JEE Main 2024 — 9 April, Shift 2

$\operatorname{Lim}_{x \rightarrow 0} \frac{e-(1+2 x)^{\frac{1}{2 x}}}{x}$ is equal to :

Consider the line L passing through the points $(1,2,3)$ and $(2,3,5)$. The distance of the point $\left(\frac{11}{3}, \frac{11}{3}, \frac{19}{3}\right)$ from the line L along the line $\frac{3 x-11}{2}=\frac{3 y-11}{1}=\frac{3 z-19}{2}$ is equal to :

Let $\int_{0}^{x} \sqrt{1-\left(y^{\prime}(t)\right)^{2}} d t=\int_{0}^{x} y(t) d t, 0 \leq x \leq 3, y \geq 0$ $\mathrm{y}(0)=0$. Then at $\mathrm{x}=2, \mathrm{y}^{\prime \prime}+\mathrm{y}+1$ is equal

Let $z$ be a complex number such that the real part of $\frac{z-2 i}{z+2 i}$ is zero. Then, the maximum value of $|\mathrm{z}-(6+8 \mathrm{i})|$ is equal to :

The area (in square units) of the region enclosed by the ellipse $x^{2}+3 y^{2}=18$ in the first quadrant below the line $\mathrm{y}=\mathrm{x}$ is :

Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E: \frac{(x-1)^{2}}{100}+\frac{(y-1)^{2}}{75}=1$ and the eccentricity

of the hyperbola H be the reciprocal of the eccentricity of the ellipse E. If the length of the transverse axis of $H$ is $\alpha$

and the length of its conjugate axis is $\beta$, then $3 \alpha^{2}+2 \beta^{2}$ is equal to :

Two vertices of a triangle ABC are $\mathrm{A}(3,-1)$ and $\mathrm{B}(-2,3)$, and its orthocentre is $\mathrm{P}(1,1)$. If the coordinates of the point $C$ are $(\alpha, \beta)$ and the centre of the circle circumscribing the triangle PAB is $(h, k)$, then the value of $(\alpha+\beta)+2(h+k)$ equals :

If the variance of the frequency distribution is 160 , then the value of $\mathrm{c} \in \mathrm{N}$ is

Let the range of the function $f(x)=\frac{1}{2+\sin 3 x+\cos 3 x}, x \in \operatorname{IR}$ be $[a, b]$. If $\alpha$ and $\beta$ are respectively the A.M. and the G.M. of $a$ and $b$, then $\frac{\alpha}{\beta}$ is equal to :

Between the following two statements :

Statement-I : Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$ and $\vec{b}=2 \hat{i}+\hat{j}-\hat{k}$. Then the vector $\vec{r}$ satisfying $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{r}}=0$ is of magnitude $\sqrt{10}$.

Statement-II : In a triangle $\mathrm{ABC}, \cos 2 \mathrm{~A}+\cos 2 \mathrm{~B}$ $+\cos 2 C \geq-\frac{3}{2}$.

In the light of the above statements, choose the correct answer from the options given below :

$\lim _{x \rightarrow \frac{x}{2}}\left(\frac{\int_{x^{3}}^{(\pi / 2)^{3}}\left(\sin \left(2 t^{1 / 3}\right)+\cos \left(t^{1 / 3}\right)\right) \mathrm{dt}}{\left(x-\frac{\pi}{2}\right)^{2}}\right)$ is equal to :

The sum of the coefficient of $x^{2 / 3}$ and $x^{-2 / 5}$ in the binomial expansion of $\left(x^{2 / 3}+\frac{1}{2} x^{-2 / 5}\right)^{9}$ is :

LetB=\left[ \begin{array}{*{35}{l}}1 & 3 \\1 & 5 \\\end{array} \right] and $A$ be a $2 \times 2$ matrix such that $\mathrm{AB}^{-1}=\mathrm{A}^{-1}$. If $\mathrm{BCB}^{-1}=\mathrm{A}$ and $\mathrm{C}^{4}+\alpha \mathrm{C}^{2}+\beta \mathrm{I}=\mathrm{O}$, then $2 \beta-\alpha$ is equal to :

If $\log _{e} y=3 \sin ^{-1} x$, then $(1-x)^{2} y^{\prime \prime}-x y^{\prime}$ at $x=\frac{1}{2}$ is equal to :

