Let f: mathbb R rightarrow mathbb R be a function defined by f(x)= \ begin array cl x 2 sin (frac π x 2 ) & ; if x neq 0 \\ 0 & ; if x=0 end array . Then which of the following statements is TRUE?

f(x)=0 has more than 25 solutions in the interval (frac 1 π 2 , 1/π )

f(x)=0 has infinitely many solutions in the interval [frac 1 10 10 , ∞ )

f(x)=0 has no solutions in the interval [1/π, ∞ )

The set of solutions of f(x)=0 in the interval (0, frac 1 10 10 ) is finite

Mathematics · Functions

JEE Advanced 2024 — Paper 2 — Question 4

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by

f(x)=\left\{\begin{array}{cl} x^{2} \sin \left(\frac{\pi}{x^{2}}\right) & ; \text { if } x \neq 0 \\ 0 & ; \text { if } x=0 \end{array}\right.

Then which of the following statements is TRUE?

Option A:
$f(x)=0$ has infinitely many solutions in the interval $\left[\frac{1}{10^{10}}, \infty\right)$
Option B:
$f(x)=0$ has no solutions in the interval $\left[\frac{1}{\pi}, \infty\right)$
Option C:
The set of solutions of $f(x)=0$ in the interval $\left(0, \frac{1}{10^{10}}\right)$ is finite
Option D:
$f(x)=0$ has more than 25 solutions in the interval $\left(\frac{1}{\pi^{2}}, \frac{1}{\pi}\right)$
Correct

Answer: D

Step-by-step solution

$x^{2} \sin \left(\frac{\pi}{x^{2}}\right)=0 \Rightarrow \frac{\pi}{x^{2}}=n \pi \Rightarrow x= \pm \frac{1}{\sqrt{n}}$ $\frac{1}{\pi^{2}}<\frac{1}{\sqrt{n}}<\frac{1}{\pi} \Rightarrow \pi^{2}<\mathrm{n}<\pi^{4}$ $f(x)=0$ has more than 25 solutions

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Exam: JEE Advanced 2024
Paper: Paper 2
Subject: Mathematics
Chapter: Functions
Topic: Periodic Function

Answer: D

Step-by-step solution

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