Mathematics - Trigonometric Equation Question with Solution | TestHub

Option-1 : $\mathrm{f}(\mathrm{x})={\mathrm{x}}^2\sin \frac{\pi }{{\mathrm{x}}^2}$ $f(x)=0\Rightarrow \sin \frac{\pi }{x^2}=0\Rightarrow \frac{\pi }{x^2}=n\pi ,n\in N$ ${\mathrm{x}}^2=\frac{1}{\mathrm{n}}\Rightarrow \mathrm{x}=\frac{1}{\sqrt{\mathrm{n}}}$ $\frac{1}{\sqrt{\mathrm{n}}}\ge \frac{1}{10^{10}}\Rightarrow 10^{10}\ge \sqrt{\mathrm{n}}\Rightarrow \mathrm{n}\le 10^{20}$, finite number of solutions Option-2 : $\mathrm{x}=\frac{1}{\sqrt{\mathrm{n}}}\Rightarrow \frac{1}{\sqrt{\mathrm{n}}}>\frac{1}{\pi }\Rightarrow \pi >\sqrt{\mathrm{n}}\Rightarrow \mathrm{n}<{\pi }^2$, Number of solutions is 9 Option-3 $:\mathrm{x}=\frac{1}{\sqrt{\mathrm{n}}},\frac{1}{\sqrt{\mathrm{n}}}<\frac{1}{10^{10}}\Rightarrow \sqrt{\mathrm{n}}>10^{10}\Rightarrow \mathrm{n}>10^{20}$, Infinite number of solutions Option-4 : $\frac{1}{{\pi }^2}<\frac{1}{\sqrt{\mathrm{n}}}<\frac{1}{\pi }\Rightarrow \sqrt{\mathrm{n}}\in \left(\pi ,{\pi }^2\right)\Rightarrow \mathrm{n}\in \left({\pi }^2,{\pi }^4\right)$, Definitely more than 25 solutions