The integral $\int_{1 / 4}^{3 / 4} \cos \left(2 \cot ^{-1} \sqrt{\frac{1-x}{1+x}}\right) d x$ is equal to:

Let $a$, ar, $a r^{2}$, ........be an infinite G.P. If $\sum_{n=0}^{\infty} a^{n}=57$ and $\sum_{n=0}^{\infty} a^{3} r^{3 n}=9747$, then $a+18 r$ is equal to :

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $i^{\text {th }}$ roll than the number obtained

in the $(\mathrm{i}-1)^{\text {th }}$ roll, $\mathrm{i}=2,3$, is equal to :

The value of the integral $\int_{-1}^{2} \log _{e}\left(x+\sqrt{x^{2}+1}\right) d x$ is :

Let $\alpha, \beta ; \alpha>\beta$, be the roots of the equation $x^{2}-\sqrt{2} x-\sqrt{3}=0$.

Let $P_{n}=\alpha^{n}-\beta^{n}, n \in N$. Then $(11 \sqrt{3}-10 \sqrt{2}) \mathrm{P}_{10}+(11 \sqrt{2}+10) \mathrm{P}_{11}-11 \mathrm{P}_{12}$ is equal to :

Let $\quad \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\alpha \hat{j}+\hat{\mathrm{k}}, \quad \dot{\mathrm{b}}=-\hat{\mathrm{i}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{c}}=\beta \hat{j}-\hat{\mathrm{k}}$where $\alpha$ and $\beta$ are integers and $\alpha \beta=-6$.

Let the values of the ordered pair $(\alpha, \beta)$ for which the area of the parallelogram of diagonals $\vec{a}+\vec{b}$ and $\vec{b}+\vec{c}$ is $\frac{\sqrt{21}}{2}$, be

$\left(\alpha_{1}, \beta_{1}\right)$ and $\left(\alpha_{2}, \beta_{2}\right)$. Then $\alpha_{1}^{2}+\beta_{1}^{2}-\alpha_{2} \beta_{2}$ is equal to

Consider the circle $C: x^{2}+y^{2}=4$ and the parabola $P: y^{2}=8 x$.If the set of all values of $\alpha$, for which three chords of the circle C on three distinct lines passing through the point $(\alpha, 0)$ are bisected by the parabola $P$ is the interval $(p, q)$, then $(2 q-p)^{2}$ is equal to $\qquad$

Let the set of all values of $p$, for which $f(x)=\left(p^{2}-6 p+8\right)\left(\sin ^{2} 2 x-\cos ^{2} 2 x\right)+2(2-p) x+7$ does not

have any critical point, be the interval $(a, b)$. Then 16 ab is equal to $\qquad$ .

For a differentiable function $\mathrm{f}: I R \rightarrow I R$, suppose $f^{\prime}(x)=3 f(x)+\alpha$, where $\alpha \in \operatorname{IR}, f(0)=1$ and $\lim _{x \rightarrow-\infty} f(x)=7$. Then $9 \mathrm{f}\left(-\log _{\mathrm{e}} 3\right)$ is equal to $\qquad$ .

The number of integers, between 100 and 1000 having the sum of their digits equals to 14 , is $\qquad$ .

Let $A=\{(x, y): 2 x+3 y=23, x, y \in N\}$ and

$B=\{x:(x, y) \in A\}$. Then the number of one-one functions from $A$ to $B$ is equal to $\qquad$

Let $\mathrm{A}, \mathrm{B}$ and C be three points on the parabola $y^{2}=6 x$ and let the line segment $A B$ meet the line $L$ through $C$ parallel to the x -axis at the point D . Let M and N respectively be the feet of the perpendiculars from $A$ and $B$ on $L$. Then $\left(\frac{A M \cdot B N}{C D}\right)^{2}$ is equal to $\qquad$

The square of the distance of the image of the point $(6,1,5)$ in the line $\frac{x-1}{3}=\frac{y}{2}=\frac{z-2}{4}$, from the origin is $\qquad$

