Rightarrow -\sqrt{9+16}\leq 3\cos\theta+4\sin\theta\leq \sqrt{9+16}$
$\Rightarrow -\sqrt{25}\leq 3\cos\theta+4\sin\theta\leq \sqrt{25}$
$\Rightarrow -5\leq 3\cos\theta+4\sin\theta\leq 5$
Hence maximum value is at 5
Hence (c) is the correct answer.
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