JEE Main 26 June 2022 Shift 2 Question Paper

Question 1

Let f : ℝ → ℝ be defined as f(x) = x-1 and g : ℝ − {1, −1} → ℝ be defined as $g(x)=\frac{x^2}{x^2-1}$.
Then the function fog is :

Accepted Answer

(D) neither one-one nor onto function

Answer

(A) one-one but not onto function

Answer

(B) onto but not one-one function

Answer

(C) both one-one and onto function

Question 2

If the system of equations αx + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = β,
has infinitely many solutions, then the ordered pair (α, β) is equal to :

Accepted Answer

(C) (1, 3)

Answer

(A) (1,–3)

Answer

(B) (–1, 3)

Answer

(D) (–1, –3)

Question 3

If $A=\sum_{n=1}^{\infty}\frac{1}{\left(3+(-1)^n\right)^n}$ and $B=\sum_{n=1}^{\infty}\frac{(-1)^n}{\left(3+(-1)^n\right)^n}$, then $\frac{A}{B}$ is equal to :

Accepted Answer

(C) $-\frac{11}{9}$

Answer

(A) $\frac{11}{9}$

Answer

(B) 1

Answer

(D) $-\frac{11}{3}$

Question 4

$\lim_{x	o 0}\frac{\cos(\sin x)-\cos x}{x^4}$ is equal to :

Accepted Answer

(C) $\frac{1}{6}$

Answer

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

Answer

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

Answer

(D) $\frac{1}{12}$

Question 5

Let f (x) = min {1, 1 + x sin x}, 0 ≤ x ≤ 2π. If m is the number of points, where f is not differentiable and n is the number of points, where f is not continuous, then the ordered pair (m, n) is equal to

Accepted Answer

(B) (1, 0)

Answer

(A) (2, 0)

Answer

(C) (1, 1)

Answer

(D) (2, 1)

Question 6

Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum of their surface areas is a constant k, then the ratio x : r, for which the sum of their volumes is maximum, is :

Accepted Answer

(B) 19:45

Answer

(A) 2 : 5

Answer

(C) 3 : 8

Answer

(D) 19 : 15

Question 7

The area of the region bounded by $y^2=8x$ and $y^2=16(3-x)$ is equal to :-

Accepted Answer

(C) 16

Answer

(A) $\frac{32}{3}$

Answer

(B) $\frac{40}{3}$

Answer

(D) 19

Question 8

If $\int \frac{1}{x}\sqrt{\frac{1-x}{1+x}}\,dx = g\left(x\right) + c$, $g\left(1\right) = 0$, then $g\left(\frac{1}{2}\right)$ is equal to :

Accepted Answer

(A) $\log_{e}\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right) + \frac{\pi}{3}$

Answer

(B) $\log_{e}\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right) + \frac{\pi}{3}$

Answer

(C) $\log_{e}\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right) - \frac{\pi}{3}$

Answer

(D) $\frac{1}{2}\log_{e}\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right) - \frac{\pi}{6}$

Question 9

If y = y(x) is the solution of the differential equation $x\frac{dy}{dx}+2y=xe^x,\ y(1)=0$, then the local maximum value of the function $z(x)=x^2y(x)-e^x$, x∈R is :

Accepted Answer

(D) $\frac{4}{e}-e$

Answer

(A) 1 – e

Answer

(B) 0

Answer

(C) $\frac{1}{2}$

Question 10

If the solution of the differential equation $\frac{dy}{dx}+e^x\left(x^2-2\right)y=\left(x^2-2x\right)\left(x^2-2\right)e^{2x}$ satisfies y(0) = 0, then the value of y(2) is ________ .

