JEE Advanced 2024 — Paper 2 — MATH-I solution

JEE Advanced2024Paper 2MATH-I

Let $f : R \to R$ be a function defined by

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

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 $(\frac{1}{π ^{2}}, \frac{1}{π})$ .

Solution

For $x \neq = 0$ , $f (x) = 0$ iff $sin (\frac{π}{x ^{2}}) = 0$ , i.e. $\frac{π}{x ^{2}} = nπ$ for some positive integer $n$ , giving $x = \frac{1}{n}$ .

Check each interval:

(A) On $[\frac{1}{1 0 ^{10}}, \infty)$ : $\frac{1}{n} \geq \frac{1}{1 0 ^{10}}$ means $n \leq 1 0^{20}$ , which is finite. So (A) is false.

(B) On $[\frac{1}{π}, \infty)$ : $\frac{1}{n} \geq \frac{1}{π}$ means $n \leq π^{2} \approx 9.87$ , so $n \in {1, 2, \dots, 9}$ — nine solutions exist, so (B) is false.

(C) On $(0, \frac{1}{1 0 ^{10}})$ : $\frac{1}{n} < \frac{1}{1 0 ^{10}}$ means $n > 1 0^{20}$ , infinitely many. So (C) is false.

(D) On $(\frac{1}{π ^{2}}, \frac{1}{π})$ : we need $π < n < π^{2}$ , i.e. $π^{2} < n < π^{4}$ . Numerically, $9.87 < n < 97.41$ , giving $n \in {10, 11, \dots, 97}$ , which is 88 values — more than 25. So (D) is TRUE.

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