JEE Main 2024 — 29 January, Shift 1

If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to

In an A.P., the sixth terms $\mathrm{a}_{6}=2$. If the $a_{1} a_{4} a_{5}$ is the greatest, then the common difference of the A.P., is equal to

A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is

If $z=\frac{1}{2}-2 i$, is such that $|\mathrm{z}+1|=\alpha \mathrm{z}+\beta(1+\mathrm{i}), \mathrm{i}=\sqrt{-1}$ and $\alpha, \beta \in \mathrm{R} \quad$, then $\alpha+\beta$ is equal to

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero vectors such that $\vec{b}$ and $\vec{c}$ are non-collinear if $\vec{a}+5 \vec{b}$ is collinear with $\vec{c}, \vec{b}+6 \vec{c}$ is collinear with $\vec{a}$ and $\vec{a}+\alpha \vec{b}+\beta \vec{c}=\overrightarrow{0}$, then $\alpha+\beta$ is equal to

Let $\left(5, \frac{\mathrm{a}}{4}\right)$, be the circumcenter of a triangle with vertices $A(a,-2), B(a, 6)$ and $C\left(\frac{a}{4},-2\right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha+\beta+\gamma$ is

For $\mathrm{x} \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, if $y(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname{cosec} x \sec x+\tan x \sin ^{2} x} d x$ and $\lim _{x \rightarrow\left(\frac{\pi}{2}\right)^{-}} y(x)=0$ then $y\left(\frac{\pi}{4}\right)$ is equal to

If $\alpha ,-\frac{\pi }{2}<\alpha <\frac{\pi }{2}$ is the solution of $4\cos \theta +5\sin \theta =1,$ then the value of $\tan \alpha$ is

A function $y=f(x)$ satisfies $f(x) \sin 2 x+\sin x-\left(1+\cos ^{2} x\right) f^{\prime}(x)=0$ with condition $f(0)=0$. Then $f\left(\frac{\pi}{2}\right)$ is equal to

Let $O$ be the origin and the position vector of $A$ and $B$ be $2 \hat{i}+2 \hat{j}+\hat{k}$ and $2 \hat{i}+4 \hat{j}+4 \hat{k}$ respectively. If the internal bisector of $\angle A O B$ meets the line $A B$ at C , then the length of OC is

Consider the function $\mathrm{f}:\left[\frac{1}{2}, 1\right] \rightarrow \mathrm{R}$ defined by $f(x)=4 \sqrt{2} x^{3}-3 \sqrt{2} x-1$. Consider the statements (I) The curve $y=f(x)$ intersects the $x$-axis exactly at one point (II) The curve $y=f(x)$ intersects the $x$-axis at $\mathrm{x}=\cos \frac{\pi}{12}$

Let $A=\left[\begin{array}{lll}1 & 0 & 0\\ 0 & \alpha & \beta\\ 0 & \beta & \alpha\end{array}\right]$ and $|2 A|^{3}=2^{21}$ where $\alpha, \beta \in Z$, Then a value of $\alpha$ is

Let $P Q R$ be a triangle with $R(-1,4,2)$. Suppose $\mathrm{M}(2,1,2)$ is the mid point of PQ . The distance of the centroid of $\triangle P Q R$ from the point of intersection of the line $\frac{\mathrm{x}-2}{0}=\frac{\mathrm{y}}{2}=\frac{\mathrm{z}+3}{-1}$ and $\frac{\mathrm{x}-1}{1}=\frac{\mathrm{y}+3}{-3}=\frac{\mathrm{z}+1}{1}$ is

Let R be a relation on $\mathrm{Z} \times \mathrm{Z}$ defined by $(a, b) R(c, d)$ if and only if $\mathrm{ad}-\mathrm{bc}$ is divisible by 5 . Then $R$ is

