JEE Main 2024 Apr 9 Shift 2 — Question Paper

Let the range of the function $f(x)=\frac{1}{2+\sin 3x+\cos 3x}, x \in \mathbb{R}$ 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

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

Let $\alpha, \beta;\ \alpha>\beta$, be the roots of the equation $x^2-\sqrt{2}x-\sqrt{3}=0$. Let $\mathrm{P}_n=\alpha^n-\beta^n, n \in \mathbb{N}$. Then $\left(11\sqrt{3}-10\sqrt{2}\right)\mathrm{P}_{10}+\left(11\sqrt{2}+10\right)\mathrm{P}_{11}-11\mathrm{P}_{12}$ is equal to

Let $B=\begin{bmatrix} 1 & 3 \\ 1 & 5 \end{bmatrix}$ and $A$ be a $2\times 2$ matrix such that $AB^{-1}=A^{-1}$. If $BCB^{-1}=A$ and $C^4+\alpha C^2+\beta I=O$, then $2\beta-\alpha$ is equal to

$\lim\limits_{x \to 0} \dfrac{e-(1+2x)^{\frac{1}{2x}}}{x}$ 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

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

If $\log_e y = 3\sin^{-1}x$, then $(1-x^2)y''-xy'$ at $x=\frac{1}{2}$ is equal to

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

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

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

The value of the integral $\int_{-1}^{2} \log_e\left(x+\sqrt{x^2+1}\right)dx$ is

Let $\int_{0}^{x}\sqrt{1-\left(y'(t)\right)^2}\,dt=\int_{0}^{x} y(t)\,dt$, $0 \le x \le 3$, $y \ge 0$, $y(0)=0$. Then at $x=2$, $y''+y+1$ is equal to

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

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

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

Between the following two statements:

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

Statement II: In a triangle $ABC$, $\cos 2A+\cos 2B+\cos 2C \ge -\dfrac{3}{2}$.

Let $\vec{a}=2\hat{i}+\alpha\hat{j}+\hat{k}$, $\vec{b}=-\hat{i}+\hat{k}$, $\vec{c}=\beta\hat{j}-\hat{k}$, 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 $\dfrac{\sqrt{21}}{2}$, be $(\alpha_1, \beta_1)$ and $(\alpha_2, \beta_2)$. Then $\alpha_1^2+\beta_1^2-\alpha_2\beta_2$ is equal to

If the variance of the frequency distribution

$x$	$c$	$2c$	$3c$	$4c$	$5c$	$6c$
$f$	$2$	$1$	$1$	$1$	$1$	$1$

is 160, then the value of $c \in \mathbb{N}$ is

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

Let $A=\{(x, y): 2x+3y=23, x, y \in \mathbb{N}\}$ and $B=\{x: (x, y) \in A\}$. Then the number of one-one functions from $A$ to $B$ is equal to __________.

Consider the matrices : $A=\begin{bmatrix} 2 & -5 \\ 3 & m \end{bmatrix}$, $B=\begin{bmatrix} 20 \\ m \end{bmatrix}$ and $X=\begin{bmatrix} x \\ y \end{bmatrix}$. Let the set of all $m$, for which the system of equations $AX=B$ has a negative solution (i.e., $x<0$ and $y<0$), be the interval $(a, b)$. Then $8\int_{a}^{b}|A|\,dm$ is equal to ________.

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

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

Let the set of all values of $p$, for which $f(x)=\left(p^2-6p+8\right)\left(\sin^2 2x-\cos^2 2x\right)+2(2-p)x+7$ does not have any critical point, be the interval $(a, b)$. Then $16ab$ is equal to ________.

For a differentiable function $f: \mathbb{R} \to \mathbb{R}$, suppose $f'(x)=3f(x)+\alpha$, where $\alpha \in \mathbb{R}$, $f(0)=1$ and $\lim\limits_{x \to -\infty} f(x)=7$. Then $9f\left(-\log_e 3\right)$ is equal to ________.

Consider the circle $C: x^2+y^2=4$ and the parabola $P: y^2=8x$. 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 $(2q-p)^2$ is equal to ________.

Let $A$, $B$ and $C$ be three points on the parabola $y^2=6x$ and let the line segment $AB$ 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 $\left(\dfrac{AM \cdot BN}{CD}\right)^2$ is equal to ________.

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

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $2\sin^{-1}x+3\cos^{-1}x=\dfrac{2\pi}{5}$, is ________.

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

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

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

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

(Assume that the system is in equilibrium and $g = 10~\mathrm{m/s^2}$)

[Figure: a rope hangs from the ceiling; at its midpoint a horizontal force $\vec{F}$ acts to the left, the upper segment $T_1$ makes $\theta = 45°$ with the vertical, and the lower segment $T_2$ carries a 1 kg block]

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 ?

