Question

The number of integral values of $k$ for which the equation $\sin x(3-2{\sin }^{2}x)+\cos x(3-2{\cos }^{2}x)=k$ possesses a solution, is

• A. $4$
• B. $5$
• C. $6$
• D. $7$

Answer 1

The correct option is B $5$

$\sin x(3-2{\sin }^{2}x)+\cos x(3-2{\cos }^{2}x)=k$
Let $\sin x=s\&\cos x=c$
$\Rightarrow 3(s+c)-2(s^3+c^3)=k\Rightarrow 3(s+c)-2\left[\right(s+c\left)\right(s^2+c^2-sc\left)\right]=k\Rightarrow (s+c)[3-2(1-sc\left)\right]=k\Rightarrow (s+c)(1+2sc)=k\Rightarrow (s+c{)}^3=k\Rightarrow k\in [-2\sqrt{2},2\sqrt{2}]$