Question

If $3\sin x+4\cos ax=7$ has at least one solution and $a=\frac{pm}{qn+1};$ where $m,n\in Z,$ then the value of $p+q$ is

Answer 1

We have $3\sin x+4\cos ax=7$ which is possible only when $\sin x=1\text{and}\cos ax=1$
$\Rightarrow x=(4n+1)\frac{\pi }{2}\text{and}ax=2m\pi$
$\Rightarrow x=(4n+1)\frac{\pi }{2}=\frac{2m\pi }{a}$
$\Rightarrow a=\frac{4m}{4n+1}=\frac{pm}{qn+1}$
$\Rightarrow p+q=4+4=8$