JEE Main 2023 — 15 Apr Shift 1 (Previous Year Paper)

If the domain of the function $f(x)=\log_e(4x^2+11x+6)+\sin^{-1}(4x+3)+\cos^{-1}\left(\frac{10x+6}{3}\right)$ is $(\alpha,\beta]$, then $36|\alpha+\beta|$ is equal to

1. A.45✓
2. B.54
3. C.63
4. D.72

Correct answer: A

The number of real roots of the equation $x|x|-5|x+2|+6=0$, is

1. A.6
2. B.3✓
3. C.5
4. D.4

Correct answer: B

If the set $\left\{\mathrm{Re}\left(\frac{z-\bar z+z\bar z}{2-3z+5\bar z}\right): z\in\mathbb{C},\ \mathrm{Re}(z)=3\right\}$ is equal to the interval $(\alpha,\beta]$, then $24(\beta-\alpha)$ is equal to

1. A.27
2. B.30✓
3. C.36
4. D.42

Correct answer: B

Let the determinant of a square matrix A of order $m$ be $m-n$, where $m$ and $n$ satisfy $4m+n=22$ and $17m+4n=93$. If $\det(n\,\mathrm{adj}(\mathrm{adj}(mA)))=3^a5^b6^c$, then $a+b+c$ is equal to

1. A.84
2. B.96✓
3. C.101
4. D.109

Correct answer: B

Let the system of linear equations $-x+2y-9z=7$, $-x+3y+7z=9$, $-2x+y+5z=8$, $-3x+y+13z=\lambda$ has a unique solution $x=\alpha, y=\beta, z=\gamma$. Then the distance of the point $(\alpha,\beta,\gamma)$ from the plane $2x-2y+z=\lambda$ is

1. A.7✓
2. B.9
3. C.11
4. D.13

Correct answer: A

The total number of three-digit numbers, divisible by 3, which can be formed using the digits 1, 3, 5, 8, if repetition of digits is allowed, is

1. A.18
2. B.20
3. C.21
4. D.22✓

Correct answer: D

Let $(a+bx+cx^2)^{10}=\sum_{i=0}^{20}p_ix^i,\ a,b,c\in\mathbb{N}$. If $p_1=20$ and $p_2=210$, then $2(a+b+c)$ is equal to

1. A.6
2. B.8
3. C.12✓
4. D.15

Correct answer: C

Let $A_1$ and $A_2$ be two arithmetic means and $G_1, G_2, G_3$ be three geometric means of two distinct positive numbers. Then $G_1^4+G_2^4+G_3^4+G_1^2G_3^2$ is equal to

1. A.$2(A_1+A_2)G_1G_3$
2. B.$(A_1+A_2)G_1^2G_3^2$
3. C.$(A_1+A_2)^2G_1G_3$✓
4. D.$2(A_1+A_2)G_1^2G_3^2$

Correct answer: C

Let $[x]$ denote the greatest integer function and $f(x)=\max\{1+x+[x],\ 2+x,\ x+2[x]\},\ 0\le x\le2$. Let $m$ be the number of points in $[0,2]$, where $f$ is not continuous and $n$ be the number of points in $(0,2)$, where $f$ is not differentiable. Then $(m+n)^2+2$ is equal to

1. A.2
2. B.3✓
3. C.6
4. D.11

Correct answer: B

If $\int_0^1\frac{1}{(5+2x-2x^2)\left(1+e^{(2-4x)}\right)}dx=\frac{1}{\alpha}\log_e\left(\frac{\alpha+1}{\beta}\right),\ \alpha,\beta>0$, then $\alpha^4-\beta^4$ is equal to

1. A.$-21$
2. B.0
3. C.19
4. D.21✓

Correct answer: D

Let $x=x(y)$ be the solution of the differential equation $2(y+2)\log_e(y+2)\,dx+\left(x+4-2\log_e(y+2)\right)dy=0,\ y>-1$ with $x(e^4-2)=1$. Then $x(e^9-2)$ is equal to

1. A.3
2. B.$\frac{10}{3}$
3. C.$\frac{4}{9}$
4. D.$\frac{32}{9}$✓

Correct answer: D

If $(\alpha,\beta)$ is the orthocenter of the triangle ABC with vertices $A(3,-7)$, $B(-1,2)$ and $C(4,5)$, then $9\alpha-6\beta+60$ is equal to

1. A.25✓
2. B.30
3. C.35
4. D.40

Correct answer: A

The number of common tangents, to the circles $x^2+y^2-18x-15y+131=0$ and $x^2+y^2-6x-6y-7=0$, is

1. A.1
2. B.2
3. C.3✓
4. D.4

Correct answer: C

Let the foot of perpendicular of the point $P(3,-2,-9)$ on the plane passing through the points $(-1,-2,-3)$, $(9,3,4)$, $(9,-2,1)$ be $Q(\alpha,\beta,\gamma)$. Then the distance of $Q$ from the origin is

1. A.$\sqrt{35}$
2. B.$\sqrt{38}$
3. C.$\sqrt{29}$
4. D.$\sqrt{42}$✓

Correct answer: D

Let S be the set of all values of $\lambda$, for which the shortest distance between the lines $\frac{x-\lambda}{0}=\frac{y-3}{4}=\frac{z+6}{1}$ and $\frac{x+\lambda}{3}=\frac{y}{-4}=\frac{z-6}{0}$ is 13. Then $8\left|\sum_{\lambda\in S}\lambda\right|$ is equal to

1. A.302
2. B.304
3. C.306✓
4. D.308

Correct answer: C

Let $S$ be the set of all $(\lambda,\mu)$ for which the vectors $\lambda\hat i-\hat j+\hat k$, $\hat i+2\hat j+\mu\hat k$ and $3\hat i-4\hat j+5\hat k$, where $\lambda-\mu=5$, are coplanar, then $\sum_{(\lambda,\mu)\in S}80(\lambda^2+\mu^2)$ is equal to

1. A.2130
2. B.2210
3. C.2290✓
4. D.2370

Correct answer: C

Let ABCD be a quadrilateral. If E and F are the mid points of the diagonals AC and BD respectively and $(\overrightarrow{AB}-\overrightarrow{BC})+(\overrightarrow{AD}-\overrightarrow{DC})=k\,\overrightarrow{FE}$, then $k$ is equal to

1. A.$-4$✓
2. B.$-2$
3. C.2
4. D.4

Correct answer: A

A bag contains 6 white and 4 black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is

1. A.$\frac{1}{5}$✓
2. B.$\frac{11}{50}$
3. C.$\frac{9}{50}$
4. D.$\frac{1}{4}$

Correct answer: A

The mean and standard deviation of 10 observations are 20 and 8 respectively. Later on, it was observed that one observation was recorded as 50 instead of 40. Then the correct variance is

1. A.11
2. B.12
3. C.13✓
4. D.14

Correct answer: C

Negation of $p\wedge(q\wedge\sim(p\wedge q))$ is

1. A.$\sim(p\vee q)$
2. B.$p\vee q$
3. C.$(\sim(p\wedge q))\wedge q$
4. D.$(\sim(p\wedge q))\vee p$✓

Correct answer: D

Let $A=\{1,2,3,4\}$ and R be a relation on the set $A\times A$ defined by $R=\{((a,b),(c,d)): 2a+3b=4c+5d\}$. Then the number of elements in R is ________.

Correct answer: 6

A person forgets his 4-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is 7 and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct code is ________.

Correct answer: 72

The number of elements in the set $\{n\in\mathbb{N}: 10\le n\le100$ and $3^n-3$ is a multiple of 7$\}$ is ________.

Correct answer: 15

If the sum of the series $\left(\frac12-\frac13\right)+\left(\frac1{2^2}-\frac1{2\cdot3}+\frac1{3^2}\right)+\left(\frac1{2^3}-\frac1{2^2\cdot3}+\frac1{2\cdot3^2}-\frac1{3^3}\right)+\left(\frac1{2^4}-\frac1{2^3\cdot3}+\frac1{2^2\cdot3^2}-\frac1{2\cdot3^3}+\frac1{3^4}\right)+\dots$ is $\frac{\alpha}{\beta}$, where $\alpha$ and $\beta$ are co-prime, then $\alpha+3\beta$ is equal to ________.

Correct answer: 7

Consider the triangles with vertices $A(2,1)$, $B(0,0)$ and $C(t,4)$, $t\in[0,4]$. If the maximum and the minimum perimeters of such triangles are obtained at $t=\alpha$ and $t=\beta$ respectively, then $6\alpha+21\beta$ is equal to ________.

Correct answer: 48

Let $f(x)=\int\frac{dx}{(3+4x^2)\sqrt{4-3x^2}},\ |x|<\frac{2}{\sqrt3}$. If $f(0)=0$ and $f(1)=\frac{1}{\alpha\beta}\tan^{-1}\left(\frac{\alpha}{\beta}\right)$, $\alpha,\beta>0$, then $\alpha^2+\beta^2$ is equal to ________.

Correct answer: 28

If the area bounded by the curve $2y^2=3x$, lines $x+y=3$, $y=0$ and outside the circle $(x-3)^2+y^2=2$ is A, then $4(\pi+4A)$ is equal to ________.

Correct answer: 42

Let an ellipse with centre $(1,0)$ and latus rectum of length $\frac12$ have its major axis along x-axis. If its minor axis subtends an angle $60^\circ$ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to ________.

Correct answer: 9

Let the plane P contain the line $2x+y-z-3=0=5x-3y+4z+9$ and be parallel to the line $\frac{x+2}{2}=\frac{3-y}{-4}=\frac{z-7}{5}$. Then the distance of the point $A(8,-1,-19)$ from the plane P measured parallel to the line $\frac{x}{-3}=\frac{y-5}{4}=\frac{2-z}{-12}$ is equal to ________.

Correct answer: 26

If the line $x=y=z$ intersects the line $x\sin A+y\sin B+z\sin C-18=0=x\sin2A+y\sin2B+z\sin2C-9$, where $A, B, C$ are the angles of a triangle $ABC$, then $80\left(\sin\frac A2\sin\frac B2\sin\frac C2\right)$ is equal to ________.

Correct answer: 5

In a linear Simple Harmonic Motion (SHM) (A) Restoring force is directly proportional to the displacement. (B) The acceleration and displacement are opposite in direction. (C) The velocity is maximum at mean position. (D) The acceleration is minimum at extreme points. Choose the correct answer from the options given below:

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

Correct answer: A

A flask contains Hydrogen and Argon in the ratio 2:1 by mass. The temperature of the mixture is $30^\circ$C. The ratio of average kinetic energy per molecule of the two gases ($K_{argon}/K_{hydrogen}$) is: (Given: Atomic Weight of Ar = 39.9)

1. A.$\frac{39.9}{2}$
2. B.1✓
3. C.39.9
4. D.2

Correct answer: B

A thermodynamic system is taken through cyclic process. The total work done in the process is:

1. A.Zero
2. B.200 J
3. C.100 J
4. D.300 J✓

Correct answer: D

A wire of length 'L' and radius 'r' is clamped rigidly at one end. When the other end of the wire is pulled by a force $f$, its length increases by '$l$'. Another wire of same material of length '2L' and radius '2r' is pulled by a force '2f'. Then the increase in its length will be:

1. A.$l$✓
2. B.$2l$
3. C.$4l$
4. D.$l/2$

Correct answer: A

A body is released from a height equal to the radius (R) of the earth. The velocity of the body when it strikes the surface of the earth will be: (Given g = acceleration due to gravity on the earth.)

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

Correct answer: D

Two identical particles each of mass 'm' go round a circle of radius $a$ under the action of their mutual gravitational attraction. The angular speed of each particle will be:

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

Correct answer: B

A vector in $x-y$ plane makes an angle of $30^\circ$ with y-axis. The magnitude of y-component of vector is $2\sqrt3$. The magnitude of $x$-component of the vector will be:

1. A.2✓
2. B.$\sqrt3$
3. C.$\frac{1}{\sqrt3}$
4. D.6

Correct answer: A

The position of a particle related to time is given by $x=(5t^2-4t+5)$ m. The magnitude of velocity of the particle at $t=2$ s will be:

1. A.14 ms$^{-1}$
2. B.16 ms$^{-1}$✓
3. C.10 ms$^{-1}$
4. D.06 ms$^{-1}$

Correct answer: B

The speed of a wave produced in water is given by $v=\lambda^ag^b\rho^c$. Where $\lambda$, $g$ and $\rho$ are wavelength of wave, acceleration due to gravity and density of water respectively. The values of $a$, $b$ and $c$ respectively, are:

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

Correct answer: C

The position vector of a particle related to time $t$ is given by $\vec r=(10t\,\hat i+15t^2\,\hat j+7\hat k)$ m. The direction of net force experienced by the particle is:

1. A.Positive $x$-axis
2. B.Positive $y$-axis✓
3. C.Positive $z$-axis
4. D.In $x$-$y$ plane

Correct answer: B

The electric field due to a short electric dipole at a large distance ($r$) from center of dipole on the equatorial plane varies with distance as:

1. A.$\frac1r$
2. B.$\frac1{r^2}$
3. C.$\frac1{r^3}$✓
4. D.$r$

Correct answer: C

The height of transmitting antenna is 180 m and the height of the receiving antenna is 245 m. The maximum distance between them for satisfactory communication in line of sight will be: (given R = 6400 km)

1. A.48 km
2. B.56 km
3. C.96 km
4. D.104 km✓

Correct answer: D

In the given circuit, the current (I) through the battery will be

1. A.1.5 A✓
2. B.2.5 A
3. C.1A
4. D.2A

Correct answer: A

The half-life of a radioactive nucleus is 5 years. The fraction of the original sample that would decay in 15 years is:

1. A.$\frac14$
2. B.$\frac18$
3. C.$\frac34$
4. D.$\frac78$✓

Correct answer: D

The de Broglie wavelength of an electron having kinetic energy E is $\lambda$. If the kinetic energy of electron becomes $\frac E4$, then its de-Broglie wavelength will be:

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

Correct answer: B

A single slit of width $a$ is illuminated by a monochromatic light of wavelength 600 nm. The value of '$a$' for which first minimum appears at $\theta=30^\circ$ on the screen will be:

1. A.0.6 $\mu$m
2. B.1.2 $\mu$m✓
3. C.1.8 $\mu$m
4. D.3 $\mu$m

Correct answer: B

Match List I with List II of Electromagnetic waves with corresponding wavelength range: List I: (A) Microwave (B) Ultraviolet (C) X-Ray (D) Infra-red. List II: (I) 400 nm to 1 nm (II) 1 nm to $10^{-3}$ nm (III) 1 mm to 700 nm (IV) 0.1 m to 1 mm. Choose the correct answer from the options given below:

1. A.(A)-(I), (B)-(IV), (C)-(II), (D)-(III)
2. B.(A)-(IV), (B)-(II), (C)-(I), (D)-(III)
3. C.(A)-(IV), (B)-(I), (C)-(II), (D)-(III)✓
4. D.(A)-(IV), (B)-(I), (C)-(III), (D)-(II)

Correct answer: C

A 12 V battery connected to a coil of resistance 6 $\Omega$ through a switch, drives a constant current in the circuit. The switch is opened in 1 ms. The emf induced across the coil is 20 V. The inductance of the coil is:

1. A.5 mH
2. B.8 mH
3. C.10 mH✓
4. D.12 mH

Correct answer: C

For designing a voltmeter of range 50 V and an ammeter of range 10 mA using a galvanometer which has a coil of resistance 54 $\Omega$ showing a full scale deflection for 1 mA as in figure. (A) for voltmeter $R\approx50$ k$\Omega$ (B) for ammeter $r\approx0.2\,\Omega$ (C) for ammeter $r=6\,\Omega$ (D) for voltmeter $R\approx5$ k$\Omega$ (E) for voltmeter $R\approx500\,\Omega$. Choose the correct answer from the options given below:

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

Correct answer: A

Given below are two statements: Statement I: The equivalent resistance of resistors in a series combination is smaller than least resistance used in the combination. Statement II: The resistivity of the material is independent of temperature. 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.Both Statement I and Statement II are false✓
3. C.Statement I is true but Statement II is false
4. D.Statement I is false but Statement II is true

Correct answer: B

The fundamental frequency of vibration of a string stretched between two rigid support is 50 Hz. The mass of the string is 18 g and its linear mass density is 20 g/m. The speed of the transverse waves so produced in the string is ________ ms$^{-1}$

Correct answer: 90

There is an air bubble of radius 1.0 mm in a liquid of surface tension 0.075 Nm$^{-1}$ and density 1000 kg m$^{-3}$ at a depth of 10 cm below the free surface. The amount by which the pressure inside the bubble is greater than the atmospheric pressure is ________ Pa ($g=10$ ms$^{-2}$)

Correct answer: 1150

A solid sphere and a solid cylinder of same mass and radius are rolling on a horizontal surface without slipping. The ratio of their radius of gyrations respectively ($k_{sph}:k_{cyl}$) is $2:\sqrt x$. The value of $x$ is ________

Correct answer: 5

A block of mass 10 kg is moving along x-axis under the action of force $F=5x$ N. The work done by the force in moving the block from $x=2$ m to 4 m will be ________ J.

Correct answer: 30

In the given figure the total charge stored in the combination of capacitors is 100 $\mu$C. The value of 'x' is ________.

Correct answer: 5

As per given figure A, B and C are the first, second and third excited energy levels of hydrogen atom respectively. If the ratio of the two wavelengths (i.e. $\frac{\lambda_1}{\lambda_2}$) is $\frac{7}{4n}$, then the value of $n$ will be ________.

Correct answer: 5

The refractive index of a transparent liquid filled in an equilateral hollow prism is $\sqrt2$. The angle of minimum deviation for the liquid will be ________ $^\circ$.

Correct answer: 30

A 20 cm long metallic rod is rotated with 210 rpm about an axis normal to the rod passing through its one end. The other end of the rod is in contact with a circular metallic ring. A constant and uniform magnetic field 0.2T parallel to the axis exists everywhere. The emf developed between the centre and the ring is ________ mV. Take $\pi=\frac{22}{7}$

Correct answer: 88

An electron in a hydrogen atom revolves around its nucleus with a speed of $6.76\times10^6$ ms$^{-1}$ in an orbit of radius 0.52 Å. The magnetic field produced at the nucleus of the hydrogen atom is ________ T.

Correct answer: 40

A network of four resistances is connected to 9 V battery, as shown in figure. The magnitude of voltage difference between the points A and B is ________ V.

Correct answer: 3

Which of the following expressions is correct in case of a CsCl unit cell (edge length 'a')?

1. A.$r_{Cs^+}+r_{Cl^-}=a$
2. B.$r_{Cs^+}+r_{Cl^-}=\frac{a}{\sqrt2}$
3. C.$r_{Cs^+}+r_{Cl^-}=\frac{\sqrt3}{2}a$✓
4. D.$r_{Cs^+}+r_{Cl^-}=\frac{a}{2}$