JEE Main 2024 — 29 Jan Shift 1 (Previous Year Paper)

If $\alpha$, $-\frac\pi2<\alpha<\frac\pi2$ is the solution of $4\cos\theta+5\sin\theta=1$, then the value of $\tan\alpha$ is:

1. A.$\frac{10-\sqrt{10}}{12}$
2. B.$\frac{\sqrt{10}-10}{6}$
3. C.$\frac{10-\sqrt{10}}{6}$
4. D.$\frac{\sqrt{10}-10}{12}$✓

Correct answer: D

Consider the function $f:\left[\frac12,1\right]\to\mathbb{R}$ defined by $f(x)=4\sqrt2x^3-3\sqrt2x-1$. 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 $x=\cos\frac{\pi}{12}$ Then:

1. A.Both (I) and (II) are incorrect
2. B.Only (II) is correct
3. C.Only (I) is correct
4. D.Both (I) and (II) are correct✓

Correct answer: D

A function $y=f(x)$ satisfies $f(x)\sin2x+\sin x-(1+\cos^2x)f'(x)=0$ with condition $f(0)=0$. Then, $f\left(\frac\pi2\right)$ is equal to:

1. A.1✓
2. B.0
3. C.−1
4. D.2

Correct answer: A

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

1. A.Reflexive and symmetric but not transitive✓
2. B.Reflexive but neither symmetric nor transitive
3. C.Reflexive and transitive but not symmetric
4. D.Reflexive, symmetric and transitive

Correct answer: A

$\lim_{x\to\frac\pi2}\left(\frac{1}{\left(x-\frac\pi2\right)^2}\int_{x^3}^{\left(\frac\pi2\right)^3}\cos\left(t^{\frac13}\right)dt\right)$ is equal to:

1. A.$\frac{3\pi^2}{4}$
2. B.$\frac{3\pi}{4}$
3. C.$\frac{3\pi}{8}$
4. D.$\frac{3\pi^2}{8}$✓

Correct answer: D

Let $A=\begin{bmatrix}1&0&0\\0&\alpha&\beta\\0&\beta&\alpha\end{bmatrix}$ and $|2A|^3=2^{21}$ where $\alpha,\beta\in\mathbb{Z}$, then a value of $\alpha$ is:

1. A.3
2. B.17
3. C.5✓
4. D.9

Correct answer: C

Let $PQR$ be a triangle with $R(-1,4,2)$. Suppose $M(2,1,2)$ is the mid point of $PQ$. The distance of the centroid of $\Delta PQR$ from the point of intersection of the lines $\frac{x-2}{0}=\frac y2=\frac{z+3}{-1}$ and $\frac{x-1}{1}=\frac{y+3}{-3}=\frac{z+1}{1}$ is:

1. A.$\sqrt{69}$✓
2. B.69
3. C.9
4. D.$\sqrt{99}$

Correct answer: A

Let $A$ be a square matrix such that $AA^T=I$. Then $\frac12A\left[\left(A+A^T\right)^2+\left(A-A^T\right)^2\right]$ is equal to:

1. A.$A^3+I$
2. B.$A^3+A^T$✓
3. C.$A^2+I$
4. D.$A^2+A^T$

Correct answer: B

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

1. A.$\frac5{11}$✓
2. B.$\frac56$
3. C.$\frac16$
4. D.$\frac6{11}$

Correct answer: A

Let $\left(5,\frac a4\right)$ be the circumcenter of a triangle with vertices $A(a,-2)$, $B(a,6)$ and $C\left(\frac a4,-2\right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha+\beta+\gamma$ is:

1. A.53✓
2. B.60
3. C.30
4. D.62

Correct answer: A

In a $\Delta ABC$, suppose $y=x$ is the equation of the bisector of the angle $B$ and the equation of the side $AC$ is $2x-y=2$. If $2AB=BC$ and the points $A$ and $B$ are respectively $(4,6)$ and $(\alpha,\beta)$, then $\alpha+2\beta$ is equal to:

1. A.45
2. B.39
3. C.42✓
4. D.48

Correct answer: C

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 $\vec a+\alpha\vec b+\beta\vec c=\vec0$, then $\alpha+\beta$ is equal to:

1. A.−30
2. B.−25
3. C.35✓
4. D.30

Correct answer: C

If the value of the integral $\int_{-\pi/2}^{\pi/2}\left(\frac{x^2\cos x}{1+\pi^x}+\frac{1+\sin^2x}{1+e^{\sin x^{2023}}}\right)dx=\frac\pi4(\pi+a)-2$, then the value of $a$ is:

1. A.3✓
2. B.$\frac32$
3. C.$-\frac32$
4. D.2

Correct answer: A

If $z=\frac12-2i$ is such that $|z+1|=\alpha z+\beta(1+i)$, $i=\sqrt{-1}$ and $\alpha,\beta\in\mathbb{R}$, then $\alpha+\beta$ is equal to:

1. A.−1
2. B.3✓
3. C.2
4. D.−4

Correct answer: B

For $x\in\left(-\frac\pi2,\frac\pi2\right)$, if $y(x)=\int\frac{\operatorname{cosec}x+\sin x}{\operatorname{cosec}x\sec x+\tan x\sin^2x}dx$, and $\lim_{x\to\left(\frac\pi2\right)^-}y(x)=0$ then $y\left(\frac\pi4\right)$ is equal to:

1. A.$\frac1{\sqrt2}\tan^{-1}\left(-\frac12\right)$✓
2. B.$-\frac1{\sqrt2}\tan^{-1}\left(\frac1{\sqrt2}\right)$
3. C.$\frac12\tan^{-1}\left(\frac1{\sqrt2}\right)$
4. D.$\tan^{-1}\left(\frac1{\sqrt2}\right)$

Correct answer: A

If $f(x)=\begin{cases}2+2x, & -1\le x<0\\ 1-\frac x3, & 0\le x\le3\end{cases}$; $g(x)=\begin{cases}-x, & -3\le x\le0\\ x, & 0<x\le1\end{cases}$, then range of $(f\circ g)(x)$ is:

1. A.(0, 1]
2. B.[0, 3)
3. C.[0, 1)
4. D.[0, 1]✓

Correct answer: D

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

1. A.0
2. B.$\sqrt\pi$✓
3. C.$\pi$
4. D.$\frac\pi2$

Correct answer: B

Let $O$ be the origin and the position vectors 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 $\angle AOB$ meets the line $AB$ at $C$, then the length of $OC$ is:

1. A.$\frac32\sqrt{31}$
2. B.$\frac32\sqrt{34}$
3. C.$\frac23\sqrt{31}$
4. D.$\frac23\sqrt{34}$✓

Correct answer: D

In an A.P., the sixth term $a_6=2$. If the product $a_1a_4a_5$ is the greatest, then the common difference of the A.P. is equal to:

1. A.$\frac58$
2. B.$\frac23$
3. C.$\frac32$
4. D.$\frac85$✓

Correct answer: D

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:

1. A.7
2. B.4
3. C.6✓
4. D.5

Correct answer: C

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

Correct answer: 13

If the points of intersection of two distinct conics $x^2+y^2=4b$ and $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ lie on the curve $y^2=3x^2$, then $3\sqrt3$ times the area of the rectangle formed by intersection points is ______.

Correct answer: 432

Equations of two diameters of a circle are $2x-3y=5$ and $3x-4y=7$. The line joining the points $\left(-\frac{22}{7},-4\right)$ and $\left(-\frac17,3\right)$ intersects the circle at only one point $P(\alpha,\beta)$. Then $17\beta-\alpha$ is equal to ______.

Correct answer: 2

If the solution curve $y=y(x)$ of the differential equation $(1+y^2)(1+\log_ex)dx+x\,dy=0$, $x>0$ passes through the point $(1,1)$ and $y(e)=\frac{\alpha-\tan\left(\frac32\right)}{\beta+\tan\left(\frac32\right)}$, then $\alpha+2\beta$ is ______.

Correct answer: 3

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 ______.

Correct answer: 6344

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

Correct answer: 65

If $\frac{^{11}C_1}{2}+\frac{^{11}C_2}{3}+\ldots+\frac{^{11}C_9}{10}=\frac nm$ with $\gcd(n,m)=1$, then $n+m$ is equal to ______.

Correct answer: 2041

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

Correct answer: 5

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 ______.