If $\left(\frac{1}{\alpha+1}+\frac{1}{\alpha+2}+\ldots .+\frac{1}{\alpha+1012}\right)$ $-\left(\frac{1}{2 \cdot 1}+\frac{1}{4 \cdot 3}+\frac{1}{6 \cdot 5}+\ldots .+\frac{1}{2024 \cdot 2023}\right)$ $=\frac{1}{2024}$, then $\alpha$

is equal to-

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation

$2 \sin ^{-1} x+3 \cos ^{-1} x=\frac{2 \pi}{5}$, is $\qquad$

Consider the matrices : \text{A}=\left[ \begin{matrix}2 & -5 \\3 & \text{ }\!\!~\!\!\text{ m} \\\end{matrix} \right],\text{B}=\left[ \begin{array}{*{35}{l}}20 \\ \text{ }\!\!~\!\!\text{ m} \\\end{array} \right]

and X=\left[ \begin{array}{*{35}{l}}x \\ y \\\end{array}\right]. Let the set of all $m$, for which the system of equations $\mathrm{AX}=\mathrm{B}$ has a negative solution

(i.e., $\mathrm{x}<0$ and $\mathrm{y}<0$ ), be the interval (a, b). Then $8 \int_{a}^{b}|\mathrm{~A}| \mathrm{dm}$ is equal to $\qquad$ .

A nucleus at rest disintegrates into two smaller nuclei with their masses in the ratio of $2: 1$. After disintegration they will move

The temperature of a gas is $-78^{\circ} \mathrm{C}$ and the average translational kinetic energy of its molecules is K . The temperature at which the average translational kinetic energy of the molecules of the same gas becomes 2 K is :

A hydrogen atom in ground state is given an energy of 10.2 eV. How many spectral lines will be emitted due to transition of electrons?

The magnetic field in a plane electromagnetic wave is $\mathrm{B}_{\mathrm{y}}=\left(3.5 \times 10^{-7}\right) \sin \left(1.5 \times 10^{3} \mathrm{x}+0.5\right.$ $\left.\times 10^{11} \mathrm{t}\right) \mathrm{T}$.

The corresponding electric field will be

A square loop of side 15 cm being moved towards right at a constant speed of $2 \mathrm{~cm} / \mathrm{s}$ as shown in figure. The front edge enters the 50 cm wide magnetic field at $t=0$. The value of induced emf in the loop at $\mathrm{t}=10 \mathrm{~s}$ will be :

Two cars are travelling towards each other at speed of $20 \mathrm{~m} \mathrm{~s}^{-1}$ each. When the cars are 300 m apart, both the drivers apply brakes and the cars retard at the rate of $2 \mathrm{~m} \mathrm{~s}^{-2}$. The distance between them when they come to rest is :

The $I-V$ characteristics of an electronic device shown in the figure. The device is :

The excess pressure inside a soap bubble is thrice the excess pressure inside a second soap bubble. The ratio between the volume of the first and the second bubble is :

The de-Broglie wavelength associated with a particle of mass $m$ and energy $E$ is $\mathrm{h} / \sqrt{2 m E}$. The dimensional formula for Planck's constant is :

A satellite of $10^{3} \mathrm{~kg}$ mass is revolving in circular orbit of radius 2 R . If $\frac{10^{4} \mathrm{R}}{6} J$ energy is supplied to the satellite, it would revolve in a new circular orbit of radius : (use $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}, \mathrm{R}=$ radius of earth)

A proton and a deutron $(\mathrm{q}=+\mathrm{e}, m=2.0 \mathrm{u})$ having same kinetic energies enter a region of uniform magnetic field $\overrightarrow{\mathrm{B}}$, moving perpendicular to $\overrightarrow{\mathrm{B}}$. The ratio of the radius $r_{d}$ of deutron path to the radius $r_{p}$ of the proton path is :

UV light of 4.13 eV is incident on a photosensitive metal surface having work function 3.13 eV . The maximum kinetic energy of ejected photoelectrons will be :

The energy released in the fusion of 2 kg of hydrogen deep in the sun is $\mathrm{E}_{\mathrm{H}}$ and the energy released in the fission of 2 kg of

${ }^{235} \mathrm{U}$ is $E_{U}$. The ratio $\frac{E_{H}}{E_{U}}$ is approximately : (Consider the fusion reaction as

