JEE Main 2024 — 31 Jan Shift 1 (Previous Year Paper)

Let $g(x)$ be a linear function and $f(x)=\begin{cases}g(x), & x\le0\\ \left(\frac{1+x}{2+x}\right)^{\frac1x}, & x>0\end{cases}$, is continuous at $x=0$. If $f'(1)=f(-1)$, then the value $g(3)$ is:

1. A.$\frac13\log_e\left(\frac{4}{9e^{1/3}}\right)$
2. B.$\log_e\left(\frac49\right)-1$
3. C.$\log_e\left(\frac{4}{9e^{1/3}}\right)$✓
4. D.$\frac13\log_e\left(\frac49\right)+1$

Correct answer: C

Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is:

1. A.$\frac4{25}$
2. B.$\frac2{25}$
3. C.$\frac4{75}$✓
4. D.$\frac23$

Correct answer: C

Let $\alpha,\beta,\gamma,\delta\in\mathbb{Z}$ and let $A(\alpha,\beta)$, $B(1,0)$, $C(\gamma,\delta)$ and $D(1,2)$ be the vertices of a parallelogram ABCD. If $AB=\sqrt{10}$ and the points A and C lie on the line $3y=2x+1$, then $2(\alpha+\beta+\gamma+\delta)$ is equal to:

1. A.10
2. B.5
3. C.12
4. D.8✓

Correct answer: D

Let $S$ be the set of positive integral values of $a$ for which $\frac{ax^2+2(a+1)x+9a+4}{x^2-8x+32}<0$, $\forall x\in\mathbb{R}$. Then, the number of elements in $S$ is:

1. A.3
2. B.0✓
3. C.1
4. D.∞

Correct answer: B

For $0<c<b<a$, let $(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$ and $\alpha\ne1$ be one of its root. Then, among the two statements: (I) If $\alpha\in(-1,0)$, then $b$ cannot be the geometric mean of $a$ and $c$ (II) If $\alpha\in(0,1)$, then $b$ may be the geometric mean of $a$ and $c$

1. A.Only (II) is true
2. B.Both (I) and (II) are true✓
3. C.Only (I) is true
4. D.Neither (I) nor (II) is true

Correct answer: B

For $\alpha,\beta,\gamma\ne0$, if $\sin^{-1}\alpha+\sin^{-1}\beta+\sin^{-1}\gamma=\pi$ and $(\alpha+\beta+\gamma)(\alpha-\gamma+\beta)=3\alpha\beta$, then $\gamma$ equals:

1. A.$\sqrt3$
2. B.$\frac{\sqrt3}{2}$✓
3. C.$\frac1{\sqrt2}$
4. D.$\frac{\sqrt3-1}{2\sqrt2}$

Correct answer: B

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=\frac{(\tan x)+y}{\sin x(\sec x-\sin x\tan x)}$, $x\in\left(0,\frac\pi2\right)$ satisfying the condition $y\left(\frac\pi4\right)=2$. Then, $y\left(\frac\pi3\right)$ is:

1. A.$\frac{\sqrt3}{2}(2+\log_e3)$
2. B.$\sqrt3(2+\log_e\sqrt3)$✓
3. C.$\sqrt3(1+2\log_e3)$
4. D.$\sqrt3(2+\log_e3)$

Correct answer: B

The area of the region $\left\{(x,y):y^2\le4x,\ x<4,\ \frac{xy(x-1)(x-2)}{(x-3)(x-4)}>0,\ x\ne3\right\}$ is:

1. A.$\frac83$
2. B.$\frac{64}{3}$
3. C.$\frac{32}{3}$✓
4. D.$\frac{16}{3}$

Correct answer: C

Let a be the sum of all coefficients in the expansion of $(1-2x+2x^2)^{2023}(3-4x^2+2x^3)^{2024}$ and $b=\lim_{x\to0}\left(\frac{\int_0^x\frac{\log(1+t)}{t^{2024}+1}dt}{x^2}\right)$. If the equations $cx^2+dx+e=0$ and $2bx^2+ax+4=0$ have a common root, where $c,d,e\in\mathbb{R}$, then $d:c:e$ equals:

