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

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Solution

The correct option is B 8
$- \sqrt{a^{2} + b^{2}} \leq a c o s θ + b s i n θ \leq \sqrt{a^{2} + b^{2}} \Rightarrow - \sqrt{74} \leq 7 c o s x + 5 s i n x \leq \sqrt{74} \Rightarrow - \sqrt{74} \leq 2 k + 1 \leq - \sqrt{74} \Rightarrow - 8 \leq 2 k + 1 \leq 8 \Rightarrow - 4.5 \leq k \leq 3.5$
Considering only intergral values, which means k can take eight integral values.

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