Accepted Answer

(C) 0

Answer

(A) –1

Answer

(B) 1

Answer

(D) e

Question 11

If m is the slope of a common tangent to the curves $\frac{x^2}{16}+\frac{y^2}{9}=1$ and $x^2+y^2=12$, then $12m^2$ is equal to :

Accepted Answer

(B) 9

Answer

(A) 6

Answer

(C) 10

Answer

(D) 12

Question 12

The locus of the mid point of the line segment joining the point (4, 3) and the points on the ellipse $x^2+2y^2=4$ is an ellipse with eccentricity :

Accepted Answer

(C) $\frac{1}{\sqrt{2}}$

Answer

(A) $\frac{\sqrt{3}}{2}$

Answer

(B) $\frac{1}{2\sqrt{2}}$

Answer

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

Question 13

The normal to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{9}=1$ at the point $\left(8,3\sqrt{3}\right)$ on it passes through the point :

Accepted Answer

(C) $\left(-1,9\sqrt{3}\right)$

Answer

(A) $\left(15,-2\sqrt{3}\right)$

Answer

(B) $\left(9,2\sqrt{3}\right)$

Answer

(D) $\left(-1,6\sqrt{3}\right)$

Question 14

If the plane 2x + y – 5z = 0 is rotated about its line of intersection with the plane 3x – y + 4z – 7 = 0 by an angle of $\frac{\pi}{2}$, then the plane after the rotation passes through the point :

Accepted Answer

(C) (1, 0, 2)

Answer

(A) (2, –2, 0)

Answer

(B) (–2, 2, 0)

Answer

(D) (–1, 0, –2)

Question 15

If the lines $\vec{r}=\left(\hat{i}-\hat{j}+\hat{k}\right)+\lambda\left(3\hat{j}-\hat{k}\right)$ and $\vec{r}=\left(\alpha\hat{i}-\hat{j}\right)+\mu\left(2\hat{i}-3\hat{k}\right)$ are co-planar, then distance of the plane containing these two lines from the point (α, 0, 0) is :

Accepted Answer

(B) $\frac{2}{11}$

Answer

(A) $\frac{2}{9}$

Answer

(C) $\frac{4}{11}$

Answer

(D) 2

Question 16

Let $\vec{a}=\hat{i}+\hat{j}+2\hat{k}$, $\vec{b}=2\hat{i}-3\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+\hat{k}$ be three given vectors. Let $\vec{v}$ be a vector in the plane of $\vec{a}$ and $\vec{b}$ whose projection on $\vec{c}$ is $\frac{2}{\sqrt{3}}$. If $\vec{v}\cdot\hat{j}=7$, then $\vec{v}\cdot\left(\hat{i}+\hat{k}\right)$ is equal to :

Accepted Answer

(D) 9

Answer

(A) 6

Answer

(B) 7

Answer

(C) 8

Question 17

The mean and standard deviation of 50 observations are 15 and 2 respectively. It was found that one incorrect observation was taken such that the sum of correct and incorrect observations is 70. If the correct mean is 16, then the correct variance is equal to :

Accepted Answer

(C) 43

Answer

(A) 10

Answer

(B) 36

Answer

(D) 60

Question 18

$16\sin(20^\circ)\,\sin(40^\circ)\,\sin(80^\circ)$ is equal to :

Accepted Answer

(B) $2\sqrt{3}$

Answer

(A) $\sqrt{3}$

Answer

(C) 3

Answer

(D) $4\sqrt{3}$

Question 19

If the inverse trigonometric functions take principal values, then $\cos^{-1}\left(\frac{3}{10}\cos\left(	an^{-1}\left(\frac{4}{3}ight)ight)+\frac{2}{5}\sin\left(	an^{-1}\left(\frac{4}{3}ight)ight)ight)$ is equal to :

Accepted Answer

(C) $\frac{\pi}{3}$

Answer

(A) 0

Answer

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

Answer

(D) $\frac{\pi}{6}$

Question 20

Let r ∈ {p, q, ~p, ~q} be such that the logical statement $r\vee(\sim p)\Rightarrow(p\wedge q)\vee r$ is a tautology. Then 'r' is equal to :

Accepted Answer

(C) ~p

Answer

(A) p

Answer

(B) q

Answer

(D) ~q

Question 21

Let $f:\mathbb{R}	o\mathbb{R}$ satisfy $f(x+y)=2^{x}f(y)+4^{y}f(x)$, ∀x, y ∈ $\mathbb{R}$. If $f(2)=3$, then $14\cdot\frac{f'(4)}{f'(2)}$ is equal to ______.

