JEE Advanced 2024 · Paper 1 · Mathematics Q17

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MathematicsList match+3 / -1Question 17 of 51

Let $f : \mathbb{R} \to \mathbb{R}$ and $g : \mathbb{R} \to \mathbb{R}$ be functions defined by

$f(x) = \begin{cases} x|x|\sin\left(\dfrac{1}{x}\right), & x \neq 0, \\ 0, & x = 0, \end{cases} \quad\text{and}\quad g(x) = \begin{cases} 1 - 2x, & 0 \le x \le \dfrac{1}{2}, \\ 0, & \text{otherwise}. \end{cases}$

Let $a, b, c, d \in \mathbb{R}$ . Define the function $h : \mathbb{R} \to \mathbb{R}$ by

$h(x) = a\,f(x) + b\left(g(x) + g\left(\frac{1}{2} - x\right)\right) + c\,(x - g(x)) + d\,g(x),\quad x \in \mathbb{R}.$

Match each entry in List-I to the correct entry in List-II.

The correct option is

List-I

P
If $a = 0$ , $b = 1$ , $c = 0$ , and $d = 0$ , then
Q
If $a = 1$ , $b = 0$ , $c = 0$ , and $d = 0$ , then
R
If $a = 0$ , $b = 0$ , $c = 1$ , and $d = 0$ , then
S
If $a = 0$ , $b = 0$ , $c = 0$ , and $d = 1$ , then

List-II

1
$h$ is one-one.
2
$h$ is onto.
3
$h$ is differentiable on $\mathbb{R}$ .
4
the range of $h$ is $[0, 1]$ .
5
the range of $h$ is $\{0, 1\}$ .

P → 4; Q → 3; R → 1; S → 2

P → 5; Q → 2; R → 4; S → 3

P → 5; Q → 3; R → 2; S → 4

P → 4; Q → 2; R → 1; S → 3

Answer and worked solution

Official answer: C

(P) $h(x) = g(x) + g\left(\frac12 - x\right)$ . On $\left[0, \frac12\right]$ both arguments lie in $\left[0,\frac12\right]$ : $h = (1-2x) + 2x = 1$ . Outside that interval both terms are 0. Range $\{0, 1\}$ . (P)→(5).

(Q) $h = f$ . For $x \ne 0$ it is differentiable, and $f'(0) = \lim_{h\to0}|h|\sin\frac1h = 0$ . So $f$ is differentiable on $\mathbb{R}$ . (Q)→(3).

(R) $h(x) = x - g(x)$ :

$x < 0$ : $h = x$ , values $(-\infty, 0)$ ;
$0 \le x \le \frac12$ : $h = 3x - 1$ , values $\left[-1, \frac12\right]$ ;
$x > \frac12$ : $h = x$ , values $\left(\frac12, \infty\right)$ .

The union is $\mathbb{R}$ , so $h$ is onto. It is not one-one, e.g. $h\left(-\frac12\right) = h\left(\frac16\right) = -\frac12$ . (R)→(2).

(S) $h = g$ , range $[0, 1]$ . (S)→(4).

Option (C).

Topics:Types of functions Differentiability Domain and range

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