MHT CET · Maths · Trigonometric Equations
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
- A 4
- B 8
- C 10
- D 2
Answer & Solution
Correct Answer
(B) 8
Step-by-step Solution
Detailed explanation
\(\begin{aligned}
& -\sqrt{7^2+5^2} \leq(7 \cos x+5 \sin x) \leq \sqrt{7^2+5^2} \\
& \Rightarrow-\sqrt{74} \leq(7 \cos x+5 \sin x) \leq \sqrt{74} \\
& \Rightarrow-8.6 \leq 2 \mathrm{k}+1 \leq 8.6 \\
& \Rightarrow-4.8 \leq \mathrm{k} \leq 3.8
\end{aligned}\)
Integral values of k are \(-4,-3,-2,-1,0,1,2,3\)
Number of integral values of \(k=8\)
& -\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 \(k=8\)
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