The number of integral values of $k$ for which the equation $3 sin x + 4 cos x = k + 1$ has a solution, $k \in R$ is

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Solution

$3 sin x + 4 cos x = k + 1$
The equation has a solution if
$- \sqrt{3^{2} + 4^{2}} \leq k + 1 \leq \sqrt{3^{2} + 4^{2}}$
$\Rightarrow - 5 \leq k + 1 \leq 5$
$\Rightarrow - 6 \leq k \leq 4$
$\Rightarrow k = - 6, - 5, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4$
$∴$ There are $11$ possible integral values.

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