JEE Main 2024 — 4 April, Shift 2

If the function f\left( x \right)=\left\{ \begin{array}{*{35}{l}}\frac{{{72}^{x}}-{{9}^{x}}-{{8}^{x}}+1}{\sqrt{2}-\sqrt{1+\text{cos}x}} & ,x\ne 0 \\\text{lo}{{\text{g}}_{e}}2\text{lo}{{\text{g}}_{e}}3 & ,x=0 \\\end{array} \right. is continuous at $x=0$, then the value of $a^{2}$ is equal to :

If $\lambda>0$, let $\theta$ be the angle between the vectors $\vec{a}=\hat{i}+\lambda \hat{j}-3 \hat{k}$ and $\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}$. If the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ are mutually perpendicular, then the value of $(14 \cos \theta)^{2}$ is equal to

Let C be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line $x+y=2$ intersects the circle C at the points P and Q . Let MN be a chord of C of length 2 unit and slope -1 . Then, a distance (in units) between the chord PQ and the chord MN is

Let a relation R on $\mathbb{N} \times \mathbb{N}$ be defined as : $\left(x_{1}, y_{1}\right) R\left(x_{2}, y_{2}\right)$ if and only if $x_{1} \leq x_{2}$ or $y_{1} \leq y_{2}$ Consider the two statements :

(I) R is reflexive but not symmetric.

(II) R is transitive

Then which one of the following is true?

Let three real numbers $a, b, c$ be in arithmetic progression and $\mathrm{a}+1, \mathrm{~b}, \mathrm{c}+3$ be in geometric progression. If $\mathrm{a}>10$ and the arithmetic mean of $\mathrm{a}, \mathrm{b}$ and c is 8 , then the cube of the geometric mean of $a, b$ and $c$ is

Let \text{A}=\left[ \begin{array}{*{35}{l}}1 & 2 \\0 & 1 \\\end{array} \right] and $\mathrm{B}=\mathrm{I}+\operatorname{adj}(\mathrm{A})+(\operatorname{adj} \mathrm{A})^{2}+\ldots+$ $(\operatorname{adj} \mathrm{A})^{10}$.

Then, the sum of all the elements of the matrix $B$ is :

The value of $\frac{1 \times 2^{2}+2 \times 3^{2}+\ldots+100 \times(101)^{2}}{1^{2} \times 2+2^{2} \times 3+\ldots+100^{2} \times 101}$ is

Let $f(x)=\int_{0}^{x}\left(t+\sin \left(1-e^{t}\right)\right) d t, x \in \mathbb{R}$. Then $\lim _{x \rightarrow 0} \frac{f(x)}{x^{3}}$ is equal to

The area (in sq. units) of the region described by $\left\{(\mathrm{x}, \mathrm{y}): \mathrm{y}^{2} \leq 2 \mathrm{x}\right.$, and $\left.\mathrm{y} \geq 4 \mathrm{x}-1\right\}$ is

The area (in sq. units) of the region $\mathrm{S}=\{\mathrm{z} \in \mathbb{C} ;|\mathrm{z}-1| \leq 2 ;(\mathrm{z}+\overline{\mathrm{z}})+\mathrm{i}(\mathrm{z}-\overline{\mathrm{z}}) \leq 2, \operatorname{lm}(z) \geq 0\}$ is

If the value of the integral $\int_{-1}^{1} \frac{\cos \alpha x}{1+3^{x}} d x$ is $\frac{2}{\pi}$. Then, a value of $\alpha$ is

Let $\mathrm{f}(\mathrm{x})=3 \sqrt{\mathrm{x}-2}+\sqrt{4-\mathrm{x}}$ be a real valued function. If $\alpha$ and $\beta$ are respectively the minimum and the

maximum values of $f$, then $\alpha^{2}+2 \beta^{2}$ is equal to

If the coefficients of $x^{4}, x^{5}$ and $x^{6}$ in the expansion of $(1+x)^{\mathrm{n}}$ are in the arithmetic progression, then the maximum value of n is :