1. A.4 : 1 : 4
2. B.1 : 1 : 4✓
3. C.1 : 2 : 4
4. D.2 : 1 : 4

Correct answer: B

Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable $x$ to be the number of rotten apples in a draw of two apples, the variance of $x$ is:

1. A.$\frac{40}{153}$✓
2. B.$\frac{37}{153}$
3. C.$\frac{47}{153}$
4. D.$\frac{57}{153}$

Correct answer: A

If one of the diameters of the circle $x^2+y^2-10x+4y+13=0$ is a chord of another circle C, whose centre is the point of intersection of the lines $2x+3y=12$ and $3x-2y=5$, then the radius of the circle C is:

1. A.6✓
2. B.$3\sqrt2$
3. C.$\sqrt{20}$
4. D.4

Correct answer: A

Let $\vec a=3\hat i+\hat j-2\hat k$, $\vec b=4\hat i+\hat j+7\hat k$ and $\vec c=\hat i-3\hat j+4\hat k$ be three vectors. If a vectors $\vec p$ satisfies $\vec p\times\vec b=\vec c\times\vec b$ and $\vec p\cdot\vec a=0$, then $\vec p\cdot(\hat i-\hat j-\hat k)$ is equal to:

1. A.28
2. B.24
3. C.32✓
4. D.36

Correct answer: C

The sum of the series $\frac{1}{1-3\cdot1^2+1^4}+\frac{2}{1-3\cdot2^2+2^4}+\frac{3}{1-3\cdot3^2+3^4}+\ldots$ up to 10 terms is:

1. A.$\frac{55}{109}$
2. B.$-\frac{45}{109}$
3. C.$-\frac{55}{109}$✓
4. D.$\frac{45}{109}$

Correct answer: C

The solution curve of the differential equation $y\frac{dx}{dy}=x(\log_ex-\log_ey+1)$, $x>0,y>0$ passing through the point $(e,1)$ is:

1. A.$2\left|\log_e\frac xy\right|=y+1$
2. B.$\left|\log_e\frac yx\right|=x$
3. C.$\left|\log_e\frac xy\right|=y$✓
4. D.$\left|\log_e\frac yx\right|=y^2$

Correct answer: C

$\lim_{x\to0}\frac{e^{2|\sin x|}-2|\sin x|-1}{x^2}$

1. A.is equal to −1
2. B.is equal to 1
3. C.does not exist
4. D.is equal to 2✓

Correct answer: D

If the system of linear equations $x-2y+z=-4$ $2x+\alpha y+3z=5$ $3x-y+\beta z=3$ has infinitely many solutions, then $12\alpha+13\beta$ is equal to:

1. A.60
2. B.54
3. C.64
4. D.58✓

Correct answer: D

If $f(x)=\frac{4x+3}{6x-4}$, $x\ne\frac23$ and $(f\circ f)(x)=g(x)$, where $g:\mathbb{R}-\left\{\frac23\right\}\to\mathbb{R}-\left\{\frac23\right\}$, $(g\circ g\circ g)(4)$ is equal to:

1. A.$-\frac{19}{20}$
2. B.4✓
3. C.−4
4. D.$\frac{19}{20}$

Correct answer: B

if $f(x)=\begin{vmatrix}x^3&2x^2+1&1+3x\\3x^2+2&2x&x^3+6\\x^3-x&4&x^2-2\end{vmatrix}$ for all $x\in\mathbb{R}$, then $2f(0)+f'(0)$ is equal to:

1. A.48
2. B.18
3. C.42✓
4. D.24

Correct answer: C

The distance of the point Q(0, 2, −2) from the line passing through the point P(5, −4, 3) and perpendicular to the lines $\vec r=(-3\hat i+2\hat k)+\lambda(2\hat i+3\hat j+5\hat k)$, $\lambda\in\mathbb{R}$ and $\vec r=(\hat i-2\hat j+\hat k)+\mu(-\hat i+3\hat j+2\hat k)$, $\mu\in\mathbb{R}$ is:

1. A.$\sqrt{54}$
2. B.$\sqrt{86}$
3. C.$\sqrt{74}$✓
4. D.$\sqrt{20}$

Correct answer: C

If the foci of a hyperbola are same as that of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ and eccentricity of the hyperbola is $\frac{15}{8}$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt2,\frac{14}{3}\sqrt{\frac25}\right)$ on the hyperbola, is equal to:

1. A.$14\sqrt{\frac25}-\frac43$
2. B.$14\sqrt{\frac25}-\frac{16}{3}$
3. C.$7\sqrt{\frac25}-\frac83$✓
4. D.$7\sqrt{\frac25}+\frac83$

Correct answer: C

If the integral $525\int_0^{\pi/2}\sin2x\cos^{\frac{11}{2}}x\left(1+\cos^{\frac52}x\right)^{\frac12}dx$ is equal to $(n\sqrt2-64)$, then $n$ is equal to ______.

Correct answer: 176

Let $\vec a$ and $\vec b$ be two vectors such that $|\vec a|=1$, $|\vec b|=4$, and $\vec a\cdot\vec b=2$. If $\vec c=(2\vec a\times\vec b)-3\vec b$ and the angle between $\vec b$ and $\vec c$ is $\alpha$, then $192\sin^2\alpha$ is equal to ______.

Correct answer: 48

Let $S=(-1,\infty)$ and $f:S\to\mathbb{R}$ be defined as $f(x)=\int_{-1}^{x}(e^t-1)^{11}(2t-1)^5(t-3)^{12}(2t-10)^{61}dt$. Let $p$ = Sum of squares of the values of $x$, where $f(x)$ attains local maxima on $S$, and $q$ = Sum of the values of $x$, where $f(x)$ attains local minima on $S$. Then, the value of $p^2+2q$ is ______.

Correct answer: 27

The total number of words (with or without meaning) that can be formed out of the letters of the word 'DISTRIBUTION' taken four at a time, is equal to ______.

Correct answer: 3734

Let the foci and length of the latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $a>b$ be $(\pm5,0)$ and $\sqrt{50}$, respectively. Then, the square of the eccentricity of the hyperbola $\frac{x^2}{b^2}-\frac{y^2}{a^2b^2}=1$ equals:

Correct answer: 51

Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x)=\frac{4^x}{4^x+2}$ and $M=\int_{f(a)}^{f(1-a)}x\sin^4(x(1-x))dx$, $N=\int_{f(a)}^{f(1-a)}\sin^4(x(1-x))dx$; $a\ne\frac12$. If $\alpha M=\beta N$, $\alpha,\beta\in\mathbb{N}$, then the least value of $\alpha^2+\beta^2$ is equal to ______.

Correct answer: 5

In the expansion of $(1+x)(1-x^2)\left(1+\frac3x+\frac3{x^2}+\frac1{x^3}\right)^5$, $x\ne0$, the sum of the coefficients of $x^3$ and $x^{-13}$ is equal to ______.

Correct answer: 118

Let $A=\{1,2,3,4\}$ and $R=\{(1,2),(2,3),(1,4)\}$ be a relation on A. Let S be the equivalence relation on A such that $R\subset S$ and the number of elements in S is n. Then, the minimum value of $n$ is:

Correct answer: 16

Let Q and R be the feet of perpendiculars from the point $P(a,a,a)$ on the lines $x=y,z=1$ and $x=-y,z=-1$ respectively. If $\angle QPR$ is a right angle, then $12a^2$ is equal to:

Correct answer: 12

If $\alpha$ denotes the number of solutions of $|1-i|^x=2^x$ and $\beta=\left(\frac{|z|}{\arg(z)}\right)$, where $z=\frac\pi4(1+i)^4\left[\frac{1-\sqrt\pi i}{\sqrt\pi+i}+\frac{\sqrt\pi-i}{1+\sqrt\pi i}\right]$, $i=\sqrt{-1}$, then the distance of the point $(\alpha,\beta)$ from the line $4x-3y=7$ is ______.