$4{ }_{1}^{1} \mathrm{H}+2 \mathrm{e}^{-} \rightarrow{ }_{2}^{4} \mathrm{He}+2 \mathrm{v}+6 \gamma+26.7 \mathrm{MeV}$, energy released in the fission reaction of ${ }^{235} \mathrm{U}$ is 200 MeV per

fission nucleus and $\mathrm{N}_{\mathrm{A}}=6.023 \times 10^{23}$ )

A 1 kg mass is suspended from the ceiling by a rope of length 4 m . A horizontal force ' $F$ ' is applied at the mid point of the rope so that the rope makes an angle of $45^{\circ}$ with respect to the vertical axis as shown in figure. The magnitude of F is :

A spherical ball of radius $1 \times 10^{-4} \mathrm{~m}$ and density $10^{5}$ $\mathrm{kg} / \mathrm{m}^{3}$ falls freely under gravity through a distance $h$ before entering a tank of water, If after entering in water the velocity of the ball does not change, then the value of $h$ is approximately : (The coefficient of viscosity of water is $9.8 \times 10^{-6}$ $\left.\mathrm{N} \mathrm{s} / \mathrm{m}^{2}\right)$

A straight magnetic strip has a magnetic moment of $44 \mathrm{Am}^{2}$. If the strip is bent in a semicircular shape, its magnetic moment will be $\qquad$ $\mathrm{Am}^{2}$ (Given $\pi=\frac{22}{7}$ )

A particle of mass 0.50 kg executes simple harmonic motion under force $\mathrm{F}=-50\left(\mathrm{Nm}^{-1}\right) \mathrm{x}$. The time period of oscillation is $\frac{x}{35} \mathrm{~s}$. The value of x is (Given $\pi=\frac{22}{7}$ )

A capacitor of reactance $4 \sqrt{3} \Omega$ and a resistor of resistance $4 \Omega$ are connected in series with an ac source of peak value $8 \sqrt{2} \mathrm{~V}$. The power dissipation in the circuit is $\qquad$ W.

An electric field $\overrightarrow{\mathrm{E}}=(2 x \hat{\mathrm{i}}) \mathrm{NC}^{-1}$ exists in space. A cube of side 2 m is placed in the space as per figure given below.

The electric flux through the cube is $\ldots \ldots . . . . . . . . . . \mathrm{Nm}^{2} / \mathrm{C}$.

A circular disc reaches from top to bottom of an inclined plane of length $l$. When it slips down the plane, if takes t s . When it rolls down the plane then it takes $\left(\frac{\alpha}{2}\right)^{1 / 2} \mathrm{t} s$, where $\alpha$ is

To determine the resistance (R) of a wire, a circuit is designed below, The V-I characteristic curve for this circuit is plotted for the voltmeter and the ammeter readings as shown in figure. The value of R is .... $\qquad$ $\Omega$.

The resultant of two vectors $\vec{A}$ and $\vec{B}$ is perpendicular to $\overrightarrow{\mathrm{A}}$ and its magnitude is half that of $\vec{B}$. The angle between vectors $\vec{A}$ and $\vec{B}$ is

Monochromatic light of wavelength 500 nm is used in Young's double slit experiment. An interference pattern is obtained on a screen When one of the slits is covered with a very thin glass plate (refractive index $=1.5$ ), the central maximum is shifted to a position previously occupied by the $4^{\text {th }}$ bright fringe. The thickness of the glass-plate is $\qquad$ .$\mu \mathrm{m}$.

A force $\left(3 x^{2}+2 x-5\right) N$ displaces a body from $\mathrm{x}=2 \mathrm{~m}$ to $\mathrm{x}=4 \mathrm{~m}$. Work done by this force is ............J.

At room temperature $\left(27^{\circ} \mathrm{C}\right)$, the resistance of a heating element is $50 \Omega$. The temperature coefficient of the material is $2.4 \times 10^{-4}{ }^{\circ} \mathrm{C}^{-1}$. The temperature of the element, when its resistance is $62 \Omega$, is $\qquad$ ${ }^{\circ} \mathrm{C}$.