If the value of the integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{\mathrm{x}^{2} \cos \mathrm{x}}{1+\pi^{\mathrm{x}}}+\frac{1+\sin ^{2} \mathrm{x}}{1+\mathrm{e}^{\sin \mathrm{sin}^{2023}}}\right) \mathrm{dx}=\frac{\pi}{4}(\pi+\mathrm{a})-2$, then the value of a is

Suppose$f(x)=\frac{\left(2^{x}+2^{-x}\right) \tan x \sqrt{\tan ^{-1}\left(x^{2}-x+1\right)}}{\left(7 x^{2}+3 x+1\right)^{3}}$Then the value of $f^{\prime}(0)$ is equal to

Let $A$ be a square matrix such that $A A^{T}=I$. Then $\frac{1}{2} A\left[\left(A+A^{T}\right)^{2}+\left(A-A^{T}\right)^{2}\right]$ is equal to …

Equation of two diameters of a circle are $2 x-3 y=5$ and $3 x-4 y=7$. The line joining the points $\left(-\frac{22}{7},-4\right)$ and $\left(-\frac{1}{7}, 3\right)$ intersects the circle at only one point $P(\alpha, \beta)$. Then $17 \beta-\alpha$ is equal to

All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word "GTWENTY" IS

Let $\alpha, \beta$ be the roots of the equation $x^{2}-x+2=0$ with $\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$. Then $\alpha^{6}+\alpha^{4}+\beta^{4}-5 \alpha^{2}$ is equal to

Let $f(x)=2^{x}-x^{2}, x \in R$. If $m$ and $n$ are respectively the number of points at which the curves $\mathrm{y}=\mathrm{f}(\mathrm{x})$ and $\mathrm{y}=\mathrm{f}^{\prime}(\mathrm{x})$ intersects the x -axis, then the value of $m+n$ is

If the points of intersection of two distinct conics $x^{2}+y^{2}=4 b$ and $\frac{x^{2}}{16}+\frac{y^{2}}{b^{2}}=1$ lie on the curve $y^{2}=3 x^{2}$, then $3 \sqrt{3}$ times the area of the rectangle formed by the intersection points is $\qquad$

If the solution curve $y=y(x)$ of the differential equation $\left(1+y^{2}\right)\left(1+\log _{e} x\right) d x+x d y=0, x>0$ passes through the point $(1,1)$ and $y(e)=\frac{\alpha-\tan \left(\frac{3}{2}\right)}{\beta+\tan \left(\frac{3}{2}\right)}$, then $\alpha+2 \beta$ is

If the mean and variance of the data $65,68,58,44$, $48,45,60, \alpha, \beta, 60$ where $\alpha>\beta$ are 56 and 66.2 respectively, then $\alpha^{2}+\beta^{2}$ is equal to

The area (in sq. units) of the part of circle $x^{2}+y^{2}=169$ which is below the line $5 x-y=13$ is $\frac{\pi \alpha}{2 \beta}-\frac{65}{2}+\frac{\alpha}{\beta} \sin ^{-1}\left(\frac{12}{13}\right) \quad$ where $\alpha, \beta \quad$ are coprime numbers. Then $\alpha+\beta$ is equal to

If $\frac{{ }^{11} \mathrm{C}_{1}}{2}+\frac{{ }^{11} \mathrm{C}_{2}}{3}+\ldots . .+\frac{{ }^{11} \mathrm{C}_{9}}{10}=\frac{n}{\mathrm{~m}}$ with $\operatorname{gcd}(n, m)=1$, then $n+m$ is equal to

A line with direction ratios $2,1,2$ meets the lines $\mathrm{x}=\mathrm{y}+2=\mathrm{z}$ and $\mathrm{x}+2=2 \mathrm{y}=2 \mathrm{z}$ respectively at the point P and Q . if the length of the perpendicular from the point $(1,2,12)$ to the line PQ is $l$, then $l^{2}$ is

In the given circuit, the breakdown voltage of the Zener diode is 3.0 V . What is the value of $\mathrm{I}_{z}$ ?