Consider a hyperbola H having centre at the origin and foci and the x -axis. Let $\mathrm{C}_{1}$ be the circle touching the hyperbola H and having the centre at the origin. Let $\mathrm{C}_{2}$ be the circle touching the hyperbola H at its vertex and having the centre at one of its foci. If areas (in sq. units) of $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ are $36 \pi$ and $4 \pi$, respectively, then the length (in units) of latus rectum of H is

If the mean of the following probability distribution of a random variable X ;

is $\frac{46}{9}$, then the variance of the distribution is

Let PQ be a chord of the parabola $y^{2}=12 x$ and the midpoint of PQ be at $(4,1)$. Then, which of the following point lies on the line passing through the points P and Q ?

Given the inverse trigonometric function assumes principal values only. Let $\mathrm{x}, \mathrm{y}$ be any two real numbers in $[-1,1]$

such that $\cos ^{-1} \mathrm{x}-\sin ^{-1} \mathrm{y}=\alpha, \frac{-\pi}{2} \leq \alpha \leq \pi$. Then, the minimum value of $x^{2}+y^{2}+2 x y \sin \alpha$ is

Let $y=y(x)$ be the solution of the differential equation $\left(x^{2}+4\right)^{2} d y+\left(2 x^{3} y+8 x y-2\right) d x=0$. If $y(0)=0$,

then $y(2)$ is equal to

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $\vec{c}=x \hat{i}+2 \hat{j}+3 \hat{k}, x \in \mathbb{R}$. If $\vec{d}$ is the unit vector in the direction of

$\vec{b}+\vec{c}$ such that $\vec{a} \cdot \vec{d}=1$, then $(\vec{a} \times \vec{b}) \cdot \vec{c}$ is equal to

Let $P$ the point of intersection of the lines $\frac{x-2}{1}=\frac{y-4}{5}=\frac{z-2}{1} \quad$ and $\quad \frac{x-3}{2}=\frac{y-2}{3}=\frac{z-3}{2}$.

Then, the shortest distance of P from the line $4 \mathrm{x}=2 \mathrm{y}=\mathrm{z}$ is

Let $S=\left\{\sin ^{2} 2 \theta:\left(\sin ^{4} \theta+\cos ^{4} \theta\right) x^{2}+(\sin 2 \theta) \mathrm{x}+\right.$ $\left(\sin ^{6} \theta+\cos ^{6} \theta\right)=0$ has real roots $\}$.

If $\alpha$ and $\beta$ be the smallest and largest elements of the set S, respectively, then $3\left((\alpha-2)^{2}+(\beta-1)^{2}\right)$ equals.....

If $\int \operatorname{cosec}^{5} x d x=\alpha \cot x \operatorname{cosec} x\left(\operatorname{cosec}^{2} x+\frac{3}{2}\right)+\beta \log _{e}\left|\tan \frac{x}{2}\right|+C$ where $\alpha, \beta \in \mathbb{R}$ and C is constant

of integration , then the value of $8(\alpha+\beta)$ equals

Let $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ be a thrice differentiable function such that $f(0)=0, f(1)=1, f(2)=-1, f(3)=2$ and $f(4)=-2$. Then, the minimum number of zeros of $\left(3 f^{\prime} f^{\prime \prime}+f f^{\prime \prime \prime}\right)(x)$ is

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=\frac{2 x}{\sqrt{1+9 x^{2}}}$. If the composition of

$\mathrm{f}, \underbrace{(\text { f of of o...o } \mathrm{f})}_{10 \text { times }}(\mathrm{x})=\frac{2^{10} \mathrm{x}}{\sqrt{1+9 \alpha \mathrm{x}^{2}}}$, then the value of $\sqrt{3 \alpha+1}$ is equal to .....