Correct answer: 553

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

Correct answer: 171

The explosive in a Hydrogen bomb is a mixture of ${}_1H^2$, ${}_1H^3$ and ${}_3Li^6$ in some condensed form. The chain reaction is given by: ${}_3Li^6+{}_0n^1\to{}_2He^4+{}_1H^3$ ${}_1H^2+{}_1H^3\to{}_2He^4+{}_0n^1$ During the explosion the energy released is approximately: [Given: M(Li) = 6.01690 amu, M(${}_1H^2$) = 2.01471 amu, M(${}_2He^4$) = 4.00388 amu and 1 amu = 931.5 MeV]

1. A.22.22 MeV✓
2. B.12.64 MeV
3. C.16.48 MeV
4. D.28.12 MeV

Correct answer: A

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:

1. A.$\frac14$
2. B.$\frac11$
3. C.$\frac18$✓
4. D.$\frac81$

Correct answer: C

Match List I with List II: List I: A. $\oint\vec B\cdot d\vec l=\mu_0i_c+\mu_0\epsilon_0\frac{d\phi_E}{dt}$; B. $\oint\vec E\cdot d\vec l=\frac{d\phi_B}{dt}$; C. $\oint\vec E\cdot d\vec A=\frac{Q}{\epsilon_0}$; D. $\oint\vec B\cdot d\vec A=0$ List II: I. Gauss' law for electricity; II. Gauss' law for magnetism; III. Faraday law; IV. Ampere – Maxwell law Choose the correct answer from the options given below:

1. A.A – IV, B – III, C – I, D – II✓
2. B.A – II, B – III, C – I, D – IV
3. C.A – IV, B – I, C – III, D – II
4. D.A – I, B – II, C – III, D – IV

Correct answer: A

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:

1. A.+16 cm
2. B.−16 cm
3. C.−160 cm✓
4. D.+160 cm

Correct answer: C

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

1. A.6
2. B.4✓
3. C.8
4. D.2

Correct answer: B

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

1. A.15 cm
2. B.−45 cm
3. C.45 cm
4. D.−15 cm✓

Correct answer: D

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:

1. A.$2:\sqrt3$
2. B.$\sqrt3:2$✓
3. C.$\sqrt3:1$
4. D.$1:\sqrt3$

Correct answer: B

A body starts moving from rest with constant acceleration covers displacement $S_1$ in first $(p-1)$ seconds and $S_2$ in first $p$ seconds. The displacement $S_1+S_2$ will be made in time:

1. A.$(2p^2-2p+1)$ s
2. B.$(2p+1)$ s
3. C.$\sqrt{(2p^2-2p+1)}$ s✓
4. D.$(2p-1)$ s

Correct answer: C

A galvanometer having coil resistance 10 Ω 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:

1. A.$3.75\times10^{-3}\ \Omega$✓
2. B.$2.75\times10^{-3}\ \Omega$
3. C.$4.85\times10^{-3}\ \Omega$
4. D.$3\times10^{-3}\ \Omega$

Correct answer: A

The resistance $R=\frac VI$ where $V=(200\pm5)$ V and $I=(20\pm0.2)$ A, the percentage error in the measurement of R is:

1. A.3%
2. B.5.5%
3. C.3.5%✓
4. D.7%

Correct answer: C

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

1. A.8
2. B.16✓
3. C.32
4. D.4

Correct answer: B

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:

1. A.3900
2. B.4200
3. C.4500
4. D.4000✓

Correct answer: D

The electric current through a wire varies with time as $I=I_0+\beta t$, where $I_0=20$ A and $\beta=3$ A/s. The amount of electric charge crossed through a section of the wire in 20 s is:

1. A.80 C
2. B.800 C
3. C.1000 C✓
4. D.1600 C

Correct answer: C

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

1. A.$\frac{\sqrt3R-R}{2}$
2. B.$\sqrt5R-R$
3. C.$\frac{\sqrt5R-R}{2}$✓
4. D.$\frac R2$

Correct answer: C

Two charges of $5Q$ and $-2Q$ are situated at the points $(3a,0)$ and $(-5a,0)$ respectively. The electric flux through a sphere of radius '4a' having center at origin is:

1. A.$\frac{5Q}{\epsilon_0}$✓
2. B.$\frac{7Q}{\epsilon_0}$
3. C.$\frac{2Q}{\epsilon_0}$
4. D.$\frac{3Q}{\epsilon_0}$

Correct answer: A

In the given circuit, the breakdown voltage of the Zener diode is 3.0 V. What is the value of $I_z$?

1. A.10 mA
2. B.7 mA
3. C.3.3 mA
4. D.5.5 mA✓

Correct answer: D

Given below are two statements: 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.

1. A.Both Statement I and Statement II are false
2. B.Both Statement I and Statement II are true
3. C.Statement I is true but Statement II is false✓
4. D.Statement I is false but Statement II is true

Correct answer: C

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

1. A.12 Ω
2. B.100 Ω
3. C.96 Ω✓
4. D.48 Ω

Correct answer: C

A capacitor of capacitance 100 μF 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:

1. A.1.5 A✓
2. B.1.2 A
3. C.2.0 A
4. D.3.2 A

Correct answer: A

When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is $\frac x8$, where $x=$ ______.

