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  4. Number of real roots of the equation | sin x x 1 1|| cos x x x 1|=0, is

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Q. Number of real roots of the equation $\begin{vmatrix}\sin x & x \\ 1 & 1\end{vmatrix}\begin{vmatrix}\cos x & x \\ x & 1\end{vmatrix}=0$, is

Determinants

Solution:

We have $\begin{vmatrix}\sin x & x \\ 1 & 1\end{vmatrix}\begin{vmatrix}\cos x & x \\ x & 1\end{vmatrix}=0$
image
$\Rightarrow(\sin x-x)\left(\cos x-x^{2}\right)=0$
$\therefore $ Either $\sin x=x$ or $\cos x=x^{2}$
Hence, number of real roots are $3$ .