Let A be a $2 \times 2$ symmetric matrix such that A\left[ \begin{array}{*{35}{l}}1 \\1 \\\end{array} \right]=\left[ \begin{array}{*{35}{l}}3 \\7 \\\end{array} \right]and the determinant of $A$ be 1. If $\mathrm{A}^{-1}=\alpha \mathrm{A}+\beta \mathrm{I}$, where I is an identity matrix of order $2 \times 2$, then $\alpha+\beta$ equals $\ldots$.

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $\frac{1}{3}$ and $\frac{2}{3}$ respectively. Let $x$ be the number of matches that the team wins, and $y$ be the number of matches that team loses. If the probability $\mathrm{P}(|\mathrm{x}-\mathrm{y}| \leq 2)$ is $p$, then $3^{9} \mathrm{p}$ equals......

Consider a triangle ABC having the vertices $\mathrm{A}(1,2), \mathrm{B}(\alpha, \beta)$ and $\mathrm{C}(\gamma, \delta)$ and angles $\angle \mathrm{ABC}=\frac{\pi}{6}$ and $\angle \mathrm{BAC}=\frac{2 \pi}{3}$. If the points B and C lie on the line $y=x+4$, then $\alpha^{2}+\gamma^{2}$ is equal to $\ldots .$.

Consider a line $L$ passing through the points $\mathrm{P}(1,2,1)$ and $\mathrm{Q}(2,1,-1)$. If the mirror image of the point $A(2,2,2)$ in the line $L$ is $(\alpha, \beta, \gamma)$, then $\alpha+\beta+6 \gamma$ is equal to .

Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $(x+y+2)^{2} d x=d y, y(0)=-2$.

Let the maximum and minimum values of the function $y=y(x)$ in $\left[0, \frac{\pi}{3}\right]$ be $\alpha$ and $\beta$, respectively.

If $(3 \alpha+\pi)^{2}+\beta^{2}=\gamma+\delta \sqrt{3}, \gamma, \delta \in \mathbb{Z}$, then $\gamma+\delta$ equals

The translational degrees of freedom ($f_t$ ) and rotational degrees of freedom ( $\mathrm{f}_{\text {r }}$ ) of $\mathrm{CH}_{4}$ molecule are :

The magnetic moment of a bar magnet is $0.5 \mathrm{Am}^{2}$. It is suspended in a uniform magnetic field of $8 \times 10^{-2} \mathrm{~T}$. The work done in rotating it from its most stable to most unstable position is :

Which of the diode circuit shows correct biasing used for the measurement of dynamic resistance of $\mathrm{p}-\mathrm{n}$ junction diode :

Arrange the following in the ascending order of wavelength : (A) Gamma rays $\left(\lambda_{1}\right)$ (B) x-ray $\left(\lambda_{2}\right)$ (C) Infrared waves $\left(\lambda_{3}\right)$ (D) Microwaves $\left(\lambda_{4}\right)$ Choose the most appropriate answer from the options given below :

Identify the logic gate given in the circuit :

The width of one of the two slits in a Young's double slit experiment is 4 times that of the other slit. The ratio of the maximum of the minimum intensity in the interference pattern is :

Correct formula for height of a satellite from earths surface is :

Match List I with List II

List-I		List-II
(A) Purely capacitive circuit	I.
(B) Purely inductive circuit	II.
(C) Series LCR circuit at resonance	III.
(D) Series LCR circuit	IV.

Choose the correct answer from the options given below :

Given below are two statements :

Statement I : The contact angle between a solid and a liquid is a property of the material of the solid and liquid as well.

Statement II : The rise of a liquid in a capillary tube does not depend on the inner radius of the tube.

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

A body of $m \mathrm{~kg}$ slides from rest along the curve of vertical circle from point A to B in friction less path. The velocity of the

body at $B$ is : (given, $\mathrm{R}=14 \mathrm{~m}, \mathrm{~g}=10 \mathrm{~m} / \mathrm{s}^{2}$ and $\sqrt{2}=1.4$ )

An electric bulb rated $50 \mathrm{~W}-200 \mathrm{~V}$ is connected across a 100 V supply. The power dissipation of the bulb is :

A 2 kg brick begins to slide over a surface which is inclined at an angle of $45^{\circ}$ with respect to horizontal axis. The co-efficient of static friction between their surfaces is :

In simple harmonic motion, the total mechanical energy of given system is E. If mass of oscillating particle $P$ is doubled then

the new energy of the system for same amplitude is :

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.