A real gas within a closed chamber at $27°C$ undergoes the cyclic process as shown in figure. The gas obeys $PV^3 = RT$ equation for the path $A$ to $B$. The net work done in the complete cycle is (assuming $R = 8$ J/mol K) :

[Figure: P–V diagram with $P$ in $\mathrm{N/m^2}$ and $V$ in $\mathrm{m^3}$; curve from A (2, 20) to B (4, 10), horizontal line from B back to C (2, 10), and vertical line from C up to A]

The energy released in the fusion of 2 kg of hydrogen deep in the sun is $E_H$ 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 $4{}^{1}_{1}\mathrm{H} + 2e^- \rightarrow {}^{4}_{2}He + 2\nu + 6\gamma + 26.7$ MeV, energy released in the fission reaction of $^{235}\mathrm{U}$ is 200 $MeV$ per fission nucleus and $\mathrm{N_A} = 6.023\times10^{23}$)

A satellite of $10^3$ kg mass is revolving in circular orbit of radius $2R$. If $\frac{10^4 R}{6}$ J energy is supplied to the satellite, it would revolve in a new circular orbit of radius :

(use $g = 10~\mathrm{m/s^2}$, $R$ = radius of earth)

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:

A spherical ball of radius $1\times10^{-4}$ m and density $10^5~\mathrm{kg/m^3}$ 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 $9.8\times10^{-6}~N~s/m^2$)

The temperature of a gas is $-78°\mathrm{C}$ 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 2K is :

Five charges $+q$, $+5q$, $-2q$, $+3q$ and $-4q$ are situated as shown in the figure. The electric flux due to this configuration through the surface $S$ is :

[Figure: an irregular closed surface S enclosing charges $+5q$, $q$ and $-2q$; charges $+3q$ and $-4q$ lie outside the surface]

The effective resistance between $A$ and $B$, if resistance of each resistor is $R$, will be :

[Figure: resistor network — from A a resistor R leads to a diamond of resistors (two R in the upper arms, two R in the lower arms) with two vertical bridge resistors R and two horizontal middle resistors R meeting at the centre, then a resistor R leads to B]

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

A square loop of side 15 cm being moved towards right at a constant speed of 2 cm/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 $t = 10$ s will be :

[Figure: a 15 cm square loop moving right at 2 cm/s towards a 50 cm wide region of uniform magnetic field $B = 1.0~\mathrm{T}$ directed into the page]

The following figure represents two biconvex lenses $L_1$ and $L_2$ having focal length 10 cm and 15 cm respectively. The distance between $L_1$ & $L_2$ is :

[Figure: parallel rays enter biconvex lens $L_1$, converge and then pass through biconvex lens $L_2$, emerging parallel to the axis]

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:

[Figure: logic circuit — inputs A and B feed an AND gate whose output goes to one input of a NOR gate giving output E; A and B also each pass through NOT gates into a second AND gate whose output goes to the other input of the NOR gate]

Truth table:

A	B	E
0	0	0
0	1	X
1	0	Y
1	1	0

In the truth table of the above circuit the value of X and Y are :

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

[Figure: I–V curve — small current for positive V rising near the origin, and for negative V a nearly constant small reverse current of $5(\mu A)$ that drops sharply at a large negative (breakdown) voltage]

The magnetic field in a plane electromagnetic wave is $\mathrm{B_y} = (3.5\times10^{-7})\sin(1.5\times10^3 x + 0.5\times10^{11} t)$ T. The corresponding electric field will be :

A force $(3x^2 + 2x - 5)$ N displaces a body from $x = 2$ m to $x = 4$ m. Work done by this force is ________ $J$.

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 $4^{th}$ bright fringe. The thickness of the glass-plate is ________ $\mu$m.

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 $8\sqrt{2}$ V. The power dissipation in the circuit is ________ W.

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 ________ $\mathrm{Am^2}$.

(given $\pi = \frac{22}{7}$)

At room temperature ($27°C$), the resistance of a heating element is $50~\Omega$. The temperature coefficient of the material is $2.4\times10^{-4}~°C^{-1}$. The temperature of the element, when its resistance is $62~\Omega$, is ________ $°C$.

An electric field $\vec{E} = (2x\hat{i})\,\mathrm{NC^{-1}}$ exists in space. A cube of side 2m is placed in the space as per figure given below. The electric flux through the cube is ________ $\mathrm{Nm^2/C}$.

[Figure: a cube of side 2 m placed with one face in the y–z plane at a distance 2 m from the origin along the x-axis]

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

(Given $\pi = \frac{22}{7}$)

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

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

[Figure: circuit — a voltmeter V of internal resistance 10 kΩ connected in parallel with resistance R, in series with a milliammeter (mA) and a variable source E; below it a graph of $I$(mA) versus $V$(volt) showing a straight line through the origin passing through points (4, 2), (6, 3) and (8, 4)]

The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 'A' $\times\ 10^{12}$ hertz and that has a radiant intensity in that direction of $\frac{1}{\text{'B'}}$ watt per steradian.

