Question

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

• A. $4$
• B. $8$
• C. $10$
• D. $12$

Answer 1

Answer: (B)

$8$

$|7\cos x+5\sin x|\le \sqrt{7^2+5^2}$

$\Rightarrow -\sqrt{7^2+5^2}\le (7\cos x+5\sin x)\le \sqrt{7^2+5^2}$

$\Rightarrow -\sqrt{74}\le (2K+1)\le \sqrt{74}$

$\Rightarrow -8.6\le (2K+1)\le 8.6$

$\Rightarrow -9.6\le 2K\le 7.6$

$\Rightarrow -4.8\le K\le 3.8$

So, integral values of $K$ are

$-4,-3,-2,-1,0,1,2,3$ (eight values)

Hence, option B is correct.