JEE Main 10 April 2023 Shift 1 Question Paper

Question 1

Let O be the origin and the position vector of the point P be $-\hat i-2\hat j+3\hat k$. If the position vectors of the points A, B and C are $-2\hat i+\hat j-3\hat k$, $2\hat i+4\hat j-2\hat k$ and $-4\hat i+2\hat j-\hat k$ respectively, then the projection of the vector $\overrightarrow{OP}$ on a vector perpendicular to the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ is

Accepted Answer

(A) $3$

Answer

(B) $\dfrac83$

Answer

(C) $\dfrac{10}{3}$

Answer

(D) $\dfrac73$

Question 2

Let the ellipse E: $x^2+9y^2=9$ intersect the positive x- and y-axes at the points A and B respectively. Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. If the area of the triangle with vertices A, P and the origin O is $\dfrac{m}{n}$, where m and n are coprime, then $m-n$ is equal to

Accepted Answer

(C) $17$

Answer

(A) $18$

Answer

(B) $16$

Answer

(D) $15$

Question 3

If $f(x)=\dfrac{(	an 1°)x+\log_e(123)}{x\log_e(1234)-(	an 1°)},\ x>0$, then the least value of $f(f(x))+f\left(f\left(\dfrac4xight)ight)$ is

Accepted Answer

(B) $4$

Answer

(A) $8$

Answer

(C) $2$

Answer

(D) $0$

Question 4

A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm$^2$) is equal to

Accepted Answer

(C) $800$

Answer

(A) $675$

Answer

(B) $1025$

Answer

(D) $900$

Question 5

Let $f$ be a differentiable function such that $x^2 f(x)-x=4\int_0^x t\,f(t)\,dt$, $f(1)=\dfrac23$. Then $18\,f(3)$ is equal to

Accepted Answer

(A) $160$

Answer

(B) $210$

Answer

(C) $180$

Answer

(D) $150$

Question 6

A line segment AB of length $\lambda$ moves such that the points A and B remain on the periphery of a circle of radius $\lambda$. Then the locus of the point, that divides the line segment AB in the ratio $2:3$, is a circle of radius

Accepted Answer

(D) $\dfrac{\sqrt{19}}{5}\lambda$

Answer

(A) $\dfrac35\lambda$

Answer

(B) $\dfrac{\sqrt{19}}{7}\lambda$

Answer

(C) $\dfrac23\lambda$

Question 7

Let the complex number $z=x+iy$ be such that $\dfrac{2z-3i}{2z+i}$ is purely imaginary. If $x+y^2=0$, then $y^4+y^2-y$ is equal to:

Accepted Answer

(D) $\dfrac34$

Answer

(A) $\dfrac32$

Answer

(B) $\dfrac43$

Answer

(C) $\dfrac23$

Question 8

$96\cos\dfrac{\pi}{33}\cos\dfrac{2\pi}{33}\cos\dfrac{4\pi}{33}\cos\dfrac{8\pi}{33}\cos\dfrac{16\pi}{33}$ is equal to

Accepted Answer

(A) $3$

Answer

(B) $2$

Answer

(C) $4$

Answer

(D) $1$

Question 9

Accepted Answer

(A) $3^{11}\cdot6^{10}$

Answer

(B) $3^{12}\cdot6^{10}$

Answer

(C) $3^{10}\cdot6^{11}$

Answer

(D) $3^{12}\cdot6^{11}$

Question 10

The slope of tangent at any point $(x,y)$ on a curve $y=y(x)$ is $\dfrac{x^2+y^2}{2xy}$, $x>0$. If $y(2)=0$, then a value of $y(8)$ is

Accepted Answer

(B) $4\sqrt3$

Answer

(A) $-2\sqrt3$

Answer

(C) $2\sqrt3$

Answer

(D) $-4\sqrt2$

Question 11

For the system of linear equations $2x-y+3z=5$, $3x+2y-z=7$, $4x+5y+\alpha z=\beta$, which of the following is NOT correct?