Assertion A : Number of photons increases with increase in frequency of light.

Reason R : Maximum kinetic energy of emitted electrons increases with the frequency of incident radiation.

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

According to Bohr's theory, the moment of momentum of an electron revolving in $4^{\text {th }}$ orbit of hydrogen atom is :

A sample of gas at temperature T is adiabatically expanded to double its volume. Adiabatic constant for the gas is $\gamma=3 / 2$. The work done by the gas in the process is : $(\mu=1$ mole $)$

A charge $q$ is placed at the center of one of the surface of a cube. The flux linked with the cube is :-

Applying the principle of homogeneity of dimensions, determine which one is correct. where T is time period, G is gravitational constant, $M$ is mass, $r$ is radius of orbit.

A 90 kg body placed at 2 R distance from surface of earth experiences gravitational pull of : ( $\mathrm{R}=$ Radius of earth, $\mathrm{g}=10 \mathrm{~ms}^{-2}$ )

The displacement of a particle executing SHM is given by $x=10 \sin \left(\omega t+\frac{\pi}{3}\right) \mathrm{m}$. The time period of motion is 3.14 s . The velocity of the particle at $t=0$ is $\qquad$ $\mathrm{m} / \mathrm{s}$.

A bus moving along a straight highway with speed of $72 \mathrm{~km} / \mathrm{h}$ is brought to halt within 4 s after applying the brakes. The distance travelled by the bus during this time (Assume the retardation is uniform) is $\qquad$ m .

A parallel plate capacitor of capacitance 12.5 pF is charged by a battery connected between its plates to potential difference of 12.0 V . The battery is now disconnected and a dielectric slab $\left(\epsilon_{r}=6\right)$ is inserted between the plates. The change in its potential energy after inserting the dielectric slab is $\qquad$ $\times 10^{-12} \mathrm{~J}$.

In a system two particles of masses $\mathrm{m}_{1}=3 \mathrm{~kg}$ and $\mathrm{m}_{2}=2 \mathrm{~kg}$ are placed at certain distance from each other. The particle of mass $m_{1}$ is moved towards the center of mass of the system through a distance 2 cm . In order to keep the center of mass of the system at the original position, the particle of mass $\mathrm{m}_{2}$ should move towards the center of mass by the distance $\qquad$ cm .

The disintegration energy $Q$ for the nuclear fission of ${ }^{235} \mathrm{U} \rightarrow{ }^{140} \mathrm{Ce}+{ }^{94} \mathrm{Zr}+\mathrm{n}$ is $\qquad$ MeV .

Given atomic masses of ${ }^{235} \mathrm{U}: 235.0439 \mathrm{u},{ }^{140} \mathrm{Ce} ; 139.9054 \mathrm{u}$, ${ }^{94} \mathrm{Zr}: 93.9063 \mathrm{u} ; \mathrm{n}: 1.0086 \mathrm{u}$

Value of $\mathrm{c}^{2}=931 \mathrm{MeV} / \mathrm{u}$

A light ray is incident on a glass slab of thickness $4 \sqrt{3} \mathrm{~cm}$ and refractive index $\sqrt{2}$. The angle of incidence is equal to the critical angle for the glass slab with air. The lateral displacement of ray after passing through glass slab is $\qquad$ cm . (Given $\sin 15^{\circ}=0.25$ )

A rod of length 60 cm rotates with a uniform angular velocity $20 \mathrm{rad} \mathrm{s}^{-1}$ about its perpendicular bisector, in a uniform magnetic field 0.5 T . The direction of magnetic field is parallel to the axis of rotation. The potential difference between the two ends of the rod is $\qquad$ V.