The electric current through a wire varies with time as $I=I_{0}+\beta$ t. where $I_{0}=20 \mathrm{~A}$ and $\beta=3 \mathrm{~A} / \mathrm{s}$. The amount of electric charge crossed through a section of the wire in 20 s is :

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

Statement –I: If a capillary tube is immersed first in cold water and then in hot water, the height of capillary rise will be smaller in hot water.

Statement –II: If a capillary tube is immersed first in cold water and then in hot water, the height of capillary rise will be smaller in cold water.

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

A convex mirror of radius of curvature 30 cm forms an image that is half the size of the object. The object distance is :

A body starts moving from rest with constant acceleration covers displacement $S_{1}$ in first ( $p-1$ ) seconds and $\mathrm{S}_{2}$ in first p seconds. The displacement $\mathrm{S}_{1}+\mathrm{S}_{2}$ will be made in time :

The potential energy function (in $J$ ) of a particle in a region of space is given as $U=\left(2 x^{2}+3 y^{3}+2 z\right)$. Here $x, y$ and $z$ are in meter. The magnitude of x - component of force (in N ) acting on the particle at point $\mathrm{P}(1,2,3) \mathrm{m}$ is :

The resistance $\mathrm{R}=\frac{\mathrm{V}}{\mathrm{I}}$ where $\mathrm{V}=(200 \pm 5) \mathrm{V}$ and $I=(20 \pm 0.2) A$, the percentage error in the measurement of R is :

A block of mass 100 kg slides over a distance of 10 m on a horizontal surface. If the co-efficient of friction between the surfaces is 0.4 , then the work done against friction (in J ) is :

Match List I with List II

Chose the correct answer from the options given below

If the radius of curvature of the path of two particles of same mass are in the ratio 3:4, then in order to have constant centripetal force, their velocities will be in the ratio of:

A galvanometer having coil resistance $10 \Omega$ shows a full scale deflection for a current of 3 mA . For it to measure a current of 8 A , the value of the shunt should be:

The de-Broglie wavelength of an electron is the same as that of a photon. If velocity of electron is $25 \%$ of the velocity of light, then the ratio of K.E. of electron and K.E. of photon will be:

The deflection in moving coil galvanometer falls from 25 divisions to 5 division when a shunt of $24 \Omega$ is applied. The resistance of galvanometer coil will be :

A biconvex lens of refractive index 1.5 has a focal length of 20 cm in air. Its focal length when immersed in a liquid of refractive index 1.6 will be

A thermodynamic system is taken from an original state A to an intermediate state B by a linear process as shown in the figure. It's volume is then reduced to the original value from B to C by an isobaric process. The total work done by the gas from A to B and B to C would be :

At what distance above and below the surface of the earth a body will have same weight, (take radius of earth as R.)

A capacitor of capacitance $100 \mu \mathrm{~F}$ is charged to a potential of 12 V and connected to a 6.4 mH inductor to produce oscillations. The maximum current in the circuit would be :

The explosive in a Hydrogen bomb is a mixture of ${ }_{1} \mathrm{H}^{2},{ }_{1} \mathrm{H}^{3}$ and ${ }_{3} \mathrm{Li}^{6}$ in some condensed form. The chain reaction is given by

${ }_{3} \mathrm{Li}^{6}+{ }_{0} \mathrm{n}^{1} \rightarrow{ }_{2} \mathrm{He}^{4}+{ }_{1} \mathrm{H}^{3}$

${ }_{1} \mathrm{H}^{2}+{ }_{1} \mathrm{H}^{3} \rightarrow{ }_{2} \mathrm{He}^{4}+{ }_{0} \mathrm{n}^{1}$

During the explosion the energy released is approximately [Given : $\mathrm{M}(\mathrm{Li})=6.01690 \mathrm{amu} . \mathrm{M}\left({ }_{1} \mathrm{H}^{2}\right)=2.01471$ amu. $\mathrm{M}\left({ }_{2} \mathrm{He}^{4}\right)=4.00388 \mathrm{amu}$, and $1 \mathrm{amu}=931.5$ $\mathrm{MeV}]$

