JEE Main 2024 — 27 Jan Shift 1 (Previous Year Paper)

Consider the function $f(x)=\begin{cases}\dfrac{a(7x-12-x^2)}{b\,|x^2-7x+12|}, & x<3\\[4pt] 2^{\frac{\sin(x-3)}{x-[x]}}, & x>3\\[4pt] b, & x=3\end{cases}$ where $[x]$ denotes the greatest integer less than or equal to $x$. If $S$ denotes the set of all ordered pairs $(a,b)$ such that $f(x)$ is continuous at $x=3$, then the number of elements in $S$ is :

1. A.2
2. B.Infinitely many
3. C.4
4. D.1✓

Correct answer: D

The distance, of the point $(7,-2,11)$ from the line $\frac{x-6}{1}=\frac{y-4}{0}=\frac{z-8}{3}$ along the line $\frac{x-5}{2}=\frac{y-1}{-3}=\frac{z-5}{6}$, is:

1. A.12
2. B.14✓
3. C.21
4. D.18

Correct answer: B

If the shortest distance of the parabola $y^2=4x$ from the centre of the circle $x^2+y^2-4x-16y+64=0$ is $d$, then $d^2$ is equal to :

1. A.36
2. B.20✓
3. C.24
4. D.16

Correct answer: B

If $(a,b)$ be the orthocentre of the triangle whose vertices are $(1,2)$, $(2,3)$ and $(3,1)$, and $I_1=\int_a^b x\sin(4x-x^2)\,dx$, $I_2=\int_a^b \sin(4x-x^2)\,dx$, then $36\frac{I_1}{I_2}$ is equal to:

1. A.88
2. B.72✓
3. C.66
4. D.80

Correct answer: B

The portion of the line $4x+5y=20$ in the first quadrant is trisected by the lines $L_1$ and $L_2$ passing through the origin. The tangent of an angle between the lines $L_1$ and $L_2$ is :

1. A.$\frac{8}{5}$
2. B.$\frac{2}{5}$
3. C.$\frac{25}{41}$
4. D.$\frac{30}{41}$✓

Correct answer: D

Let $a_1,a_2,\ldots,a_{10}$ be 10 observations such that $\sum_{k=1}^{10}a_k=50$ and $\sum_{\forall k<j}a_k\cdot a_j=1100$. Then the standard deviation of $a_1,a_2,\ldots,a_{10}$ is equal to:

1. A.5
2. B.10
3. C.$\sqrt{5}$✓
4. D.$\sqrt{115}$

Correct answer: C

${}^{n-1}C_r=(k^2-8)\,{}^{n}C_{r+1}$ if and only if:

1. A.$2\sqrt{3}<k<3\sqrt{3}$
2. B.$2\sqrt{3}<k\le3\sqrt{2}$
3. C.$2\sqrt{2}<k<2\sqrt{3}$
4. D.$2\sqrt{2}<k\le3$✓

Correct answer: D

The number of common terms in the progressions $4,9,14,19,\ldots$ up to $25^{th}$ term and $3,6,9,12,\ldots$ up to $37^{th}$ term is:

1. A.9
2. B.7✓
3. C.5
4. D.8

Correct answer: B

The length of the chord of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, whose mid-point is $\left(1,\frac{2}{5}\right)$, is equal to:

1. A.$\frac{\sqrt{1741}}{5}$
2. B.$\frac{\sqrt{2009}}{5}$
3. C.$\frac{\sqrt{1691}}{5}$✓
4. D.$\frac{\sqrt{1541}}{5}$

Correct answer: C

If the shortest distance between the lines $\frac{x-4}{1}=\frac{y+1}{2}=\frac{z}{-3}$ and $\frac{x-\lambda}{2}=\frac{y+1}{4}=\frac{z-2}{-5}$ is $\frac{6}{\sqrt5}$, then the sum of all possible values of $\lambda$ is:

1. A.7
2. B.5
3. C.8✓
4. D.10

Correct answer: C

Let $S=\{1,2,3,\ldots,10\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $R=\{(A,B):A\cap B\ne\phi;\ A,B\in M\}$ is:

1. A.symmetric and reflexive only
2. B.symmetric only✓
3. C.reflexive only
4. D.symmetric and transitive only

Correct answer: B

If $a=\lim_{x\to0}\frac{\sqrt{1+\sqrt{1+x^4}}-\sqrt2}{x^4}$ and $b=\lim_{x\to0}\frac{\sin^2x}{\sqrt2-\sqrt{1+\cos x}}$, then the value of $ab^3$ is:

1. A.32✓
2. B.36
3. C.25
4. D.30

Correct answer: A

If $\int_0^1\frac{1}{\sqrt{3+x}+\sqrt{1+x}}\,dx=a+b\sqrt2+c\sqrt3$, where $a,b,c$ are rational numbers, then $2a+3b-4c$ is equal to:

1. A.4
2. B.8✓
3. C.7
4. D.10

Correct answer: B

Consider the matrix $f(x)=\begin{bmatrix}\cos x&-\sin x&0\\\sin x&\cos x&0\\0&0&1\end{bmatrix}$. Given below are two statements: Statement I: $f(-x)$ is the inverse of the matrix $f(x)$. Statement II: $f(x)f(y)=f(x+y)$. In the light of the above statements, choose the correct answer from the options given below:

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

Correct answer: A

The function $f:\mathbb{N}-\{1\}\to\mathbb{N}$; defined by $f(n)=$ the highest prime factor of $n$, is :

1. A.both one-one and onto
2. B.onto only
3. C.neither one-one nor onto✓
4. D.one-one only

Correct answer: C

Let $\vec a=\hat i+2\hat j+\hat k$, $\vec b=3(\hat i-\hat j+\hat k)$. Let $\vec c$ be the vector such that $\vec a\times\vec c=\vec b$ and $\vec a\cdot\vec c=3$. Then $\vec a\cdot\left((\vec c\times\vec b)-\vec b-\vec c\right)$ is equal to:

1. A.24✓
2. B.36
3. C.32
4. D.20

Correct answer: A

If A denotes the sum of all the coefficients in the expansion of $(1-3x+10x^2)^n$ and B denotes the sum of all the coefficients in the expansion of $(1+x^2)^n$, then:

1. A.$A=B^3$✓
2. B.$A=3B$
3. C.$B=A^3$
4. D.$3A=B$

Correct answer: A

Four distinct points $(2k,3k)$, $(1,0)$, $(0,1)$ and $(0,0)$ lie on a circle for $k$ equal to :

1. A.$\frac{3}{13}$
2. B.$\frac{5}{13}$✓
3. C.$\frac{1}{13}$
4. D.$\frac{2}{13}$

Correct answer: B

Let $x=x(t)$ and $y=y(t)$ be solutions of the differential equations $\frac{dx}{dt}+ax=0$ and $\frac{dy}{dt}+by=0$ respectively, $a,b\in\mathbb{R}$. Given that $x(0)=2$; $y(0)=1$ and $3y(1)=2x(1)$. The value of $t$, for which $x(t)=y(t)$, is:

1. A.$\log_{4/3}2$✓
2. B.$\log_4 3$
3. C.$\log_{2/3}2$
4. D.$\log_3 4$

Correct answer: A

If $S=\{z\in\mathbb{C}:|z-i|=|z+i|=|z-1|\}$, then, $n(S)$ is:

1. A.3
2. B.0
3. C.1✓
4. D.2

Correct answer: C

If $8=3+\frac14(3+p)+\frac{1}{4^2}(3+2p)+\frac{1}{4^3}(3+3p)+\ldots\infty$, then the value of $p$ is ______.

Correct answer: 9

If $\alpha$ satisfies the equation $x^2+x+1=0$ and $(1+\alpha)^7=A+B\alpha+C\alpha^2$, $A,B,C\ge0$, then $5(3A-2B-C)$ is equal to ______.

Correct answer: 5

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required and let $a=P(X=3)$, $b=P(X\ge3)$ and $c=P(X\ge6\mid X>3)$. Then $\frac{b+c}{a}$ is equal to ______.

Correct answer: 12

If the solution of the differential equation $(2x+3y-2)dx+(4x+6y-7)dy=0$, $y(0)=3$, is $\alpha x+\beta y+3\log_e|2x+3y-\gamma|=6$, then $\alpha+2\beta+3\gamma$ is equal to ______.

Correct answer: 29

Let for a differentiable function $f:(0,\infty)\to\mathbb{R}$, $f(x)-f(y)\ge\log_e\left(\frac{x}{y}\right)+x-y$, $\forall\,x,y\in(0,\infty)$. Then $\sum_{n=1}^{20}f'\left(\frac{1}{n^2}\right)$ is equal to ______.

Correct answer: 2890

The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha\hat i-2\hat j+2\hat k$ and $\alpha\hat i+2\alpha\hat j-2\hat k$ is acute, is ______.

Correct answer: 5

Let the area of the region $\{(x,y):x-2y+4\ge0,\ x+2y^2\ge0,\ x+4y^2\le8,\ y\ge0\}$ be $\frac{m}{n}$, where $m$ and $n$ are coprime numbers. Then $m+n$ is equal to ______.

Correct answer: 119

Let $f(x)=x^3+x^2f'(1)+xf''(2)+f'''(3)$, $x\in\mathbb{R}$. Then $f'(10)$ is equal to ______.

Correct answer: 202

Let $A=\begin{bmatrix}2&0&1\\1&1&0\\1&0&1\end{bmatrix}$, $B=[B_1,B_2,B_3]$, where $B_1,B_2,B_3$ are column matrices, and $AB_1=\begin{bmatrix}1\\0\\0\end{bmatrix}$, $AB_2=\begin{bmatrix}2\\3\\0\end{bmatrix}$, $AB_3=\begin{bmatrix}3\\2\\1\end{bmatrix}$. If $\alpha=|B|$ and $\beta$ is the sum of all the diagonal elements of $B$, then $\alpha^3+\beta^3$ is equal to ______.

Correct answer: 28

Let the set of all $a\in\mathbb{R}$ such that the equation $\cos2x+a\sin x=2a-7$ has a solution be $[p,q]$ and $r=\tan9^\circ-\tan27^\circ-\frac{1}{\cot63^\circ}+\tan81^\circ$, then $pqr$ is equal to ______.

Correct answer: 48

A plane electromagnetic wave propagating in $x$-direction is described by $E_y=(200\ \text{V m}^{-1})\sin[1.5\times10^7t-0.05x]$; The intensity of the wave is : (Use $\epsilon_0=8.85\times10^{-12}\ \text{C}^2\text{N}^{-1}\text{m}^{-2}$)

1. A.$106.2\ \text{W m}^{-2}$
2. B.$53.1\ \text{W m}^{-2}$✓
3. C.$35.4\ \text{W m}^{-2}$
4. D.$26.6\ \text{W m}^{-2}$

Correct answer: B

The radius of third stationary orbit of electron for Bohr's atom is R. The radius of fourth stationary orbit will be:

1. A.$\frac34R$
2. B.$\frac{9}{16}R$
3. C.$\frac43R$
4. D.$\frac{16}{9}R$✓

Correct answer: D

0.08 kg air is heated at constant volume through 5°C. The specific heat of air at constant volume is 0.17 kcal/kg°C and J = 4.18 joule/cal. The change in its internal energy is approximately.

1. A.298 J
2. B.318 J
3. C.284 J✓
4. D.142 J

Correct answer: C

Position of an ant (S in metres) moving in Y-Z plane is given by $\vec S=2t^2\hat j+5\hat k$ (where t is in second). The magnitude and direction of velocity of the ant at t = 1 s will be :

1. A.16 m/s in y-direction
2. B.4 m/s in y-direction✓
3. C.4 m/s in x-direction
4. D.9 m/s in z-direction

Correct answer: B

Given below are two statements : Statement (I) : Planck's constant and angular momentum have same dimensions. Statement (II) : Linear momentum and moment of force have same dimensions. In the light of the above statements, choose the correct answer from the options given below :

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

Correct answer: D

A body of mass 1000 kg is moving horizontally with a velocity 6 m/s. If 200 kg extra mass is added, the final velocity (in m/s) is :

1. A.3
2. B.5✓
3. C.6
4. D.2

Correct answer: B

Which of the following circuits is reverse - biased ?

1. A.Circuit (1): ground — diode (anode at ground) — resistor — $-5$ V
2. B.Circuit (2): $+2$ V — resistor — diode (anode towards +2 V) — ground
3. C.Circuit (3): $+2$ V — diode (anode at +2 V) — resistor — $+4$ V✓
4. D.Circuit (4): $+2$ V — diode (anode side) — resistor R — $-10$ V

Correct answer: C

A wire of length 10 cm and radius $\sqrt7\times10^{-4}$ m connected across the right gap of a meter bridge. When a resistance of 4.5 Ω is connected on the left gap by using a resistance box, the balance length is found to be at 60 cm from the left end. If the resistivity of the wire is $R\times10^{-7}$ Ω m, then value of R is :

1. A.66✓
2. B.70
3. C.63
4. D.35

Correct answer: A

A convex lens of focal length 40 cm forms an image of an extended source of light on a photo electric cell. A current I is produced. The lens is replaced by another convex lens having the same diameter but focal length 20 cm. The photoelectric current now is :

1. A.$\frac{I}{2}$
2. B.$I$✓
3. C.$2I$
4. D.$4I$

Correct answer: B

Given below are two statements : Statement (I) : Viscosity of gases is greater than that of liquids. Statement (II) : Surface tension of a liquid decreases due to the presence of insoluble impurities. In the light of the above statements, choose the most appropriate answer from the options given below :

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

Correct answer: D

A wire of resistance R and length L is cut into 5 equal parts. If these parts are joined parallely, then resultant resistance will be :

1. A.$\frac15R$
2. B.$\frac{1}{25}R$✓
3. C.$5R$
4. D.$25R$

Correct answer: B

If the refractive index of the material of a prism is $\cot\left(\frac{A}{2}\right)$ where $A$ is the angle of prism then the angle of minimum deviation will be :

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

Correct answer: B

A rectangular loop of length 2.5 m and width 2 m is placed at 60° to a magnetic field of 4 T. The loop is removed from the field in 10 s. The average emf induced in the loop during this time is :

1. A.−2 V
2. B.−1 V
3. C.+2 V
4. D.+1 V✓

Correct answer: D

The acceleration due to gravity on the surface of earth is g. If the diameter of earth reduces to half of its original value and mass remains constant, then acceleration due to gravity on the surface of earth would be :

1. A.$4g$✓
2. B.$2g$
3. C.$\frac g4$
4. D.$\frac g2$

Correct answer: A

An electric charge $10^{-6}\ \mu C$ is placed at origin (0, 0) m of X — Y co-ordinate system. Two points P and Q are situated at $(\sqrt3,\sqrt3)$ m and $(\sqrt6,0)$ m respectively. The potential difference between the points P and Q will be :

1. A.$\sqrt6$ V
2. B.0 V✓
3. C.3 V
4. D.$\sqrt3$ V

Correct answer: B

Two bodies of mass 4 g and 25 g are moving with equal kinetic energies. The ratio of magnitude of their linear momentum is :

1. A.3 : 5
2. B.4 : 5
3. C.5 : 4
4. D.2 : 5✓

Correct answer: D

A train is moving with a speed of 12 m/s on rails which are 1.5 m apart. To negotiate a curve of radius 400 m, the height by which the outer rail should be raised with respect to the inner rail is: (Given, g = 10 m/s²)

1. A.5.4 cm✓
2. B.4.8 cm
3. C.4.2 cm
4. D.6.0 cm

Correct answer: A

A proton moving with a constant velocity passes through a region of space without any change in its velocity. If $\vec E$ and $\vec B$ represent the electric and magnetic fields respectively, then the region of space may have : (A) E = 0, B = 0 (B) E = 0, B ≠ 0 (C) E ≠ 0, B = 0 (D) E ≠ 0, B ≠ 0 Choose the most appropriate answer from the options given below :

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

Correct answer: B

Identify the physical quantity that cannot be measured using spherometer :

1. A.Thickness of thin plates
2. B.Radius of curvature of concave surface
3. C.Radius of curvature of convex surface
4. D.Specific rotation of liquids✓

Correct answer: D

The average kinetic energy of a monoatomic molecule is 0.414 eV at temperature: (Use $K_B=1.38\times10^{-23}$ J/mol-K)

1. A.1500 K
2. B.3000 K
3. C.3200 K✓
4. D.1600 K

Correct answer: C

If average depth of an ocean is 4000 m and the bulk modulus of water is $2\times10^9$ N m$^{-2}$, then fractional compression $\frac{\Delta V}{V}$ of water at the bottom of ocean is $\alpha\times10^{-2}$. The value of $\alpha$ is ______. (Given, $g=10$ m s$^{-2}$, $\rho=1000$ kg m$^{-3}$)

Correct answer: 2

A particle starts from origin at $t=0$ with a velocity $5\hat i$ m/s and moves in x-y plane under action of a force which produces a constant acceleration of $(3\hat i+2\hat j)$ m/s². If the x-coordinate of the particle at that instant is 84 m, then the speed of the particle at this time is $\sqrt\alpha$ m/s. The value of $\alpha$ is ______.