Accepted Answer

(C) The system has infinitely many solutions for $\alpha=-6$ and $\beta=9$

Answer

(A) The system has infinitely many solutions for $\alpha=-5$ and $\beta=9$

Answer

(B) The system has a unique solution for $\alpha
eq-5$ and $\beta=8$

Answer

(D) The system is inconsistent for $\alpha=-5$ and $\beta=8$

Question 12

Let N denotes the sum of the numbers obtained when two dice are rolled. If the probability that $2^N<N!$ is $\dfrac{m}{n}$, where m and n are coprime, then $4m-3n$ is equal to

Accepted Answer

(A) $8$

Answer

(B) $16$

Answer

(C) $10$

Answer

(D) $12$

Question 13

Let P be the point of intersection of the line $\dfrac{x+3}{3}=\dfrac{y+2}{1}=\dfrac{1-z}{2}$ and the plane $x+y+z=2$. If the distance of the point P from the plane $3x-4y+12z=32$ is q, then q and 2q are the roots of the equation

Accepted Answer

(C) $x^2-18x+72=0$

Answer

(A) $x^2-18x-72=0$

Answer

(B) $x^2+18x+72=0$

Answer

(D) $x^2+18x-72=0$

Question 14

The negation of the statement $(p\lor q)\land(q\lor(\sim r))$ is

Accepted Answer

(A) $((\sim p)\lor r)\land(\sim q)$

Answer

(B) $((\sim p)\lor(\sim q))\land(\sim r)$

Answer

(C) $((\sim p)\lor(\sim q))\lor(\sim r)$

Answer

(D) $(p\lor r)\land(\sim q)$

Question 15

If the coefficient of $x^7$ in $\left(ax-\dfrac{1}{bx^2}\right)^{13}$ and the coefficient of $x^{-5}$ in $\left(ax+\dfrac{1}{bx^2}\right)^{13}$ are equal, then $a^4 b^4$ is equal to:

Accepted Answer

(B) $22$

Answer

(A) $44$

Answer

(C) $11$

Answer

(D) $33$

Question 16

Let two vertices of triangle ABC be $(2,4,6)$ and $(0,-2,-5)$, and its centroid be $(2,1,-1)$. If the image of third vertex in the plane $x+2y+4z=11$ is $(\alpha,\beta,\gamma)$, then $\alpha\beta+\beta\gamma+\gamma\alpha$ is equal to

Accepted Answer

(B) $74$

Answer

(A) $72$

Answer

(C) $76$

Answer

(D) $70$

Question 17

The shortest distance between the lines $\dfrac{x+2}{1}=\dfrac{y}{-2}=\dfrac{z-5}{2}$ and $\dfrac{x-4}{1}=\dfrac{y-1}{2}=\dfrac{z+3}{0}$ is

Accepted Answer

(B) $9$

Answer

(A) $6$

Answer

(C) $7$

Answer

(D) $8$

Question 18

If $I(x)=\int e^{\sin^2 x}(\cos x\sin 2x-\sin x)\,dx$ and $I(0)=1$, then $I\left(\dfrac{\pi}{3}\right)$ is equal to

Accepted Answer

(C) $\dfrac12 e^{\frac34}$

Answer

(A) $-\dfrac12 e^{\frac34}$

Answer

(B) $e^{\frac34}$

Answer

(D) $-e^{\frac34}$

Question 19

Let the first term $a$ and the common ratio $r$ of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to

Accepted Answer

(A) $231$

Answer

(B) $210$

Answer

(C) $220$

Answer

(D) $241$

Question 20

An arc PQ of a circle subtends a right angle at its centre O. The mid point of the arc PQ is R. If $\overrightarrow{OP}=\vec u$, $\overrightarrow{OR}=\vec v$ and $\overrightarrow{OQ}=\alpha\vec u+\beta\vec v$, then $\alpha,\beta^2$ are the roots of the equation

Accepted Answer

(A) $x^2-x-2=0$

Answer

(B) $3x^2+2x-1=0$

Answer

(C) $x^2+x-2=0$

Answer

(D) $3x^2-2x-1=0$

Question 21

The coefficient of $x^7$ in $\left(1-x+2x^3\right)^{10}$ is _________.

Accepted Answer

960

Question 22

Let $f:(-2,2)\to\mathbb{R}$ be defined by $f(x)=\begin{cases}x[x] & ,-2<x<0\(x-1)[x] & ,0\le x<2\end{cases}$ where $[x]$ denotes the greatest integer function. If m and n respectively are the number of points in $(-2,2)$ at which $y=|f(x)|$ is not continuous and not differentiable, then $m+n$ is equal to _________.

Accepted Answer

4

Question 23

The sum of all those terms, of the arithmetic progression $3,8,13,\ldots,373$, which are not divisible by 3, is equal to _________.

Accepted Answer

9525

Question 24

Let a common tangent to the curves $y^2=4x$ and $(x-4)^2+y^2=16$ touch the curves at the points P and Q. Then $(PQ)^2$ is equal to _________.

Accepted Answer

32

Question 25

The number of permutations, of the digits $1,2,3,\ldots,7$ without repetition, which neither contain the string 153 nor the string 2467, is _________.

Accepted Answer

4898

Question 26

Let a, b, c be three distinct positive real numbers such that $(2a)^{\log_e a}=(bc)^{\log_e b}$ and $b^{\log_e 2}=a^{\log_e c}$. Then $6a+5bc$ is equal to _________.

Accepted Answer

8

Question 27

Let $y=p(x)$ be the parabola passing through the points $(-1,0),(0,1)$ and $(1,0)$. If the area of the region $\{(x,y):(x+1)^2+(y-1)^2\le1,\ y\le p(x)\}$ is A, then $12(\pi-4A)$ is equal to _________.

Accepted Answer

16

Question 28

If the mean of the frequency distribution — Class: 0-10, 10-20, 20-30, 30-40, 40-50; Frequency: 2, 3, x, 5, 4 — is 28, then its variance is _________.

Accepted Answer

151

Question 29

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total number of persons who participated in the tournament is _________.

Accepted Answer

16

Question 30

The number of elements in the set $\{n\in\mathbb{Z}:|n^2-10n+19|<6\}$ is _________.

Accepted Answer

6

Question 31

A physical quantity P is given as $P=\dfrac{a^2 b^3}{c\sqrt d}$. The percentage error in the measurement of a, b, c and d are $1\%, 2\%, 3\%$ and $4\%$ respectively. The percentage error in the measurement of quantity P will be

Accepted Answer

(A) 13%

Answer

(B) 14%

Answer

(C) 15%

Answer

(D) 16%

Question 32

Assuming the earth to be a sphere of uniform mass density, the weight of a body at a depth $\dfrac{R}{2}$ from the surface of earth, if its weight on the surface of earth is 200 N, will be: (Given R = Radius of earth)

Accepted Answer

(D) 100 N

Answer

(A) 400 N

Answer

(B) 500 N

Answer

(C) 300 N

Question 33

A zener diode of power rating 1.6 W is to be used as voltage regulator. If the zener diode has a breakdown of 8V and it has to regulate voltage fluctuating between 3V and 10 V, the value of resistance R for safe operation of diode will be:

Accepted Answer

(C) $10\,\Omega$

Answer

(A) $13.3\,\Omega$

Answer

(B) $12\,\Omega$

Answer

(D) $11\,\Omega$

Question 34

The range of the projectile projected at an angle of $15°$ with horizontal is 50 m. If the projectile is projected with same velocity at an angle of $45°$ with horizontal, then the range will be

Accepted Answer

(C) 100 m

Answer

(A) 50 m

Answer

(B) $50\sqrt2$ m

Answer

(D) $100\sqrt2$ m

Question 35

A carrier wave of amplitude 15V is modulated by a sinusoidal base band signal of amplitude 3V. The ratio of maximum amplitude to minimum amplitude in an amplitude modulated wave is:

Accepted Answer

(B) $\dfrac32$

Answer

(A) 2

Answer

(C) 5

Answer

(D) $\dfrac12$

Question 36

The angular momentum for the electron in Bohr's orbit is L. If the electron is assumed to revolve in second orbit of hydrogen atom, then the change in angular momentum will be:

Accepted Answer

(C) L

Answer

(A) $\dfrac{L}{2}$

Answer

(B) zero

Answer

(D) 2L

Question 37

A particle of mass m moving with velocity v collides with a stationary particle of mass 2m. After collision, they stick together and continue to move together with velocity

Accepted Answer

(C) $\dfrac{v}{3}$

Answer

(A) v

Answer

(B) $\dfrac{v}{2}$

Answer

(D) $\dfrac{v}{4}$

Question 38

Given below are two statements:
Statement I: If the number of turns in the coil of a moving coil galvanometer is doubled then the current sensitivity becomes double.
Statement II: Increasing current sensitivity of a moving coil galvanometer by only increasing the number of turns in the coil will also increase its voltage sensitivity in the same ratio. In the light of the above statement, choose the correct answer from the options given below:

Accepted Answer

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

Answer

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

Answer

(B) Both Statement I and Statement II are true

Answer

(C) Both Statement I and Statement II are false

Question 39

Match List I (degrees of freedom) with List II (type of gas). Choose the correct answer from the options given below:

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 40

The equivalent resistance of the circuit shown below between points a and b is:

Accepted Answer

(B) $3.2\,\Omega$

Answer

(A) $24\,\Omega$

Answer

(C) $20\,\Omega$

Answer

(D) $16\,\Omega$

Question 41

Consider two containers A and B containing monoatomic gases at the same Pressure (P), Volume (V) and Temperature (T). The gas in A is compressed isothermally to $\dfrac18$ of its original volume while the gas in B is compressed adiabatically to $\dfrac18$ of its original volume. The ratio of final pressure of gas in B to that of gas in A is:

Accepted Answer

(D) 4

Answer

(A) 8

Answer

(B) $\dfrac18$

Answer

(C) $\dfrac14$

Question 42

Given below are two statements:
Statement I: Maximum power is dissipated in a circuit containing an inductor, a capacitor and a resistor connected in series with an AC source, when resonance occurs.
Statement II: Maximum power is dissipated in a circuit containing pure resistor due to zero phase difference between current and voltage.
In the light of the above statements, choose the correct answer from the options given below:

Accepted Answer

(C) Both Statement I and Statement II are true

Answer

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

Answer

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

Answer

(D) Both Statement I and Statement II are false

Question 43

Two satellites of masses m and 3m revolve around the earth in circular orbits of radii r and 3r respectively. The ratio of orbital speeds of the satellites respectively is:

Accepted Answer

(C) $\sqrt3:1$

Answer

(A) 1 : 1

Answer

(B) 2 : 3

Answer

(D) 9 : 1

Question 44

Given below are two statements:
Statement I: Pressure in a reservoir of water is same at all points at the same level of water.
Statement II: The pressure applied to enclosed water is transmitted in all directions equally.
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 true

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 false

Question 45

The equivalent capacitance of the combination shown is:

Accepted Answer

(C) 3C

Answer

(A) $\dfrac{C}{2}$

Answer

(B) 4C

Answer

(D) $\dfrac53 C$

Question 46

The energy of an electromagnetic wave contained in a small volume oscillates with

Accepted Answer

(C) double the frequency of the wave

Answer

(A) zero frequency

Answer

(B) half the frequency of the wave

Answer

(D) the frequency of the wave

Question 47

An object is placed at a distance of 12 cm in front of a plane mirror. The virtual and erect image is formed by the mirror. Now the mirror is moved by 4 cm towards the stationary object. The distance by which the position of image would be shifted, will be:

Accepted Answer

(B) 8 cm towards mirror

Answer

(A) 4 cm towards mirror

Answer

(C) 8 cm away from mirror

Answer

(D) 4 cm away from mirror

Question 48

The de Broglie wavelength of a molecule in a gas at room temperature (300 K) is $\lambda_1$. If the temperature of the gas is increased to 600 K, then the de Broglie wavelength of the same gas molecule becomes

Accepted Answer

(A) $\dfrac{1}{\sqrt2}\lambda_1$

Answer

(B) $2\lambda_1$

Answer

(C) $\dfrac12\lambda_1$

Answer

(D) $\sqrt2\lambda_1$

Question 49

A particle executes S.H.M. of amplitude A along x-axis. At $t=0$, the position of the particle is $x=\dfrac{A}{2}$ and it moves along positive x-axis. The displacement of particle in time t i.e. $x=A\sin(\omega t+\delta)$, then the value of $\delta$ will be:

Accepted Answer

(A) $\dfrac{\pi}{6}$

Answer

(B) $\dfrac{\pi}{3}$

Answer

(C) $\dfrac{\pi}{4}$

Answer

(D) $\dfrac{\pi}{2}$

Question 50

The position-time graphs for two students A and B returning from the school to their homes are shown in figure. [Conclusions: (A) A lives closer to the school; (B) B lives closer to the school; (C) A takes lesser time to reach home; (D) A travels faster than B; (E) B travels faster than A.] Choose the correct answer from the options given below:

Accepted Answer

(A) (A) and (E) only

Answer

(B) (B) and (E) only

Answer

(C) (A), (C) and (E) only

Answer

(D) (A), (D) and (E) only

Question 51

Unpolarised light of intensity $32\,W m^{-2}$ passes through the combination of three polaroids such that the pass axis of the last polaroid is perpendicular to that of the pass axis of first polaroid. If intensity of emerging light is $3\,W m^{-2}$, then the angle between pass axis of first two polaroids is _________ °.

Accepted Answer

30

Question 52

A closed circular tube of average radius 15 cm, whose inner walls are rough, is kept in vertical plane. A block of mass 1 kg just fits inside the tube. The speed of block is 22 m/s when it is introduced at the top of tube. After completing five oscillations, the block stops at the bottom region of tube. The work done by the tube on the block is _________ J. [Given $g=10$ m/s$^2$]

Accepted Answer

245

Question 53

If the earth suddenly shrinks to $\dfrac{1}{64}$th of its original volume with its mass remaining the same, the period of rotation of earth becomes $\dfrac{24}{x}$ h. The value of x is _________.

