Question

The number of solutions of the equation $\left|\begin{array}{cc}{x}^2+\sin x\cos x& x(1+\sin x)\\ x+\cos x& x+1\end{array}\right|=0$ is

• A. $1$
• B. $2$
• C. $3$
• D. Infinite

Answer 1

The correct option is C $3$

The given determinant equation can be written as
$\left|\begin{array}{cc}\sin x& x\\ 1& 1\end{array}\right|\times \left|\begin{array}{cc}\cos x& x\\ x& 1\end{array}\right|=0$
it can be simplified as ,
$(x-\sin x)\times (\cos x-x^2)=0$
$\sin x=x$ , this happens only when $x=0$

$\cos x=x^2$, if we plot both graphs together then it intersects at 2 points as shown in above graph and hence 2 roots.
So there are total $3$ solutions for this equation.