Correct answer: 673

Two long, straight wires carry equal currents in opposite directions as shown in figure. The separation between the wires is 5.0 cm. The magnitude of the magnetic field at a point P midway between the wires is ___ μT. (Given : $\mu_0=4\pi\times10^{-7}$ T m A$^{-1}$)

Correct answer: 160

The charge accumulated on the capacitor connected in the following circuit is ______ μC. (given $C=150\ \mu F$)

Correct answer: 400

Two immiscible liquids of refractive indices $\frac85$ and $\frac32$ respectively are put in a beaker as shown in the figure. The height of each column is 6 cm. A coin is placed at the bottom of the beaker. For near normal vision, the apparent depth of the coin is $\frac{\alpha}{4}$ cm. The value of $\alpha$ is ______.

Correct answer: 31

Two coils have mutual inductance 0.002 H. The current changes in the first coil according to the relation $i=i_0\sin\omega t$, where $i_0=5$ A and $\omega=50\pi$ rad/s. The maximum value of emf in the second coil is $\frac{\pi}{\alpha}$ V. The value of $\alpha$ is ______.

Correct answer: 2

Four particles each of mass 1 kg are placed at four corners of a square of side 2 m. Moment of inertia of system about an axis perpendicular to its plane and passing through one of its vertex is ______ kg m².

Correct answer: 16

A thin metallic wire having cross sectional area of $10^{-4}$ m² is used to make a ring of radius 30 cm. A positive charge of $2\pi$ C is uniformly distributed over the ring, while another positive charge of 30 pC is kept at the centre of the ring. The tension in the ring is ______ N; provided that the ring does not get deformed (neglect the influence of gravity). (given, $\frac{1}{4\pi\epsilon_0}=9\times10^9$ SI units)

Correct answer: 3

A particle executes simple harmonic motion with an amplitude of 4 cm. At the mean position, velocity of the particle is 10 cm/s. The distance of the particle from the mean position when its speed becomes 5 cm/s is $\sqrt\alpha$ cm, where $\alpha=$ ______.

Correct answer: 12

In a nuclear fission process, a high mass nuclide $(A\approx236)$ with binding energy 7.6 MeV/Nucleon dissociated into middle mass nuclides $(A\approx118)$, having binding energy of 8.6 MeV/Nucleon. The energy released in the process would be ______ MeV.

Correct answer: 236

Yellow compound of lead chromate gets dissolved on treatment with hot NaOH solution. The product of lead formed is a :

1. A.Dianionic complex with coordination number six
2. B.Tetraanionic complex with coordination number six
3. C.Neutral complex with coordination number four
4. D.Dianionic complex with coordination number four✓

Correct answer: D

Consider the following complex ions $P=[\text{FeF}_6]^{3-}$ $Q=[\text{V(H}_2\text{O)}_6]^{2+}$ $R=[\text{Fe(H}_2\text{O)}_6]^{2+}$ The correct order of the complex ions, according to their spin only magnetic moment values (in B.M.) is:

1. A.Q < P < R
2. B.R < Q < P
3. C.Q < R < P✓
4. D.R < P < Q

Correct answer: C

The electronic configuration for Neodymium is : [Atomic Number for Neodymium 60]

1. A.$[\text{Xe}]\,4f^1\,5d^1\,6s^2$
2. B.$[\text{Xe}]\,4f^6\,6s^2$
3. C.$[\text{Xe}]\,4f^4\,6s^2$✓
4. D.$[\text{Xe}]\,5f^7\,7s^2$

Correct answer: C

Given below are two statements : Statement (I) : Aqueous solution of ammonium carbonate is basic. Statement (II) : Acidic/basic nature of salt solution of a salt of weak acid and weak base depends on $K_a$ and $K_b$ value of acid and the base forming it. In the light of the above statements, choose the most appropriate answer from the options given below :

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

Correct answer: B

Given below are two statements : Statement (I) : p-nitrophenol is more acidic than m-nitrophenol and o-nitrophenol. Statement (II) : Ethanol will give immediate turbidity with Lucas reagent. In the light of the above statements, choose the correct answer from the options given below :

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