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)

$8$

We know that, $a\cos \theta +b\sin \theta =c$ has one solution only.
$|c|\le \sqrt{a^2+b^2}$
Then, for the given equation, we must have
$|2k+1|\le \sqrt{74}$
$\Rightarrow -\sqrt{74}<2k+1<\sqrt{74}$
$\Rightarrow -8<2k+1<8$
$\therefore k=-4,-3,-2,-1,0,1,2,3$
Thus, there are $8$ values which will satisfy the above inequality.