The number of integral values of $k$, for which the equation $7 \cos x+5 \sin x=2 \mathrm{k}+1$ has a…
The number of integral values of $k$, for which the equation $7 \cos x+5 \sin x=2 \mathrm{k}+1$ has a solution, is
- 4
- 8
- 10
- 2
Solution
$\begin{aligned}
& -\sqrt{7^2+5^2} \leq(7 \cos x+5 \sin x) \leq \sqrt{7^2+5^2} \\
& \Rightarrow-\sqrt{74} \leq(7 \cos x+5 \sin x) \leq \sqrt{74} \\
& \Rightarrow-8.6 \leq 2 \mathrm{k}+1 \leq 8.6 \\
& \Rightarrow-4.8 \leq \mathrm{k} \leq 3.8
\end{aligned}$
Integral values of k are $-4,-3,-2,-1,0,1,2,3$
Number of integral values of $k=8$
Asked in: MHT CET 2024 (09 May Shift 2)
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