Two wires A and B are made up of the same material and have the same mass. Wire A has radius of 2.0 mm and wire $B$ has radius of 4.0 mm . The resistance of wire B is $2 \Omega$. The resistance of wire A is $\qquad$ $\Omega$.

Two parallel long current carrying wire separated by a distance 2 r are shown in the figure. The ratio of magnetic field at $A$ to the magnetic field produced at $C$ is $\frac{x}{7}$. The value of $x$ is $\qquad$

Mercury is filled in a tube of radius 2 cm up to a height of 30 cm . The force exerted by mercury on the bottom of the tube

is ___ N. (Given, atmospheric pressure $=10^{5} \mathrm{Nm}^{-2}$, density of mercury $=1.36 \times 10^{4} \mathrm{~kg} \mathrm{~m}^{-3}, \mathrm{~g}=10 \mathrm{~ms}^{-2}$,

$\left.\pi=\frac{22}{7}\right)$

The equilibrium constant for the reaction

$\mathrm{SO}_{3}(\mathrm{~g}) \rightleftharpoons \mathrm{SO}_{2}(\mathrm{~g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g})$

is $\mathrm{K}_{\mathrm{C}}=4.9 \times 10^{-2}$.

The value of $\mathrm{K}_{\mathrm{C}}$ for the reaction given below is $2 \mathrm{SO}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{SO}_{3}(\mathrm{~g})$ is

When $\mathrm{MnO}_{2}$ and $\mathrm{H}_{2} \mathrm{SO}_{4}$ is added to a salt (A), the greenish yellow gas liberated as salt (A) is :

The correct statement/s about Hydrogen bonding is/are :

A. Hydrogen bonding exists when H is covalently bonded to the highly electro negative atom.

B. Intermolecular H bonding is present in o-nitro phenol

C. Intramolecular H bonding is present in HF .

D. The magnitude of H bonding depends on the physical state of the compound.

E. H-bonding has powerful effect on the structure and properties of compounds.

Choose the correct answer from the options given below :

In the given reaction sequence, identify the the reagents "$A$ " and " $B$ ", respectively.

Common name of Benzene-1, 2-diol is

The adsorbent used in adsorption chromatography is/are

A. silica gel

B. alumina

C. quick lime

D. magnesia

Choose the most appropriate answer from the options given below :

Correct order of stability of carbanion is

If an iron (III) complex with the formula $\left[\mathrm{Fe}\left(\mathrm{NH}_{3}\right)_{\mathrm{x}}(\mathrm{CN})_{\mathrm{y}}\right]^{-}$has no electron in its $\mathrm{e}_{\mathrm{g}}$ orbital, then the value of $x+y$ is

Fuel cell, using hydrogen and oxygen as fuels

A. has been used in spaceship

B. has as efficiency of $40 \%$ to produce electricity

C. uses aluminium as catalysts

D. is eco-friendly

E. is actually a type of Galvanic cell only

Choose the incorrect statement about Dalton's Atomic Theory:

Match List I with List II

Choose the correct answer from the options given below:

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

Statement I : The correct order of first ionization enthalpy values of $\mathrm{Li}, \mathrm{Na}, \mathrm{F}$ and Cl is $\mathrm{Na}<\mathrm{Li}<\mathrm{Cl}<\mathrm{F}$.

Statement II : The correct order of negative electron gain enthalpy values of $\mathrm{Li}, \mathrm{Na}, \mathrm{F}$ and Cl is $\mathrm{Na}<\mathrm{Li}<\mathrm{F}<\mathrm{Cl}$

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

For a strong electrolyte, a plot of molar conductivity against (concentration) ${ }^{1 / 2}$ is a straight line, with a negative slope, the correct unit for the slope is

A first row transition metal in its +2 oxidation state has a spin-only magnetic moment value of 3.86 BM . The atomic number of the metal is

The number of unpaired d-electrons in $\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}$ is $\qquad$

The number of species from the following that have pyramidal geometry around the central atom is $\qquad$

$\mathrm{S}_{2} \mathrm{O}_{3}^{2-}, \mathrm{SO}_{4}^{2-}, \mathrm{SO}_{3}^{2-}, \mathrm{S}_{2} \mathrm{O}_{7}^{2-}$

The maximum number of orbitals which can be identified with $\mathrm{n}=4$ and $\mathrm{m}_{l}=0$ is $\_\_\_\_\_$.