Accepted Answer

248

Question 22

Let p and q be two real numbers such that $p+q=3$ and $p^{4}+q^{4}=369$. Then $\left(\frac{1}{p}+\frac{1}{q}\right)^{-2}$ is equal to ________.

Accepted Answer

4

Question 23

If $z^{2} + z + 1 = 0$, $z \in \mathbb{C}$, then $\left|\sum_{n=1}^{15}\left(Z^{n} + \left(-1\right)^{n}\frac{1}{Z^{n}}\right)^{2}\right|$ is equal to ______ .

Accepted Answer

2

Question 24

Let $X=\begin{bmatrix} 0 & 1 & 0 \ 0 & 0 & 1 \ 0 & 0 & 0 \end{bmatrix}$, $Y=\alpha I+\beta X+\gamma X^{2}$ and $Z=\alpha^{2}I-\alpha\beta X+\left(\beta^{2}-\alpha\gammaight)X^{2}$, $\alpha,\beta,\gamma\in\mathbb{R}$. If $Y^{-1}=\begin{bmatrix} \frac{1}{5} & \frac{-2}{5} & \frac{1}{5} \ 0 & \frac{1}{5} & \frac{-2}{5} \ 0 & 0 & \frac{1}{5} \end{bmatrix}$, then $(\alpha-\beta+\gamma)^{2}$ is equal to ______ .

Accepted Answer

100

Question 25

The total number of 3−digit numbers, whose greatest common divisor with 36 is 2, is ______ .

Accepted Answer

150

Question 26

If $\left(^{40}C_{0}\right)+\left(^{41}C_{1}\right)+\left(^{42}C_{2}\right)+...+\left(^{60}C_{20}\right)=\frac{m}{n}\,^{60}C_{20}$, m and n are coprime, then $m+n$ is equal to _____ .

Accepted Answer

102

Question 27

If $a_{1}(>0)$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}$ are in a G.P., $a_{2}+a_{4}=2a_{3}+1$ and $3a_{2}+a_{3}=2a_{4}$, then $a_{2}+a_{4}+2a_{5}$ is equal to ____.

Accepted Answer

40

Question 28

The integral $\frac{24}{\pi}\int_{0}^{\sqrt{2}}\frac{\left(2-x^{2}\right)dx}{\left(2+x^{2}\right)\sqrt{4+x^{4}}}$ is equal to ____.

Accepted Answer

3

Question 29

Let a line $L_{1}$ be tangent to the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{4}=1$ and let $L_{2}$ be the line passing through the origin and perpendicular to $L_{1}$. If the locus of the point of intersection of $L_{1}$ and $L_{2}$ is $(x^{2}+y^{2})^{2}=\alpha x^{2}+\beta y^{2}$, then $\alpha+\beta$ is equal to ______ .

Accepted Answer

12

Question 30

If the probability that a randomly chosen 6-digit number formed by using digits 1 and 8 only is a multiple of 21 is p, then 96 p is equal to ______ .

Accepted Answer

33

Question 31

The dimension of mutual inductance is :

Accepted Answer

(C) $[ML^2T^{-2}A^{-2}]$

Answer

(A) $[ML^2 T^{-2} A^{-1}]$

Answer

(B) $[ML^2T^{-3}A^{-1}]$

Answer

(D) $[ML^2T^{-3}A^{-2}]$

Question 32

In the arrangement shown in figure $a_1$,$a_2$, $a_3$ and $a_4$ are the accelerations of masses $m_1$,$m_2$,$m_3$ and $m_4$ respectively. Which of the following relation is true for this arrangement?