The correct stability order of the following resonance structures of $\mathrm{CH}_{3}-\mathrm{CH}=\mathrm{CH}-\mathrm{CHO}$ is

The correct increasing order for bond angles among $\mathrm{BF}_{3}, \mathrm{PF}_{3}$ and $\mathrm{C} \ell \mathrm{F}_{3}$ is :

Match List I with List II

Choose the correct answer from the options given below:

Match List I with List II

Choose the correct answer from the options given below:

Match List I with List II

Choose the correct answer from the options given below:

The coordination environment of $\mathrm{Ca}^{2+}$ ion in its complex with EDTA $^{4}$ is :

The incorrect statement about Glucose is :

Which of the following compound can give positive iodoform test when treated with aqueous KOH solution followed by potassium hypoiodite.

For a sparingly soluble salt $\mathrm{AB}_{2}$, the equilibrium concentrations of $\mathrm{A}^{2+}$ ions and $\mathrm{B}^{-}$ions are $1.2 \times 10^{-4} \mathrm{M}$ and $0.24 \times 10^{-3} \mathrm{M}$, respectively. The solubility product of $\mathrm{AB}_{2}$ is :

Major product of the following reaction is

Given below are two statements, one is labelled as Statement I and the other is labelled as Statement II.

Statement I : The higher oxidation states are more stable down the group among transition elements unlike p-block elements.

Statement II : Copper can not liberate hydrogen from weak acids.

In the light of the above statements, choose the correct answer from the options given below :

The incorrect statement regarding ethyne is

Match List I with List II.

Choose the correct answer from the options given below:

Match List I with List II

Choose the correct answer from the options given below :

Which of the following compounds will give silver mirror with ammoniacal silver nitrate?

(A) Formic acid

(B) Formaldehyde

(C) Benzaldehyde

(D) Acetone

Choose the correct answer from the options given below :

Which out of the following is a correct equation to show change in molar conductivity with respect to concentration for a weak electrolyte, if the symbols carry their usual meaning :

The electronic configuration of Einsteinium is : $($ Given atomic number of Einsteinium $=99)$

Number of oxygen atoms present in chemical formula of fuming sulphuric acid is $\qquad$ .

A transition metal ' M ' among $\mathrm{Sc}, \mathrm{Ti}, \mathrm{V}, \mathrm{Cr}, \mathrm{Mn}$ and Fe has the highest second ionisation enthalpy. The spin only

magnetic moment value of $\mathrm{M}^{+}$ion is $\qquad$ BM (Near integer)

(Given atomic number $\mathrm{Sc}: 21, \mathrm{Ti}: 22, \mathrm{~V}: 23, \mathrm{Cr}$ : 24, $\mathrm{Mn}: 25, \mathrm{Fe}: 26)$

The vapour pressure of pure benzene and methyl benzene at $27^{\circ} \mathrm{C}$ is given as 80 Torr and 24 Torr, respectively. The mole fraction of methyl benzene in vapour phase, in equilibrium with an equimolar mixture of those two liquids (ideal solution) at the same temperature is $\qquad$ $\times 10^{-2}$ (nearest integer)

Consider the following test for a group-IV cation. $\mathrm{M}^{2+}+\mathrm{H}_{2} \mathrm{~S} \rightarrow \mathrm{~A}$ (Black precipitate) + byproduct $\mathrm{A}+$ aqua regia

$\rightarrow \mathrm{B}+\mathrm{NOCl}+\mathrm{S}+\mathrm{H}_{2} \mathrm{O}$ $\mathrm{B}+\mathrm{KNO}_{2}+\mathrm{CH}_{3} \mathrm{COOH} \rightarrow \mathrm{C}+$ byproduct The spin only magnetic moment

value of the metal complex C is $\qquad$ BM. (Nearest integer)

Consider the following first order gas phase reaction at constant temperature $\mathrm{A}(\mathrm{g}) \rightarrow 2 \mathrm{~B}(\mathrm{~g})+\mathrm{C}(\mathrm{g})$ If the total pressure of the gases is found to be 200 torr after 23 sec . and 300 torr upon the complete decomposition of A after a very long time, then the rate constant of the given reaction is $\qquad$ $\times 10^{-2} \mathrm{~s}^{-1}$ (nearest integer) [Given : $\left.\log _{10}(2)=0.301\right]$

In the given TLC, the distance of spot $A \& B$ are 5 cm & 7 cm , from the bottom of TLC plate, respectively. $R_{f}$ value of $B$ is $x \times 10^{-1}$ times more than $A$. The value of $x$ is $\qquad$ .