Correct answer: 9

A square loop of side 10 cm and resistance 0.7 Ω 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 $\sqrt x\times10^{-3}$ V. The value of $x$ is ______.

Correct answer: 2

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 m/s. The maximum value of $\sigma$ if the electron strikes S at $t=1$ s is $\alpha\left[\frac{m\epsilon_0}{e}\right]\frac{C}{m^2}$, the value of $\alpha$ is ______.

Correct answer: 8

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 $\sqrt x$ m s$^{-1}$ where $x=$ ______ [use $g=10$ m/s²].

Correct answer: 2

A cylinder is rolling down on an inclined plane of inclination 60°. It's acceleration during rolling down will be $\frac{x}{\sqrt3}$ m/s², where $x=$ ______ (use $g=10$ m/s²).

Correct answer: 10

When a hydrogen atoms going from $n=2$ to $n=1$ emits a photon, its recoil speed is $\frac x5$ m/s. Where $x=$ ______. (Use, mass of hydrogen atom $=1.6\times10^{-27}$ kg)

Correct answer: 17

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 $1.5\times10^{-5}$ T m. The magnetic moment of the dipole is ______ A m². (Given : $\frac{\mu_0}{4\pi}=10^{-7}$ T m A$^{-1}$)

Correct answer: 6

In a double slit experiment shown in figure, when light of wavelength 400 nm is used, dark fringe is observed at P. If D = 0.2 m, the minimum distance between the slits $S_1$ and $S_2$ is ______ mm.

Correct answer: 0

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 m/s and 65 m/s respectively. If the wing area is 2 m², the lift of the wing is ______ N. (Given density of air = 1.2 kg m$^{-3}$)

Correct answer: 810

A 16 Ω wire is bend to form a square loop. A 9 V battery with internal resistance 1 Ω is connected across one of its sides. If a 4 μF capacitor is connected across one of its diagonals, the energy stored by the capacitor will be $\frac x2$ μJ, where $x=$ ______.

Correct answer: 81

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

1. A.5, 1, 1, $+\frac12$
2. B.5, 0, 1, $+\frac12$
3. C.5, 1, 0, $+\frac12$
4. D.5, 0, 0, $+\frac12$✓

Correct answer: D

In which one of the following metal carbonyls, CO forms a bridge between metal atoms?

1. A.$[\text{Os}_3(\text{CO})_{12}]$
2. B.$[\text{Ru}_3(\text{CO})_{12}]$
3. C.$[\text{Mn}_2(\text{CO})_{10}]$
4. D.$[\text{Co}_2(\text{CO})_8]$✓

Correct answer: D

Which of the following is not correct?

1. A.$\sin\theta = -\frac{1}{5}$
2. B.$\cos\theta = 1$
3. C.$\sec\theta = \frac{1}{2}$✓
4. D.$\tan\theta = 20$

Correct answer: C

The arenium ion which is not involved in the bromination of Aniline is ______.

1. A.Iminium-form ion ($=\overset{+}{\text{N}}\text{H}_2$) with H and Br on the ortho carbon
2. B.Ion with $\text{NH}_2$, positive charge on C-2 and H, Br on C-3 (meta attack)✓
3. C.Iminium-form ion ($=\overset{+}{\text{N}}\text{H}_2$) with H and Br on the para carbon
4. D.Ion with $\text{NH}_2$, H and Br on the ortho carbon and positive charge on the ring

Correct answer: B

Chlorine undergoes disproportionation in alkaline medium as shown below: $a\text{Cl}_{2(g)}+b\text{OH}^-_{(aq)}\to c\text{ClO}^-_{(aq)}+d\text{Cl}^-_{(aq)}+e\text{H}_2\text{O}_{(l)}$ The values of a, b, c and d in a balanced redox reaction are respectively:

1. A.2, 2, 1 and 3
2. B.1, 2, 1 and 1✓
3. C.2, 4, 1 and 3
4. D.3, 4, 4 and 2

Correct answer: B

In alkaline medium, $\text{MnO}_4^-$ oxidises $\text{I}^-$ to:

1. A.$\text{IO}_3^-$✓
2. B.$\text{IO}^-$
3. C.$\text{IO}_4^-$
4. D.$\text{I}_2$

Correct answer: A

Appearance of blood red colour, on treatment of the sodium fusion extract of an organic compound with $\text{FeSO}_4$ in presence of concentrated $\text{H}_2\text{SO}_4$ indicates the presence of element/s