Question

The sum of integral values of ${}^{\prime }{n}^{\prime }$ such that equation $\sin x(2\sin x+\cos x)=n$ has at least one real solution is ?

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is A $1$

$2{\sin }^{2}x+\sin x\cos x=n$
$⟹1-\cos 2x+\frac{1}{2}\sin 2x=n$
$⟹\sin 2x-2\cos 2x=2n-1$
Range of $a\cos \theta +b\sin \theta$ is $[-\sqrt{a^2+b^2},\sqrt{a^2+b^2}]$
Therefore, range of LHS is: $[-\sqrt{5},\sqrt{5}]$
$\sqrt{5}\approx 2.22$, therefore, odd integers in the above range are: $\{-1,1\}$
Values of $n$: $\{0,1\}$
Sum: $0+1=1$