Number of compounds/species from the following with non-zero dipole moment is $\qquad$

$\mathrm{BeCl}_{2}, \mathrm{BCl}_{3}, \mathrm{NF}_{3}, \mathrm{XeF}_{4}, \mathrm{CCl}_{4}, \mathrm{H}_{2} \mathrm{O} \mathrm{H}_{2} \mathrm{~S}, \mathrm{HBr}$, $\mathrm{CO}_{2}, \mathrm{H}_{2}, \mathrm{HCl}$

From 6.55 g of aniline, the maximum amount of acetanilide that can be prepared will be $\qquad$ $\times 10^{-1} \mathrm{~g}$

Consider the following reaction, the rate expression of which is given below $\mathrm{A}+\mathrm{B} \rightarrow \mathrm{C}$ rate $=\mathrm{k}[\mathrm{A}]^{1 / 2}[\mathrm{~B}]^{1 / 2}$

The reaction is initiated by taking 1 M concentration $A$ and $B$ each. If the rate constant ( k ) is $4.6 \times 10^{-2} \mathrm{~s}^{-1}$, then the time taken for A to become 0.1 M is $\qquad$ sec. (nearest integer)

Phthalimide is made to undergo following sequence of reactions.

Phthalimide $\xrightarrow{\begin{array}{l}\text { (i) } \mathrm{KOH} \text { (ii)Benzylchloride }\end{array}}{ }^{\prime} \mathrm{P}^{\prime}$ Total number of $\pi$ bonds present in product ${ }^{\prime} \mathrm{P}^{\prime}$ is/are

The total number of 'sigma' and 'Pi' bonds in 2-oxohex-4-ynoic acid is $\qquad$

A first row transition metal with highest enthalpy of atomisation, upon reaction with oxygen at high temperature forms oxides of formula $\mathrm{M}_{2} \mathrm{O}_{\mathrm{n}}$ (where $\mathrm{n}=3,4,5$ ). The 'spin-only' magnetic moment value of the amphoteric oxide from the above oxides is $\qquad$ BM (near integer) (Given atomic number : $\mathrm{Sc}: 21, \mathrm{Ti}: 22, \mathrm{~V}: 23$, $\mathrm{Cr}: 24, \mathrm{Mn}: 25, \mathrm{Fe}: 26$, Co : 27, Ni : $28, \mathrm{Cu}: 29$, $\mathrm{Zn}: 30)$

2.7 Kg of each of water and acetic acid are mixed, The freezing point of the solution will be $-\mathrm{x}{ }^{\circ} \mathrm{C}$. Consider the acetic acid

does not dimerise in water, nor dissociates in water $\mathrm{x}=$ $\qquad$ (nearest integer)

[Given : Molar mass of water $=18 \mathrm{~g} \mathrm{~mol}^{-1}$, acetic acid $\left.=60 \mathrm{~g} \mathrm{~mol}^{-1}\right]$

${ }^{\mathrm{K}_{\mathrm{f}}} \mathrm{H}_{2} \mathrm{O}: 1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$

${ }^{\mathrm{K}_{\mathrm{f}}}$ acetic acid : $3.90 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$ freezing point : $\mathrm{H}_{2} \mathrm{O}=273 \mathrm{~K}$, acetic acid $\left.=290 \mathrm{~K}\right]$

Vanillin compound obtained from vanilla beans, has total sum of oxygen atoms and $\pi$ electrons is