Question

The number of solutions $x$ of the equation $\sin (x+x^2)-\sin (x^2)=\sin x$ in the interval $[2,3]$ is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (C)

Given,
$\sin (x+x^2)-\sin (x^2)=\sin x$
$\Rightarrow \sin (x+x^2)=\sin (x^2)+\sin x$
$\Rightarrow 2\sin \left(\frac{x+x^2}{2}\right)\cos \left(\frac{x+x^2}{2}\right)$
$=2\sin \left(\frac{x^2+x}{2}\right)\cos \left(\frac{x^2-x}{2}\right)$
$\sin \left(\frac{x+x^2}{2}\right)=0$
or $\cos \left(\frac{x^2+x}{2}\right)-\cos \left(\frac{x^2-x}{2}\right)=0$
$\frac{x^2+x}{2}=x\pi$ or $2\sin \frac{x^2}{2}\sin \frac{x}{2}=0$
$\Rightarrow \frac{x^2+x}{2}=0,\pi ,2\pi$, or $\frac{x^2}{2}=0,\pi ,\frac{n}{2}$
$=0,\pi ,2\pi$
$\Rightarrow x^2+x=2\pi$ or $x^2=2\pi$
$\Rightarrow x^2+x-2\pi =0$ or $x=\sqrt{2\pi }$
$\Rightarrow x=\frac{-1\pm \sqrt{1+8\pi }}{2}$ or $2<\sqrt{2}\pi <3$
$\Rightarrow \sqrt{1+8\pi }=\sqrt{25.14}$
$\therefore x=\frac{5.2-1}{2}$
$=\frac{4.2}{2}=2.1$
$\because$ Total numbers of solution lies between $\left(2,3\right)=2$