'A' and 'B' are respectively

The correct increasing order for bond angles among $\mathrm{BF_3}$, $\mathrm{PF_3}$ and $\mathrm{ClF_3}$ is :

For a sparingly soluble salt $\mathrm{AB_2}$, the equilibrium concentrations of $\mathrm{A^{2+}}$ ions and $\mathrm{B^-}$ ions are $1.2 \times 10^{-4}$ M and $0.24 \times 10^{-3}$ M, respectively. The solubility product of $\mathrm{AB_2}$ is :

Match List I with List II

	LIST I (Cell)		LIST II (Use/Property/Reaction)
A.	Leclanche cell	I.	Converts energy of combustion into electrical energy
B.	Ni - Cd cell	II.	Does not involve any ion in solution and is used in hearing aids
C.	Fuel cell	III.	Rechargeable
D.	Mercury cell	IV.	Reaction at anode $\mathrm{Zn \rightarrow Zn^{2+} + 2e^-}$

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)

Match List I with List II

	LIST I (Element)		LIST II (Electronic Configuration)
A.	N	I.	$\mathrm{[Ar]\ 3d^{10}\ 4s^{2}\ 4p^{5}}$
B.	S	II.	$\mathrm{[Ne]\ 3s^{2}\ 3p^{4}}$
C.	Br	III.	$\mathrm{[He]\ 2s^{2}\ 2p^{3}}$
D.	Kr	IV.	$\mathrm{[Ar]\ 3d^{10}\ 4s^{2}\ 4p^{6}}$

Choose the correct answer from the options given below:

Match List I with List II

	LIST I		LIST II
A.	Melting Point [K]	I.	$\mathrm{Tl > In > Ga > Al > B}$
B.	Ionic Radius [$\mathrm{M^{+3}}$/pm]	II.	$\mathrm{B > Tl > Al \approx Ga > In}$
C.	$\Delta_i H_1$ [kJ mol$^{-1}$]	III.	$\mathrm{Tl > In > Al > Ga > B}$
D.	Atomic Radius [pm]	IV.	$\mathrm{B > Al > Tl > In > Ga}$

Choose the correct answer from the options given below:

Give below are two statements :

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

Match List I with List II

	LIST I		LIST II
A.	$\mathrm{K_2[Ni(CN)_4]}$	I.	$\mathrm{sp^3}$
B.	$\mathrm{[Ni(CO)_4]}$	II.	$\mathrm{sp^3d^2}$
C.	$\mathrm{[Co(NH_3)_6]Cl_3}$	III.	$\mathrm{dsp^2}$
D.	$\mathrm{Na_3[CoF_6]}$	IV.	$\mathrm{d^2sp^3}$

Choose the correct answer from the options given below:

Match List I with List II

	LIST I (Test)		LIST II (Observation)
A.	$\mathrm{Br_2}$ water test	I.	Yellow orange or orange red precipitate formed
B.	Ceric ammonium nitrate test	II.	Reddish orange colour disappears
C.	Ferric chloride test	III.	Red colour appears
D.	2, 4 - DNP test	IV.	Blue, Green, Violet or Red colour appear

Choose the correct answer from the options given below:

The correct stability order of the following resonance structures of $\mathrm{CH_3-CH=CH-CHO}$ is

[Figure: three resonance structures — I: $\mathrm{CH_3-\overset{\ominus}{C}H-CH=CH-\overset{\oplus}{O}}$ (carbanion on the carbon next to $\mathrm{CH_3}$, positively charged oxygen); II: $\mathrm{CH_3-\overset{\oplus}{C}H-CH=CH-\overset{\ominus}{O}}$ (carbocation on the carbon next to $\mathrm{CH_3}$, negatively charged oxygen); III: $\mathrm{CH_3-CH=CH-CH=O}$ (neutral structure)]

The incorrect statement regarding ethyne is

[Figure: $\mathrm{C_6H_5-CH(CH_3)-CH(Br)(OCH_3)}$ (benzylic carbon bears $\mathrm{CH_3}$; adjacent carbon bears Br and $\mathrm{OCH_3}$)] $\xrightarrow[\Delta]{\mathrm{KCN\ (alc)}}$ Major Product 'P'

In the above reaction product 'P' is

Total number of stereo isomers possible for the given structure :

[Figure: $\mathrm{CH_3-C(Br)=CH-CH_2-CH(Br)-C(CH_3)=C(Br)-CH_3}$ (a diene with Br on both double bonds and on the sp$^3$ carbon between them)]