Accepted Answer

16

Question 54

The current required to be passed through a solenoid of 15 cm length and 60 turns in order to demagnetise a bar magnet of magnetic intensity $2.4	imes10^3$ Am$^{-1}$ is _________ A.

Accepted Answer

6

Question 55

A 1m long metal rod XY completes the circuit as shown in figure. The plane of the circuit is perpendicular to the magnetic field of flux density 0.15 T. If the resistance of the circuit is $5\,\Omega$, the force needed to move the rod in direction, as indicated, with a constant speed of 4 m/s will be _________ $	imes10^{-3}$ N.

Accepted Answer

18

Question 56

A transverse harmonic wave on a string is given by $y(x,t)=5\sin(6t+0.003x)$ where x and y are in cm and t in sec. The wave velocity is _________ ms$^{-1}$.

Accepted Answer

20

Question 57

The decay constant for a radioactive nuclide is $1.5	imes10^{-5}$ s$^{-1}$. Atomic weight of the substance is 60 g mole$^{-1}$, $(N_A=6	imes10^{23})$. The activity of 1.0 µg of the substance is _________ $	imes10^{10}$ Bq.

Accepted Answer

15

Question 58

Three concentric spherical metallic shells X, Y and Z of radius a, b and c respectively $[a<b<c]$ have surface charge densities $\sigma, -\sigma$ and $\sigma$ respectively. The shells X and Z are at same potential. If the radii of X and Y are 2 cm and 3 cm respectively, the radius of shell Z is _________ cm.

Accepted Answer

5

Question 59

10 resistors each of resistance $10\,\Omega$ can be connected in such a way as to get maximum and minimum equivalent resistance. The ratio of maximum and minimum equivalent resistance will be _________.

Accepted Answer

100

Question 60

Two wires each of radius 0.2 cm and negligible mass, one made of steel and other made of brass are loaded as shown in the figure. The elongation of the steel wire is _________ $	imes10^{-6}$ m. [Young's modulus for steel $=2	imes10^{11}$ Nm$^{-2}$ and $g=10$ ms$^{-2}$]

Accepted Answer

20

Question 61

Using column chromatography, mixture of two compounds 'A' and 'B' was separated. 'A' eluted first, this indicates 'B' has

Accepted Answer

(D) low R$_f$, stronger adsorption

Answer

(A) lower R$_f$, weaker adsorption

Answer

(B) high R$_f$, stronger adsorption

Answer

(C) high R$_f$, weaker adsorption

Question 62

Prolonged heating is avoided during the preparation of ferrous ammonium sulphate to

Accepted Answer

(A) prevent oxidation

Answer

(B) prevent reduction

Answer

(C) prevent hydrolysis

Answer

(D) prevent breaking

Question 63

Lime reacts exothermally with water to give 'A' which has low solubility in water. Aqueous solution of 'A' is often used for the test of CO$_2$, a test in which insoluble B is formed. If B is further reacted with CO$_2$ then soluble compound is formed. 'A' is

Accepted Answer

(B) Slaked lime

Answer

(A) Quick lime

Answer

(C) Lime water

Answer

(D) White lime

Question 64

The pair from the following pairs having both compounds with net non-zero dipole moment is

Accepted Answer

(C) CH$_2$Cl$_2$, CHCl$_3$

Answer

(A) Benzene, anisidine

Answer

(B) 1,4-Dichlorobenzene, 1,3-Dichlorobenzene

Answer

(D) cis-butene, trans-butene

List I	List II
A.3 Translational degrees of freedom	I.Monoatomic gases
B.3 Translational, 2 rotational degrees of freedom	II.Polyatomic gases
C.3 Translational, 2 rotational and 1 vibrational degrees of freedom	III.Rigid diatomic gases
D.3 Translational, 3 rotational and more than one vibrational degrees of freedom	IV.Nonrigid diatomic gases

List I (Industry)	List II (Waste Generated)
A.Steel plants	I.Gypsum
B.Thermal power plants	II.Fly ash
C.Fertilizer industries	III.Slag
D.Paper mills	IV.Bio-degradable Wastes

List I (Polymer)	List II (Type/Class)
A.Nylon-2-Nylon-6	I.Thermosetting Polymer
B.Buna-N	II.Biodegradable Polymer
C.Urea-formaldehyde resin	III.Synthetic rubber
D.Dacron	IV.Polyester

JEE Main 10 April 2023 Shift 1 Question Paper with Answers

Physics — JEE Main 10 April 2023 Shift 1

Chemistry — JEE Main 10 April 2023 Shift 1

Mathematics — JEE Main 10 April 2023 Shift 1

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