JEE Main 24 June 2022 Shift 2 Question Paper

Question 1

Let $x*y = x^{2} + y^{3}$ and $(x*1)*1 = x*(1*1)$. Then a value of $2\sin^{-1}\left(\dfrac{x^{4}+x^{2}-2}{x^{4}+x^{2}+2}\right)$ is

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 2

The sum of all the real roots of the equation $(e^{2x} - 4)(6e^{2x} - 5e^{x} + 1) = 0$ is

Accepted Answer

(B) $-\log_e 3$

Answer

(A) $\log_e 3$

Answer

(C) $\log_e 6$

Answer

(D) $-\log_e 6$

Question 3

Let the system of linear equations
$x + y + \alpha z = 2$
$3x + y + z = 4$
$x + 2z = 1$
have a unique solution $(x^*, y^*, z^*)$. If $(\alpha, x^*)$, $(y^*, \alpha)$ and $(x^*, -y^*)$ are collinear points, then the sum of absolute values of all possible values of $\alpha$ is :

Accepted Answer

(C) 2

Answer

(A) 4

Answer

(B) 3

Answer

(D) 1

Question 4

Let $x, y > 0$. If $x^{3}y^{2} = 2^{15}$, then the least value of $3x + 2y$ is

Accepted Answer

(D) 40

Answer

(A) 30

Answer

(B) 32

Answer

(C) 36

Question 5

Let $f(x) = \begin{cases} \dfrac{\sin\left(x-[x]ight)}{x-[x]} &,& x \in (-2,-1) \ \max\left\{2x, 3\left[|x|ight]ight\} &,& |x| < 1 \ 1 &,& 	ext{otherwise} \end{cases}$ where $[t]$ denotes greatest integer $\le t$. If m is the number of points where $f$ is not continuous and n is the number of points where $f$ is not differentiable, then the ordered pair (m, n) is :

Accepted Answer

(C) (2, 3)

Answer

(A) (3, 3)

Answer

(B) (2, 4)

Answer

(D) (3, 4)

Question 6

The value of the integral $\displaystyle\int_{-\pi/2}^{\pi/2} \dfrac{dx}{\left(1+e^{x}\right)\left(\sin^{6} x+\cos^{6} x\right)}$ is equal to

Accepted Answer

(C) $\pi$

Answer

(A) $2\pi$

Answer

(B) $0$

Answer

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

Question 7

$\lim_{n	o\infty}\left(\dfrac{n^{2}}{\left(n^{2}+1ight)\left(n+1ight)} + \dfrac{n^{2}}{\left(n^{2}+4ight)\left(n+2ight)} + \dfrac{n^{2}}{\left(n^{2}+9ight)\left(n+3ight)} + ... + \dfrac{n^{2}}{\left(n^{2}+n^{2}ight)\left(n+night)}ight)$ is equal to

Accepted Answer

(A) $\dfrac{\pi}{8}+\dfrac{1}{4}\log_e 2$

Answer

(B) $\dfrac{\pi}{4}+\dfrac{1}{8}\log_e 2$

Answer

(C) $\dfrac{\pi}{4}-\dfrac{1}{8}\log_e 2$

Answer

(D) $\dfrac{\pi}{8}+\log_e \sqrt{2}$

Question 8

A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with

Accepted Answer

(A) length of latus rectum 3

Answer

(B) length of latus rectum 6

Answer

(C) focus $\left(\dfrac{4}{3},0\right)$

Answer

(D) focus $\left(0,\dfrac{3}{4}\right)$

Question 9

Let the maximum area of the triangle that can be inscribed in the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{4}=1$, $a > 2$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the y-axis, be $6\sqrt{3}$. Then the eccentricity of the ellispe is :

Accepted Answer

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

Answer

(B) $\dfrac{1}{2}$

Answer

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

Answer

(D) $\dfrac{\sqrt{3}}{4}$

Question 10

Let the area of the triangle with vertices $A(1, \alpha)$, $B(\alpha, 0)$ and $C(0, \alpha)$ be 4 sq. units. If the point $(\alpha, -\alpha)$, $(-\alpha, \alpha)$ and $(\alpha^{2}, \beta)$ are collinear, then $\beta$ is equal to

Accepted Answer

(C) -64

Answer

(A) 64

Answer

(B) -8

Answer

(D) 512

Question 11

The number of distinct real roots of the equation $x^{7} - 7x - 2 = 0$ is

Accepted Answer

(D) 3

Answer

(A) 5

Answer

(B) 7

Answer

(C) 1

Question 12

The number of solutions of the equation $\cos\left(x+\dfrac{\pi}{3}\right)\cos\left(\dfrac{\pi}{3}-x\right)=\dfrac{1}{4}\cos^2 2x$, $x \in [-3\pi, 3\pi]$ is :

Accepted Answer

(D) 7

Answer

(A) 8

Answer

(B) 5

Answer

(C) 6

Question 13

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

Accepted Answer

(A) 16

Answer

(B) 6

Answer

(C) 12

Answer

(D) 15

Question 14

Let the points on the plane P be equidistant from the points $(-4, 2, 1)$ and $(2, -2, 3)$. Then the acute angle between the plane P and the plane $2x + y + 3z = 1$ is

Accepted Answer

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

Answer

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

Answer

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

Answer

(D) $\dfrac{5\pi}{12}$

Question 15

Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that $\left|\left(\hat{a}+\hat{b}ight)+2\left(\hat{a}	imes\hat{b}ight)ight|=2$. If $	heta \in (0, \pi)$ is the angle between $\hat{a}$ and $\hat{b}$, then among the statements :
(S1) : $2\left|\hat{a}	imes\hat{b}ight|=\left|\hat{a}-\hat{b}ight|$
(S2) : The projection of $\hat{a}$ on $\left(\hat{a}+\hat{b}ight)$ is $\dfrac{1}{2}$

Accepted Answer

(C) Both (S1) and (S2) are true

Answer

(A) Only (S1) is true

Answer

(B) Only (S2) is true

Answer

(D) Both (S1) and (S2) are false

Question 16

If $y = 	an^{-1}(\sec x^3 - 	an x^3)$. $\dfrac{\pi}{2} < x^3 < \dfrac{3\pi}{2}$, then

Accepted Answer

(B) $x^2 y'' - 6y + \dfrac{3\pi}{2} = 0$

Answer

(A) $xy'' + 2y' = 0$

Answer

(C) $x^2 y'' - 6y + 3\pi = 0$

Answer

(D) $xy'' - 4y' = 0$

Question 17

Consider the following statements :
A : Rishi is a judge.
B : Rishi is honest.
C : