Trigonometrical Equations 3 Question 3
3. The number of integral values of $k$ for which the equation $7 \cos x+5 \sin x=2 k+1$ has a solution, is
(2002, 1M)
(a) 4
(b) 8
(c) 10
(d) 12
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Answer:
Correct Answer: 3. (b)
Solution:
- We know that,
$$ \begin{aligned} & -\sqrt{a^{2}+b^{2}} \leq a \sin x+b \cos x \leq \sqrt{a^{2}+b^{2}} \\ & \therefore \quad-\sqrt{74} \leq 7 \cos x+5 \sin x \leq \sqrt{74} \\ & \text { i.e. } \quad-\sqrt{74} \leq 2 k+1 \leq \sqrt{74} \end{aligned} $$
Since, $k$ is integer, $-9<2 k+1<9$
$\Rightarrow-10<2 k<8 \quad \Rightarrow \quad-5<k<4$
$\Rightarrow$ Number of possible integer values of $k=8$.
