Question

Let $f\left(x\right)=\sin \left(\frac{\pi }{x}\right)$ and $S=\left\{x:f\left(x\right)>0\right\}$.Then $S$doesn't contain

• A. $\left(\frac{-1}{2},-\pi \right)$
• B. $\left(\frac{1}{4},\frac{1}{5}\right)$
• C. $\left(\frac{1}{2},\frac{1}{3}\right)$
• D. $\left(\frac{-1}{2},-1\right)$

Answer 1

The correct option is D

$\left(\frac{-1}{2},-1\right)$

Explanation for the correct option:

Finding what S a set defined does not contain.

We are given

$f\left(x\right)=\sin \left(\frac{\pi }{x}\right)[Asf(x)>0]$

For $\sin \left(\frac{\pi }{x}\right)$

$\Rightarrow \left[x:\frac{\pi }{x}\in \cup (2n\pi +1)\pi \right]\supseteq \left\{x:x\in \cup \left[\left(\frac{1}{2n+1}\right),\left(\frac{1}{2n}\right)\right]\right\}$

Now from given options we can see that,

$$\begin{gathered} D_f\supseteq \left(\frac{1}{3},\frac{1}{2}\right) \\ {D}_f\supseteq \left(\frac{1}{5},\frac{1}{4}\right) \\ {D}_f\supseteq \left(-1,\frac{1}{2}\right) \end{gathered}$$

So, the set does not contain $\left(\frac{-1}{2},-1\right)$

Hence, the correct option is (D)