1
JEE Advanced 2024 Paper 1 Online
MCQ (Single Correct Answer)
+3
-1
Change Language

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

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

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

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

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

List-I List-II
(P) If $a = 0$, $b = 1$, $c = 0$, and $d = 0$, then (1) $h$ is one-one.
(Q) If $a = 1$, $b = 0$, $c = 0$, and $d = 0$, then (2) $h$ is onto.
(R) If $a = 0$, $b = 0$, $c = 1$, and $d = 0$, then (3) $h$ is differentiable on $\mathbb{R}$.
(S) If $a = 0$, $b = 0$, $c = 0$, and $d = 1$, then (4) the range of $h$ is $[0, 1]$.
(5) the range of $h$ is $\{0, 1\}$.

The correct option is
A
$(\mathrm{P}) \rightarrow(4)$ $(\mathrm{Q}) \rightarrow(3)$ $(\mathrm{R}) \rightarrow(1)$ (S) $\rightarrow$ (2)
B
$(\mathrm{P}) \rightarrow(5)$ $(\mathrm{Q}) \rightarrow(2)$ $(\mathrm{R}) \rightarrow(4)$ (S) $\rightarrow(3)$
C
$(\mathrm{P}) \rightarrow(5)$ $(\mathrm{Q}) \rightarrow(3)$ $(\mathrm{R}) \rightarrow(2)$ $(\mathrm{S}) \rightarrow(4)$
D
$(\mathrm{P}) \rightarrow(4)$ $(\mathrm{Q}) \rightarrow(2)$ $(\mathrm{R}) \rightarrow(1)$ $(\mathrm{S}) \rightarrow(3)$
2
JEE Advanced 2024 Paper 1 Online
MCQ (Single Correct Answer)
+3
-1
Change Language

A dimensionless quantity is constructed in terms of electronic charge $e$, permittivity of free space $\varepsilon_0$, Planck's constant $h$, and speed of light $c$. If the dimensionless quantity is written as $e^\alpha \varepsilon_0{ }^\beta h^\gamma c^\delta$ and $n$ is a non-zero integer, then $(\alpha, \beta, \gamma, \delta)$ is given by :

A
$(2 n,-n,-n,-n)$
B
$(n,-n,-2 n,-n)$
C
$(n,-n,-n,-2 n)$
D
$(2 n,-n,-2 n,-2 n)$
3
JEE Advanced 2024 Paper 1 Online
MCQ (Single Correct Answer)
+3
-1
Change Language

An infinitely long wire, located on the $z$-axis, carries a current $I$ along the $+z$-direction and produces the magnetic field $\vec{B}$. The magnitude of the line integral $\int \vec{B} \cdot \overrightarrow{d l}$ along a straight line from the point $(-\sqrt{3} a, a, 0)$ to $(a, a, 0)$ is given by

[ $\mu_0$ is the magnetic permeability of free space.]

A
$${{7{\mu _0}I} \over {24}}$$
B
$${{7{\mu _0}I} \over {12}}$$
C
$${{{\mu _0}I} \over {8}}$$
D
$${{{\mu _0}I} \over {6}}$$
4
JEE Advanced 2024 Paper 1 Online
MCQ (Single Correct Answer)
+3
-1
Change Language

Two beads, each with charge $q$ and mass $m$, are on a horizontal, frictionless, non-conducting, circular hoop of radius $R$. One of the beads is glued to the hoop at some point, while the other one performs small oscillations about its equilibrium position along the hoop. The square of the angular frequency of the small oscillations is given by

[ $\varepsilon_0$ is the permittivity of free space.]

A
$${{{q^2}} \over {4\pi {\varepsilon _0}{R^3}m}}$$
B
$${{{q^2}} \over {32\pi {\varepsilon _0}{R^3}m}}$$
C
$${{{q^2}} \over {8\pi {\varepsilon _0}{R^3}m}}$$
D
$${{{q^2}} \over {16\pi {\varepsilon _0}{R^3}m}}$$
JEE Advanced Papers
EXAM MAP
Medical
NEETAIIMS
Graduate Aptitude Test in Engineering
GATE CSEGATE ECEGATE EEGATE MEGATE CEGATE PIGATE IN
Civil Services
UPSC Civil Service
Defence
NDA
Staff Selection Commission
SSC CGL Tier I
CBSE
Class 12