The number of integer values of k for which the equation $7 cos x + 5 sin x = 2 k + 1$ has a solution is

4

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8

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10

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12

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Solution

The correct option is C $8$
The term $7 cos x + 5 sin x$ belongs from $- \sqrt{74}$ to $\sqrt{74}$
Thus, $- \sqrt{74} < 2 k + 1 < \sqrt{74}$
$\Rightarrow 2 k + 1 > - 9$ or $k > - 5$ and $2 k + 1 < 8$ or $k < 3.5$
So, we can have integral values of $k$ from $- 4$ to $3$ i.e. $8$ values.

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