Two vessels $A$ and $B$ are of the same size and are at same temperature. A contains 1 g of hydrogen and B contains 1 g of oxygen. $\mathrm{P}_{\mathrm{A}}$ and $\mathrm{P}_{\mathrm{B}}$ are the pressures of the gases in $A$ and $B$ respectively, then $\frac{\mathrm{P}_{\mathrm{A}}}{\mathrm{P}_{\mathrm{B}}}$ is :

When a hydrogen atom going from $n=2$ to $n=1$ emits a photon, its recoil speed is $\frac{\mathrm{X}}{5} \mathrm{~m} / \mathrm{s}$. Where $\mathrm{x}=$ (Use : mass of hydrogen atom $\left.=1.6 \times 10^{-27} \mathrm{~kg}\right)$

A ball rolls off the top of a stairway with horizontal velocity $u$. The steps are 0.1 m high and 0.1 m wide. The minimum velocity u with which that ball just hits the step 5 of the stairway will be $\sqrt{\mathrm{x}} \mathrm{ms}^{-1}$ where $\mathrm{x}=$ $\_\_\_\_\_\_\_$ [use $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$ ].

A square loop of side 10 cm and resistance $0.7 \Omega$ is placed vertically in east-west plane. A uniform magnetic field of 0.20 T is set up across the plane in north east direction. The magnetic field is decreased to zero in 1 s at a steady rate. Then, magnitude of induced emf is $\sqrt{\mathrm{x}} \times 10^{-3} \mathrm{~V}$. The value of $x$ is $\qquad$ .

A cylinder is rolling down on an inclined plane of inclination $60^{\circ}$. It's acceleration during rolling down will be $\frac{\mathrm{x}}{\sqrt{3}} \mathrm{~m} / \mathrm{s}^{2}$, where $\mathrm{x}=$ $\_\_\_\_\_\_\_$ . (use $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$ ).

The magnetic potential due to a magnetic dipole at a point on its axis situated at a distance of 20 cm from its center is $1.5 \times 10^{-5} \mathrm{Tm}$. The magnetic moment of the dipole is $\qquad$ $\mathrm{Am}^{2}$. (Given : $\frac{\mu_{0}}{4 \pi}=10^{-7} \mathrm{TmA}^{-1}$ )

In a double slit experiment shown in figure, when light of wavelength 400 nm is used, dark fringe is observed at P . If $\mathrm{D}=0.2 \mathrm{~m}$. the minimum distance between the slits $S_{1}$ and $S_{2}$ is $\_\_\_\_\_\_\_$ mm .

A $16 \Omega$ wire is bend to form a square loop. A 9 V battery with internal resistance $1 \Omega$ is connected across one of its sides. If a $4 \mu \mathrm{~F}$ capacitor is connected across one of its diagonals, the energy stored by the capacitor will be $\frac{x}{2} \mu \mathrm{~J}$. where $\mathrm{x}=$ $\qquad$

When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is $\frac{x}{8}$, where $x=$ $\qquad$ .

An electron is moving under the influence of the electric field of a uniformly charged infinite plane sheet S having surface charge density $+\sigma$. The electron at $t=0$ is at a distance of 1 m from S and has a speed of $1 \mathrm{~m} / \mathrm{s}$. The maximum value of $\sigma$ if the electron strikes $S$ at $t=1 \mathrm{~s}$ is $\alpha\left[\frac{\mathrm{m} \epsilon_{0}}{\mathrm{e}}\right] \frac{\mathrm{C}}{\mathrm{m}^{2}}$ the value of $\alpha$ is

In a test experiment on a model aeroplane in wind tunnel, the flow speeds on the upper and lower surfaces of the wings are $70 \mathrm{~ms}^{-1}$ and $65 \mathrm{~ms}^{-1}$ respectively. If the wing area is $2 \mathrm{~m}^{2}$ the lift of the wing is $\qquad$ N . (Given density of air $=1.2 \mathrm{~kg} \mathrm{~m}^{-3}$ )

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

Assertion (A) : The first ionisation enthalpy decreases across a period.