Accepted Answer

(A) $4a_1 + 2a_2 + a_3 + a_4 = 0$

Answer

(B) $a_1 + 4a_2 + 3a_3 + a_4 = 0$

Answer

(C) $a_1 + 4a_2 + 3a_3 + 2a_4 = 0$

Answer

(D) $2a_1 + 2a_2 + 3a_3 + a_4 = 0$

Question 33

Arrange the four graphs in descending order of total work done; where $W_1$, $W_2$, $W_3$ and $W_4$ are the work done corresponding to figure a, b, c and d respectively.

Accepted Answer

(A) $W_3 > W_2 > W_1 > W_4$

Answer

(B) $W_3 > W_2 > W_4 > W_1$

Answer

(C) $W_2 > W_3 > W_4 > W_1$

Answer

(D) $W_2 > W_3 > W_1 > W_4$

Question 34

Solid spherical ball is rolling on a frictionless horizontal plane surface about its axis of symmetry. The ratio of rotational kinetic energy of the ball to its total kinetic energy is :-

Accepted Answer

(B) $\frac{2}{7}$

Answer

(A) $\frac{2}{5}$

Answer

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

Answer

(D) $\frac{7}{10}$

Question 35

Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : If we move from poles to equator, the direction of acceleration due to gravity of earth always points towards the center of earth without any variation in its magnitude.
Reason R : At equator, the direction of acceleration due to the gravity is towards the center of earth.
In the light of above statements, choose the correct answer from the options given below :

Accepted Answer

(D) A is false but R is true

Answer

(A) Both A and R are true and R is the correct explanation of A.

Answer

(B) Both A and R are true but R is NOT the correct explanation of A.

Answer

(C) A is true but R is false

Question 36

If ρ is the density and η is coefficient of viscosity of fluid which flows with a speed v in the pipe of diameter d, the correct formula for Reynolds number $R_e$ is :

Accepted Answer

(C) $R_e = \frac{\rho v d}{\eta}$

Answer

(A) $R_e = \frac{\eta d}{\rho v}$

Answer

(B) $R_e = \frac{\rho v}{\eta d}$

Answer

(D) $R_e = \frac{\eta}{\rho v d}$

Question 37

A flask contains argon and oxygen in the ratio of 3:2 in mass and the mixture is kept at 27°C. The ratio of their average kinetic energy per molecule respectively will be :

Accepted Answer

(D) 1 : 1

Answer

(A) 3 : 2

Answer

(B) 9 : 4

Answer

(C) 2 : 3

Question 38

The charge on capacitor of capacitance 15μF in the figure given below is :

Accepted Answer

(A) 60μc

Answer

(B) 130μc

Answer

(C) 260 μc

Answer

(D) 585 μc

Question 39

A parallel plate capacitor with plate area A and plate separation d=2 m has a capacitance of 4 μF. The new capacitance of the system if half of the space between them is filled with a dielectric material of dielectric constant K=3 (as shown in figure) will be :

Accepted Answer

(C) 6μF

Answer

(A) 2μF

Answer

(B) 32μF

Answer

(D) 8μF

Question 40

Sixty four conducting drops each of radius 0.02 m and each carrying a charge of 5 μC are combined to form a bigger drop. The ratio of surface density of bigger drop to the smaller drop will be :

Accepted Answer

(B) 4 : 1

Answer

(A) 1 : 4

Answer

(C) 1 : 8

Answer

(D) 8 : 1

Question 41

The equivalent resistance between points A and B in the given network is :

Accepted Answer

(C) 5Ω

Answer

(A) 65Ω

Answer

(B) 20Ω

Answer

(D) 2Ω

Question 42

A bar magnet having a magnetic moment of $2.0 	imes 10^5$ JT$^{-1}$, is placed along the direction of uniform magnetic field of magnitude B= $14 	imes 10^{-5}$ T. The work done in rotating the magnet slowly through 60° from the direction of field is :

Accepted Answer

(A) 14 J

Answer

(B) 8.4 J

Answer

(C) 4 J

Answer

(D) 1.4 J

Question 43

Two coils of self inductance $L_1$ and $L_2$ are connected in series combination having mutual inductance of the coils as M. The equivalent self inductance of the combination will be :

Accepted Answer

(D) $L_1 + L_2 - 2M$

Answer

(A) $\frac{1}{L_1} + \frac{1}{L_2} + \frac{1}{M}$

Answer

(B) $L_1 + L_2 + M$

Answer

(C) $L_1 + L_2 + 2M$

Question 44

A metallic conductor of length 1m rotates in a vertical plane parallel to east-west direction about one of its end with angular velocity 5 rad/s. If the horizontal component of earth's magnetic field is $0.2 	imes 10^{-4}$ T, then emf induced between the two ends of the conductor is :

Accepted Answer

(B) 50μV

Answer

(A) 5μV

Answer

(C) 5mV

Answer

(D) 50mV

Question 45

Which is the correct ascending order of wavelengths?

Accepted Answer

(B) $\lambda_{gamma-ray} < \lambda_{X-ray} < \lambda_{visible} < \lambda_{microwave}$

Answer

(A) $\lambda_{visible} < \lambda_{X-ray} < \lambda_{gamma-ray} < \lambda_{microwave}$

Answer

(C) $\lambda_{X-ray} < \lambda_{gamma-ray} < \lambda_{visible} < \lambda_{microwave}$

Answer

(D) $\lambda_{microwave} < \lambda_{visible} < \lambda_{gamma-ray} < \lambda_{X-ray}$

Question 46

For a specific wavelength 670 nm of light coming from a galaxy moving with velocity v, the observed wavelength is 670.7 nm.
The value of v is :

Accepted Answer

(C) 3.13 × $10^{5}$ ms$^{-1}$

Answer

(A) 3 × $10^{8}$ ms$^{-1}$

Answer

(B) 3 × $10^{10}$ ms$^{-1}$

Answer

(D) 4.48 × $10^{5}$ ms$^{-1}$

Question 47

A metal surface is illuminated by a radiation of wavelength 4500 Å. The ejected photo-electron enters a constant magnetic field of 2 mT making an angle of 90° with the magnetic field. If it starts revolving in a circular path of radius 2 mm, the work function of the metal is approximately :

Accepted Answer

(A) 1.36 eV

Answer

(B) 1.69 eV

Answer

(C) 2.78 eV

Answer

(D) 2.23 eV

Question 48

A radioactive nucleus can decay by two different processes. Half-life for the first process is 3.0 hours while it is 4.5 hours for the second process. The effective half- life of the nucleus will be :

Accepted Answer

(D) 1.80 hours

Answer

(A) 3.75 hours

Answer

(B) 0.56 hours

Answer

(C) 0.26 hours

Question 49

The positive feedback is required by an amplifier to act an oscillator. The feedback here means :

Accepted Answer

(B) A portion of the output power is returned back to the input.

Answer

(A) External input is necessary to sustain ac signal in output.

Answer

(C) Feedback can be achieved by LR network.

Answer

(D) The base-collector junction must be forward biased.

Question 50

A sinusoidal wave y(t) = 40sin(10 x $10^{6}$ πt) is amplitude modulated by another sinusoidal wave x(t) = 20sin (1000πt). The amplitude of minimum frequency component of modulated signal is :

Accepted Answer

(D) 10

Answer

(A) 0.5

Answer

(B) 0.25

Answer

(C) 20

Question 51

A ball is projected vertically upward with an initial velocity of 50 ms$^{-1}$ at t = 0s. At t = 2s. another ball is projected vertically upward with same velocity. At t =________s, second ball will meet the first ball (g =10 ms$^{-2}$).

Accepted Answer

6

Question 52

A batsman hits back a ball of mass 0.4 kg straight in the direction of the bowler without changing its initial speed of 15 ms$^{-1}$. The impulse imparted to the ball is ________________Ns.

Accepted Answer

12

Question 53

A system to 10 balls each of mass 2 kg are connected via massless and unstretchable string. The system is allowed to slip over the edge of a smooth table as shown in figure. Tension on the string between the $7^{th}$ and $8^{th}$ ball is______________N when $6^{th}$ ball just leaves the table.

Accepted Answer

36

Question 54

A geyser heats water flowing at a rate of 2.0 kg per minute from 30°C to 70°C. If geyser operates on a gas burner, the rate of combustion of fuel will be _____________g min$^{-1}$
[Heat of combustion = 8 × $10^{3}$ Jg$^{-1}$
Specific heat of water = 4.2 Jg$^{-1}$ °C$^{-1}$]

Accepted Answer

42

Question 55

A heat engine operates with the cold reservoir at temperature 324 K.
The minimum temperature of the hot reservoir, if the heat engine takes 300 J heat from the hot reservoir and delivers 180 J heat to the cold reservoir per cycle, is ____________K.

Accepted Answer

540

Question 56

A set of 20 tuning forks is arranged in a series of increasing frequencies. If each fork gives 4 beats with respect to the preceding fork and the frequency of the last fork is twice the frequency of the first, then the frequency of last fork is______Hz.

Accepted Answer

152

Question 57

Two 10 cm long, straight wires, each carrying a current of 5A are kept parallel to each other. If each wire experienced a force of $10^{-5}$ N, then separation between the wires is _________ cm.

Accepted Answer

5

Question 58

A small bulb is placed at the bottom of a tank containing water to a depth of $\sqrt{7}$ m. The refractive index of water is $\frac{4}{3}$. The area of the surface of water through which light from the bulb can emerge out is xπ m$^{2}$. The value of x is________.

Accepted Answer

9

Question 59

A travelling microscope is used to determine the refractive index of a glass slab. If 40 divisions are there in 1 cm on main scale and 50 Vernier scale divisions are equal to 49 main scale divisions, then least count of the travelling microscope is ______×$10^{-6}$ m.

Accepted Answer

5

Question 60

The stopping potential for photoelectrons emitted from a surface illuminated by light of wavelength 6630 Å is 0.42 V. If the threshold frequency is x × $10^{13}$/s, where x is _______ (nearest integer).
(Given, speed light = 3 × $10^{8}$ m/s, Planck's constant = 6.63 × $10^{-34}$ Js)

Accepted Answer

35

Question 61

The number of radial and angular nodes in 4d orbital are, respectively

Accepted Answer

(A) 1 and 2

Answer

(B) 3 and 2

Answer

(C) 1 and 0

Answer

(D) 2 and 1

Question 62

Match List I with List II.
Choose the most appropriate answer from the options given below :

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 63

Which of the following elements in considered as a metalloid?

Accepted Answer

(D) Te

Answer

(A) Sc

Answer

(B) Pb

Answer

(C) Bi

Question 64

The role of depressants in Froth Flotation method is to

Accepted Answer

(A) selectively prevent one component of the ore from coming to the froth.

Answer

(B) reduce the consumption of oil for froth formation.

Answer

(C) stabilize the froth.

Answer

(D) enhance non-wettability of the mineral particles.

List I (Enzyme)	List II (Conversion of)
A.Invertase	I.Starch into maltose
B.Zymase	II.Maltose into glucose
C.Diastase	III.Glucose into ethanol
D.Maltase	IV.Cane sugar into glucose

JEE Main 26 June 2022 Shift 2 Question Paper with Answers

Physics — JEE Main 26 June 2022 Shift 2

Chemistry — JEE Main 26 June 2022 Shift 2

Mathematics — JEE Main 26 June 2022 Shift 2

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