Correct answer: 3

In a plane EM wave, the electric field oscillates sinusoidally at a frequency of $5\times10^{10}$ Hz and an amplitude of 50 V m$^{-1}$. The total average energy density of the electromagnetic field of the wave is: [Use $\epsilon_0=8.85\times10^{-12}$ C²/N m²]

1. A.$4.425\times10^{-8}$ J m$^{-3}$
2. B.$2.212\times10^{-8}$ J m$^{-3}$
3. C.$1.106\times10^{-8}$ J m$^{-3}$✓
4. D.$2.212\times10^{-10}$ J m$^{-3}$

Correct answer: C

The parameter that remains the same for molecules of all gases at a given temperature is:

1. A.kinetic energy✓
2. B.momentum
3. C.mass
4. D.speed

Correct answer: A

The given figure represents two isobaric processes for the same mass of an ideal gas, then:

1. A.$P_1=P_2$
2. B.$P_2>P_1$
3. C.$P_2\ge P_1$
4. D.$P_1>P_2$✓

Correct answer: D

The refractive index of a prism with apex angle A is $\cot A/2$. The angle of minimum deviation is:

1. A.$\delta_m=180^\circ-2A$✓
2. B.$\delta_m=180^\circ-A$
3. C.$\delta_m=180^\circ-3A$
4. D.$\delta_m=180^\circ-4A$

Correct answer: A

The fundamental frequency of a closed organ pipe is equal to the first overtone frequency of an open organ pipe. If length of the open pipe is 60 cm, the length of the closed pipe will be:

1. A.45 cm
2. B.15 cm✓
3. C.60 cm
4. D.30 cm

Correct answer: B

A coil is placed perpendicular to a magnetic field of 5000 T. When the field is changed to 3000 T in 2s, an induced emf of 22 V is produced in the coil. If the diameter of the coil is 0.02 m, then the number of turns is the coil is:

1. A.7
2. B.35
3. C.70✓
4. D.140

Correct answer: C

A rigid wire consists of a semicircular position of radius R and two straight sections. The wire is partially immerged in a perpendicular magnetic field $B=B_0\hat k$ as shown in figure. The magnetic force on the wire if it has current $i$ is:

1. A.$2iBR\hat j$
2. B.$-iBR\hat j$
3. C.$iBR\hat j$
4. D.$-2iBR\hat j$✓

Correct answer: D

When a metal surface is illuminated by light of wavelength λ, the stopping potential is 8V. When the same surface is illuminated by light of wavelength 3λ, stopping potential is 2V. The threshold wavelength for this surface is:

1. A.9λ✓
2. B.3λ
3. C.5λ
4. D.4.5λ

Correct answer: A

Four identical particles of mass m are kept at the four corners of a square. If the gravitational force exerted on one of the masses by the other masses is $\left(\frac{2\sqrt2+1}{32}\right)\frac{Gm^2}{L^2}$, the length of the sides of the square is:

1. A.2L
2. B.4L✓
3. C.3L
4. D.$\frac L2$

Correct answer: B

Identify the logic operation performed by the given circuit.

1. A.AND
2. B.NAND
3. C.OR✓
4. D.NOR

Correct answer: C

If the percentage errors in measuring the length and the diameter of a wire are 0.1% each. The percentage error in measuring its resistance will be:

1. A.0.2%
2. B.0.1%
3. C.0.144%
4. D.0.3%✓

Correct answer: D

A small steel ball is dropped into a long cylinder containing glycerine. Which one of the following is the correct representation of the velocity time graph for the transit of the ball?

