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:

  1. 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 $k=8$.



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