Mathematics - Trigonometric Equation Question with Solution | TestHub

We have, $\sin \pi \left(x^2+x\right)=\sin \pi x^2$ $\Rightarrow \pi \left({\mathrm{x}}^2+\mathrm{x}\right)=\mathrm{n}\pi +(-1{)}^{\mathrm{n}}\pi {\mathrm{x}}^2$ $\therefore$ Either ${\mathrm{x}}^2+\mathrm{x}=2\mathrm{m}+{\mathrm{x}}^2\Rightarrow \mathrm{x}=2\mathrm{m}\in \mathrm{I}$ or ${\mathrm{x}}^2+\mathrm{x}=\mathrm{k}-{\mathrm{x}}^2,$ where $\mathrm{k}$ is an odd integer $\Rightarrow 2{\mathrm{x}}^2+\mathrm{x}-\mathrm{k}=0\Rightarrow \mathrm{x}=\frac{-1\pm \sqrt{1+8\mathrm{k}}}{4}$ For least positive non-integral solution is $x=\frac{1}{2},$ when $k=1$