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)

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