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 ?

(D) $f(x)=0$ has more than 25 solutions in the interval $\left(\frac{1}{\pi^{2}}, \frac{1}{\pi}\right)$.

(A) $f(x)=0$ has infinitely many solutions in the interval $\left[\frac{1}{10^{10}}, \infty\right)$.

(B) $f(x)=0$ has no solutions in the interval $\left[\frac{1}{\pi}, \infty\right)$.

(C) The set of solutions of $f(x)=0$ in the interval $\left(0, \frac{1}{10^{10}}\right)$ is finite.

JEE Advanced · 2024 · 26 May 2024 · Paper 2 · Trigonometric Equations

Let

f : R \to R

be a function defined by

f (x) = {x^{2} sin (\frac{π}{x ^{2}}), 0, if x \neq = 0 if x = 0.

Then which of the following statements is TRUE ?

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

Answer: (D)

For

x \neq = 0

f (x) = 0

when

sin (\frac{π}{x ^{2}}) = 0

, which gives

\frac{π}{x ^{2}} = nπ

for

n \in N

, so

x = \frac{1}{n}

.

(A) For

x \geq \frac{1}{1 0 ^{10}}

\frac{1}{n} \geq \frac{1}{1 0 ^{10}}

gives

n \leq 1 0^{20}

, which is finite.

(B) For

x \geq \frac{1}{π}

\frac{1}{n} \geq \frac{1}{π}

gives

n < π^{2} \approx 9.87

, so there are 9 solutions.

(C) For

0 < x < \frac{1}{1 0 ^{10}}

\frac{1}{n} < \frac{1}{1 0 ^{10}}

gives

n > 1 0^{20}

, which is infinite.

(D) For

\frac{1}{π ^{2}} < x < \frac{1}{π}

π < n < π^{2}

gives

π^{2} < n < π^{4}

, i.e.,

n \in (9.87, 97.4)

, which contains more than 25 integers.

Therefore, option (D) is correct.

JEE Advanced · 2024 · 26 May 2024 · Paper 2 · Single correct option