JEE Main 2021 (Online) 4th August Evening ShiftQuestion 1
The number of real solutions of the equation
$\begin{vmatrix}x^{2}+\sin x\cos x&x(1+\sin x)\\x+\cos x&x+1\end{vmatrix}=0$
is $\_\_\_\_\_\_\_\_$ .
Determinant
Answer$3$
Solution
The determinant is $(x-\sin x)(x^2-\cos x)$. $x-\sin x=0$ only at $x=0$ since $x-\sin x$ is increasing. $x^2-\cos x=0$ is even, negative at $0$ and increasing for $x>0$, so it has exactly two roots $\pm\alpha$. Total $3$ real solutions.