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

Solution :
We know that
$$ \Rightarrow \,\, - \sqrt {{a^2} + {b^2}} \leqslant a\cos \theta + b\sin \theta \leqslant \sqrt {{a^2} + {b^2}} $$           NOTE THIS STEP
$$\eqalign{ & \Rightarrow \,\, - \sqrt {74} \leqslant 7\cos x + 5\sin x \leqslant \sqrt {74} \cr & \Rightarrow \,\, - \sqrt {74} \leqslant 2k + 1 \leqslant \sqrt {74} \cr & \Rightarrow \,\, - 8.6 \leqslant 2k + 1 \leqslant 8.6 \cr & \Rightarrow \,\, - 4.8 \leqslant k \leqslant 3.8 \cr} $$
(considering only integral values)
⇒ $$k$$ can take 8 integral values.