1. A.v rises linearly then falls linearly (triangle)
2. B.v increases linearly with t (straight line)
3. C.v rises and levels off asymptotically (concave curve approaching terminal velocity)✓
4. D.v constant then decreasing

Correct answer: C

A coin is placed on a disc. The coefficient of friction between the coin and the disc is μ. If the distance of the coin from the center of the disc is r, the maximum angular velocity which can be given to the disc, so that the coin does not slip away, is:

1. A.$\sqrt{\frac{r}{\mu g}}$
2. B.$\frac{\mu}{\sqrt{rg}}$
3. C.$\frac{\mu g}{r}$
4. D.$\sqrt{\frac{\mu g}{r}}$✓

Correct answer: D

A force is represented by $F=ax^2+bt^{\frac12}$ where $x$ = distance and $t$ = time. The dimensions of $b^2/a$ are:

1. A.$[MLT^{-2}]$
2. B.$[ML^3T^{-3}]$✓
3. C.$[ML^{-1}T^{-1}]$
4. D.$[ML^2T^{-3}]$

Correct answer: B

Two conductors have the same resistance at 0°C but their temperature coefficients of resistance are $\alpha_1$ and $\alpha_2$. The respective temperature coefficients for their series and parallel combinations are:

1. A.$\alpha_1+\alpha_2,\ \frac{\alpha_1\alpha_2}{\alpha_1+\alpha_2}$
2. B.$\frac{\alpha_1+\alpha_2}{2},\ \frac{\alpha_1+\alpha_2}{2}$✓
3. C.$\frac{\alpha_1+\alpha_2}{2},\ \alpha_1+\alpha_2$
4. D.$\alpha_1+\alpha_2,\ \frac{\alpha_1+\alpha_2}{2}$

Correct answer: B

Two charges q and 3q are separated by a distance 'r' in air. At a distance x from charge q, the resultant electric field is zero. The value of x is:

1. A.$\frac{r}{3(1+\sqrt3)}$
2. B.$\frac{r}{(1+\sqrt3)}$✓
3. C.$\frac{(1+\sqrt3)}{r}$
4. D.$r(1+\sqrt3)$

Correct answer: B

The relation between time 't' and distance 'x' is $t=\alpha x^2+\beta x$, where α and β are constants. The relation between acceleration (a) and velocity (v) is:

1. A.$a=-5\alpha v^5$
2. B.$a=-4\alpha v^4$
3. C.$a=-3\alpha v^2$
4. D.$a=-2\alpha v^3$✓

Correct answer: D

If the wavelength of the first member of Lyman series of hydrogen is λ. The wavelength of the second member will be:

1. A.$\frac5{27}\lambda$
2. B.$\frac{32}{27}\lambda$
3. C.$\frac{27}{32}\lambda$✓
4. D.$\frac{27}{5}\lambda$

Correct answer: C

In the given arrangement of a doubly inclined plane two blocks of masses M and m are placed. The blocks are connected by a light string passing over an ideal pulley as shown. The coefficient of friction between the surface of the plane and the blocks is 0.25. The value of m, for which M = 10 kg will move down with an acceleration of 2 m/s², is: (take $g=10$ m/s² and $\tan37^\circ=3/4$)

1. A.9 kg
2. B.2.25 kg
3. C.6.5 kg
4. D.4.5 kg✓

Correct answer: D

An artillery piece of mass $M_1$ fires a shell of mass $M_2$ horizontally. Instantaneously after the firing, the ratio of kinetic energy of the artillery and that of the shell is:

1. A.$M_2/(M_1+M_2)$
2. B.$M_1/(M_1+M_2)$
3. C.$\frac{M_2}{M_1}$✓
4. D.$\frac{M_1}{M_2}$

Correct answer: C

Equivalent resistance of the following network is ______ Ω.