Reason (R) : The increasing nuclear charge outweighs the shielding across the period.

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

Match List I with List II

Choose the correct answer from the options given below:

The difference in energy between the actual structure and the lowest energy resonance structure for the given compound is

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

Statement I : The electronegativity of group 14 elements from Si to Pb gradually decreases.

Statement II : Group 14 contains non-metallic, metallic, as well as metalloid elements.

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

The correct set of four quantum numbers for the valence electron of rubidium atom $(Z=37)$ is:

Appearance of blood red colour, on treatment of the sodium fusion extract of an organic compound with $\mathrm{FeSO}_{4}$ in presence of concentrated $\mathrm{H}_{2} \mathrm{SO}_{4}$ indicates the presence of element/s

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

Assertion A : Aryl halides cannot be prepared by replacement of hydroxyl group of phenol by halogen atom.

Reason R : Phenols react with halogen acids violently.

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

Identify the incorrect pair from the following

The interaction between $\pi$ bond and lone pair of electrons present on an adjacent atom is responsible for

$\mathrm{KMnO}_{4}$ decomposes on heating at 513 K to form $\mathrm{O}_{2}$ along with

Type of amino acids obtained by hydrolysis of proteins is :

The final product A formed in the following multistep reaction sequence is

Which of the following is not correct?

Chlorine undergoes disproportionation in alkaline medium as shown below :

The values of $a, b, c$ and $d$ in a balanced redox reaction are respectively:

In alkaline medium. $\mathrm{MnO}_{4}^{-}$oxidises $\mathrm{I}^{-}$to

The mass of zinc produced by the electrolysis of zinc sulphate solution with a steady current of 0.015 A for 15 minutes is $\qquad$ $\times 10^{-4} \mathrm{~g}$. (Atomic mass of zinc $=65.4 \mathrm{amu})$

For a reaction taking place in three steps at same temperature, overall rate constant $K=\frac{\mathrm{K}_{1} \mathrm{~K}_{2}}{\mathrm{~K}_{3}}$. If $\mathrm{Ea}_{1}, \mathrm{Ea}_{2}$ and $\mathrm{Ea}_{3}$ are 40,50 and $60 \mathrm{~kJ} / \mathrm{mol}$ respectively, the overall Ea is $\qquad$ $\mathrm{kJ} / \mathrm{mol}$.

For the reaction $\mathrm{N}_{2} \mathrm{O}_{4}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NO}_{2}(\mathrm{~g})$, $\mathrm{K}_{\mathrm{p}}=0.492 \mathrm{~atm}$ at $300 \mathrm{~K} . \mathrm{K}_{\mathrm{c}}$

for the reaction at same temperature is $\qquad$ $\times 10^{-2}$.

(Given : $\mathrm{R}=0.082 \mathrm{~L} \mathrm{~atm}^{-1} \mathrm{Kol}^{-1}$ )

A solution of $\mathrm{H_2SO_4}$ is $31.4\%$ $\mathrm{H_2SO_4}$ by mass and has a density of $1.25\ \mathrm{g\ mL^{-1}}$. The molarity of the $\mathrm{H_2SO_4}$ solution is $\_\_\_\_\_$ $M$ (nearest integer).

[Given: molar mass of $\mathrm{H_2SO_4} = 98\ \mathrm{g\ mol^{-1}}$]

The osmotic pressure of a dilute solution is $7 \times 10^{5} \mathrm{~Pa}$ at 273 K . Osmotic pressure of the same solution at 283 K is $\qquad$ $\times 10^{4} \mathrm{Nm}^{-2}$.

The number of species from the following which are paramagnetic and with bond order equal to one is $\qquad$ .

From the compounds given below, number of compounds which give positive Fehling's test is $\qquad$ . Benzaldehyde, Acetaldehyde, Acetone, Acetophenone,Methanal, 4-nitrobenzaldehyde, cyclohexane carbaldehyde.