JEE Main 15 April 2023 Shift 1 Question Paper

Question 1

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:

Accepted Answer

(A) 22

Answer

(B) 18

Answer

(C) 21

Answer

(D) 20

Question 2

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

Accepted Answer

(C) 306

Answer

(A) 304

Answer

(B) 308

Answer

(D) 302

Question 3

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:

Accepted Answer

(B) 13

Answer

(A) 14

Answer

(C) 12

Answer

(D) 11

Question 4

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

Accepted Answer

(C) $-4$

Answer

(A) 2

Answer

(B) $-2$

Answer

(D) 4

Question 5

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

Accepted Answer

(C) $\frac{32}{9}$

Answer

(A) $\frac{4}{9}$

Answer

(B) $\frac{10}{3}$

Answer

(D) 3

Question 6

Let $[x]$ denote the greatest integer function and $f(x)=\max\{1+x+[x],\,2+x,\,x+2[x]\}$, $0\le x\le 2$. 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:

Accepted Answer

(D) 3

Answer

(A) 11

Answer

(B) 2

Answer

(C) 6

Question 7

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

Accepted Answer

(B) 3

Answer

(A) 5

Answer

(C) 6

Answer

(D) 4

Question 8

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

Accepted Answer

(B) 12

Answer

(A) 8

Answer

(C) 15

Answer

(D) 6

Question 9

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 $\det(n\,\mathrm{adj}(\mathrm{adj}(mA)))=3^a 5^b 6^c$, then $a+b+c$ is equal to:

Accepted Answer

(A) 96

Answer

(B) 101

Answer

(C) 109

Answer

(D) 84

Question 10

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. The $G_1^4+G_2^4+G_3^4+G_1^2 G_3^2$ is equal to

Accepted Answer

(B) $(A_1+A_2)^2 G_1 G_3$

Answer

(A) $2(A_1+A_2)G_1 G_3$

Answer

(C) $(A_1+A_2)G_1^2 G_3^2$

Answer

(D) $2(A_1+A_2)G_1^2 G_3^2$

Question 11

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

Accepted Answer

(D) 30

Answer

(A) 36

Answer

(B) 42

Answer

(C) 27

Question 12

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 :

Accepted Answer

(A) 3

Answer

(B) 2

Answer

(C) 1

Answer

(D) 4

Question 13

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

Accepted Answer

(D) $(\sim(p\wedge q))\vee p$

Answer

(A) $\sim(p\vee q)$

Answer

(B) $p\vee q$

Answer

(C) $(\sim(p\wedge q))\wedge q$

Question 14

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

Accepted Answer

(D) 7

Answer

(A) 9

Answer

(B) 11

Answer

(C) 13

Question 15

If $(\alpha,\beta)$ is the orthocentre 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:

Accepted Answer

(B) 25

Answer

(A) 30

Answer

(C) 40

Answer

(D) 35

Question 16

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:

Accepted Answer

(C) $\sqrt{42}$

Answer

(A) $\sqrt{29}$

Answer

(B) $\sqrt{35}$

Answer

(D) $\sqrt{38}$

Question 17

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:

Accepted Answer

(C) $\frac{1}{5}$

Answer

(A) $\frac{1}{4}$

Answer

(B) $\frac{9}{50}$

Answer

(D) $\frac{11}{50}$

Question 18

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

Accepted Answer

(A) $21$

Answer

(B) $0$

Answer

(C) $19$

Answer

(D) $-21$

Question 19

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:

Accepted Answer

(C) $2290$

Answer

(A) $2370$

Answer

(B) $2130$

Answer

(D) $2210$

Question 20

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

Accepted Answer

(B) $45$

Answer

(A) $63$

Answer

(C) $72$

Answer

(D) $54$

Question 21

If the sum of the series $\left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{2^2} - \frac{1}{2 \cdot 3} + \frac{1}{3^2}\right) + \left(\frac{1}{2^3} - \frac{1}{2^2 \cdot 3} + \frac{1}{2 \cdot 3^2} - \frac{1}{3^3}\right) + \left(\frac{1}{2^4} - \frac{1}{2^3 \cdot 3} + \frac{1}{2^2 \cdot 3^2} - \frac{1}{2 \cdot 3^3} + \frac{1}{3^4}\right) + \cdots$ is $\frac{\alpha}{\beta}$, where $\alpha$ and $\beta$ are co-prime, then $\alpha + 3\beta$ is equal to....

Accepted Answer

7

Question 22

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____

Accepted Answer

72

Question 23

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____

Accepted Answer

26

Question 24

Let an ellipse with centre $(1, 0)$ and latus rectum of length $\frac{1}{2}$ 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____

Accepted Answer

9

Question 25

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

Accepted Answer

6

Question 26

The number of elements in the set $\{n \in \mathbb{N} : 10 \le n \le 100 	ext{ and } 3^n - 3 	ext{ is a multiple of } 7\}$ is____

Accepted Answer

15

Question 27

If the line $x = y = z$ intersects the line $x\sin A + y\sin B + z\sin C - 18 = 0 = x\sin 2A + y\sin 2B + z\sin 2C - 9$, where $A, B, C$ are the angles of a triangle $ABC$, then $80\left(\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}\right)$ is equal to____

Accepted Answer

5

Question 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____

Accepted Answer

42

Question 29

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____

Accepted Answer

48

Question 30

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

Accepted Answer

28

Question 31

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 :

Accepted Answer

(C) $\frac{1}{r^{3}}$

Answer

(A) $r$

Answer

(B) $\frac{1}{r}$

Answer

(D) $\frac{1}{r^{2}}$

Question 32

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 :

Accepted Answer

(A) (A), (B) and (C) only

Answer

(B) (C) and (D) only

Answer

(C) (A), (B) and (D) only

Answer

(D) (A), (C) and (D) only

Question 33

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 :

Accepted Answer

(C) $\sqrt{\frac{Gm}{4a^{3}}}$

Answer

(A) $\sqrt{\frac{Gm}{2a^{3}}}$

Answer

(B) $\sqrt{\frac{Gm}{8a^{3}}}$

Answer

(D) $\sqrt{\frac{Gm}{a^{3}}}$

Question 34

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)

Accepted Answer

(D) 104 km

Answer

(A) 48 km

Answer

(B) 56 km

Answer

(C) 96 km

Question 35

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

Accepted Answer

(C) $\frac{7}{8}$

Answer

(A) $\frac{1}{8}$

Answer

(B) $\frac{1}{4}$

Answer

(D) $\frac{3}{4}$

Question 36

The de Broglie wavelength of an electron having kinetic energy $E$ is $\lambda$. If the kinetic energy of electron becomes $\frac{E}{4}$, then its de-Broglie wavelength will be :

Accepted Answer

(C) $2\lambda$

Answer

(A) $\frac{\lambda}{\sqrt{2}}$

Answer

(B) $\frac{\lambda}{2}$

Answer

(D) $\sqrt{2}\lambda$

Question 37

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 $R \approx 50$ k$\Omega$
(B) for ammeter $r \approx 0.2$ $\Omega$
(C) for ammeter $r \approx 6$ $\Omega$
(D) for voltmeter $R \approx 5$ k$\Omega$
(E) for voltmeter $R \approx 500$ $\Omega$
Choose the correct answer from the options given below :

Accepted Answer

(C) (A) and (C)

Answer

(A) (C) and (E)

Answer

(B) (C) and (D)

Answer

(D) (A) and (B)

Question 38

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)

Accepted Answer

(A) 1

Answer

(B) 2

Answer

(C) $\frac{39.9}{2}$

Answer

(D) 39.9

Question 39

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 :

Accepted Answer

(B) Both Statement I and Statement II are false

Answer

(A) Statement I is false but Statement II is true

Answer

(C) Statement I is true but Statement II is false

Answer

(D) Both Statement I and Statement II are true

Question 40

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.)

Accepted Answer

(A) $\sqrt{gR}$

Answer

(B) $\sqrt{4gR}$

Answer

(C) $\sqrt{2gR}$

Answer

(D) $\sqrt{\frac{gR}{2}}$

Question 41

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 :

Accepted Answer

(D) 10 mH

Answer

(A) 5 mH

Answer

(B) 12 mH

Answer

(C) 8 mH

Question 42

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 '$\ell$'. 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 :

Accepted Answer

(B) $\ell$

Answer

(A) $2\ell$

Answer

(C) $4\ell$

Answer

(D) $\ell/2$

Question 43

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 :

Accepted Answer

(C) 16 ms$^{-1}$

Answer

(A) 10 ms$^{-1}$

Answer

(B) 14 ms$^{-1}$

Answer

(D) 06 ms$^{-1}$

Question 44

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

Accepted Answer

(A) Positive y-axis

Answer

(B) Positive x-axis

Answer

(C) Positive z-axis

Answer

(D) In x-y plane

Question 45

Match List I (Electromagnetic waves) with List II (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 1mm. Choose the correct answer from the options given below :

Accepted Answer

(B) (A)-(IV), (B)-(I), (C)-(II), (D)-(III)

Answer

(A) (A)-(I), (B)-(IV), (C)-(II), (D)-(III)

Answer

(C) (A)-(IV), (B)-(II), (C)-(I), (D)-(III)

Answer

(D) (A)-(IV), (B)-(I), (C)-(III), (D)-(II)

Question 46

A vector in x-y plane makes an angle of $30^\circ$ with y-axis. The magnitude of y-component of vector is $2\sqrt{3}$. The magnitude of x-component of the vector will be :