Correct answer: 1

Two waves of intensity ratio 1:9 cross each other at a point. The resultant intensites at that point, when (a) Waves are incoherent is $I_1$ (b) Waves are coherent is $I_2$ and differ in phase by 60°. If $\frac{I_1}{I_2}=\frac{10}{x}$ then $x=$ ______.

Correct answer: 13

The depth below the surface of sea to which a rubber ball be taken so as to decrease its volume by 0.02% is ______ m. (Take density of sea water = $10^3$ kg m$^{-3}$, Bulk modulus of rubber = $9\times10^8$ N m$^{-2}$, and $g=10$ m s$^{-2}$)

Correct answer: 18

A particle performs simple harmonic motion with amplitude A. Its speed is increased to three times at an instant when its displacement is $\frac{2A}{3}$. The new amplitude of motion is $\frac{nA}{3}$. The value of n is ______.

Correct answer: 7

A small square loop of wire of side l is placed inside a large square loop of wire of side $L(L=l^2)$. The loops are coplanar and their centers coincide. The value of the mutual inductance of the system is $\sqrt x\times10^{-7}$ H, where $x=$ ______.

Correct answer: 128

A body starts falling freely from height H hits an inclined plane in its path at height h. As a result of this perfectly elastic impact, the direction of the velocity of the body becomes horizontal. The value of $\frac Hh$ for which the body will take the maximum time to reach the ground is ______.

Correct answer: 2

The mass defect in a particular reaction is 0.4g. The amount of energy liberated is $n\times10^7$ kWh, where $n=$ ______. (speed of light = $3\times10^8$ m/s)

Correct answer: 1

A parallel plate capacitor with plate separation 5 mm is charged up by a battery. It is found that on introducing a dielectric sheet of thickness 2 mm, while keeping the battery connections intact, the capacitor draws 25% more charge from the battery than before. The dielectric constant of the sheet is ______.

Correct answer: 2

A solid circular disc of mass 50 kg rolls along a horizontal floor so that its center of mass has a speed of 0.4 m/s. The absolute value of work done on the disc to stop it is:

Correct answer: 6

An electron moves through a uniform magnetic field $\vec B=B_0\hat i+2B_0\hat j$ T. At a particular instant of time, the velocity of electron is $\vec u=3\hat i+5\hat j$ m/s. If the magnetic force acting on electron is $\vec F=5e\hat k$ N, where e is the charge of electron, then the value of $B_0$ is ______ T.

Correct answer: 5

The correct statements from following are: A. The strength of anionic ligands can be explained by crystal field theory. B. Valence bond theory does not give a quantitative interpretation of kinetic stability of coordination compounds. C. The hybridization involved in formation of $[\text{Ni(CN)}_4]^{2-}$ complex is $dsp^2$. D. The number of possible isomer(s) of cis-$[\text{PtCl}_2(\text{en})_2]^{2+}$ is one Choose the correct answer from the options given below:

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

Correct answer: A

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R: Assertion A: Alcohols react both as nucleophiles and electrophiles. Reason R: Alcohols reacts with active metals such as sodium, potassium and aluminium to yield corresponding alkoxides and liberate hydrogen. In the light of the above statements, choose the correct answer from the options given below:

1. A.A is false but R is true.
2. B.Both A and R are true but R is NOT the correct explanation of A.✓
3. C.Both A and R are true and R is the correct explanation of A.
4. D.A is true but R is false.

Correct answer: B

Given below are two statements: Statement I: IUPAC name of $\text{HO-CH}_2\text{-(CH}_2)_3\text{-CH}_2\text{-COCH}_3$ is 7-hydroxyheptan-2-one. Statement II: 2-oxoheptan-7-ol is the correct IUPAC name for above compound. In the light of the above statements, choose the most appropriate answer from the options given below: