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. $8$
• B. $9$
• C. $10$
• D. $12$

Answer 1

The correct option is A $8$

$7\cos x+5\sin x=2k+1$
This is $a\cos x+b\sin x=c$ form.
So, solution is possible iff
$-1\le \frac{c}{\sqrt{a^2+b^2}}\le 1\Rightarrow -1\le \frac{2k+1}{\sqrt{7^2+5^2}}\le 1\Rightarrow -\sqrt{74}\le 2k+1\le \sqrt{74}\Rightarrow \frac{-1-\sqrt{74}}{2}\le k\le \frac{-1+\sqrt{74}}{2}\therefore k=-4,-3,-2,-1,0,1,2,3$

Hence, there are $8$ integral values of $k$.