Accepted Answer

(D) $2$

Answer

(A) $\frac{1}{\sqrt{3}}$

Answer

(B) $6$

Answer

(C) $\sqrt{3}$

Question 47

The speed of a wave produced in water is given by $v = \lambda^a g^b \rho^c$. 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 :

Accepted Answer

(A) $\frac{1}{2}, \frac{1}{2}, 0$

Answer

(B) $1, 1, 0$

Answer

(C) $1, -1, 0$

Answer

(D) $\frac{1}{2}, 0, \frac{1}{2}$

Question 48

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

Accepted Answer

(B) $300$ J

Answer

(A) $100$ J

Answer

(C) Zero

Answer

(D) $200$ J

Question 49

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 $	heta = 30^\circ$ on the screen will be :

Accepted Answer

(B) $1.2\ \mu$m

Answer

(A) $0.6\ \mu$m

Answer

(C) $1.8\ \mu$m

Answer

(D) $3\ \mu$m

Question 50

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

Accepted Answer

(A) $1.5$ A

Answer

(B) $1$ A

Answer

(C) $2.5$ A

Answer

(D) $2$ A

Question 51

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 $0.2$ T parallel to the axis exists everywhere. The emf developed between the centre and the ring is __________ mV. Take $\pi = \frac{22}{7}$.

Accepted Answer

88

Question 52

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.

Accepted Answer

3

Question 53

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$^{-1}$.

Accepted Answer

90

Question 54

As per given figure A, B and C are the first, second and third excited energy level 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 __________.

Accepted Answer

5

Question 55

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 $2 : \sqrt{x}$, then value of x is __________.

Accepted Answer

5

Question 56

The refractive index of a transparent liquid filled in an equilateral hollow prism is $\sqrt{2}$. The angle of minimum deviation for the liquid will be __________ $^\circ$.

Accepted Answer

30

Question 57

An electron in a hydrogen atom revolves around its nucleus with a speed of $6.76 	imes 10^6$ ms$^{-1}$ in an orbit of radius $0.52$ A$^\circ$. The magnetic field produced at the nucleus of the hydrogen atom is __________ T.

Accepted Answer

40

Question 58

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$^{-2}$).

Accepted Answer

1150

Question 59

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.

Accepted Answer

30

Question 60

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

Accepted Answer

5

Question 61

Match List I with List II:

Accepted Answer

(A) (A)-(III), (B)-(I), (C)-(IV), (D)-(II)

Answer

(B) (A)-(III), (B)-(IV), (C)-(II), (D)-(I)

Answer

(C) (A)-(IV), (B)-(I), (C)-(II), (D)-(III)

Answer

(D) (A)-(II), (B)-(III), (C)-(IV), (D)-(I)

Question 62

The product formed in the following multistep reaction is: $CH_3-CH=CH_2$ with i) $B_2H_6$ ii) $H_2O_2, NaOH$ iii) PCC iv) $CH_3MgBr$

Accepted Answer

(A) $CH_3-CH_2-CH(OH)-CH_3$

Answer

(B) $CH_3-CH_2-CH_2-CH_2-OH$

Answer

(C) $CH_3-CH_2-C(=O)-OCH_3$

Answer

(D) $CH_3-C(OH)(CH_3)-CH_3$

Question 63

The possibility of photochemical smog formation will be minimum at

Accepted Answer

(D) Srinagar, Jammu and Kashmir in January

Answer

(A) Kolkata in October

Answer

(B) Mumbai in May

Answer

(C) New-Delhi in August (Summer)

Question 64

Which one of the following is not an example of calcination?

Accepted Answer

(D) $2PbS + 3O_2 \xrightarrow{\Delta} 2PbO + 2SO_2$

Answer

(A) $Fe_2O_3 \cdot xH_2O \xrightarrow{\Delta} Fe_2O_3 + xH_2O$

Answer

(B) $CaCO_3 \xrightarrow{\Delta} CaO + CO_2$

Answer

(C) $CaCO_3 \cdot MgCO_3 \xrightarrow{\Delta} CaO + MgO + 2CO_2$

List-I	List-II
A.Microwave	I.400 nm to 1 nm
B.Ultraviolet	II.1 nm to $10^{-3}$ nm
C.X-Ray	IV.0.1 m to 1mm
D.Infra-red	III.1 mm to 700 nm

List I-(Monomer)	List II-(Polymer)
A.Tetrafluoroethene	I.Orlon
B.Acrylonitrile	II.Natural rubber
C.Caprolactam	III.Teflon
D.Isoprene	IV.Nylon-6

JEE Main 15 April 2023 Shift 1 Question Paper with Answers

Physics — JEE Main 15 April 2023 Shift 1

Chemistry — JEE Main 15 April 2023 Shift 1

Mathematics — JEE Main 15 April 2023 Shift 1

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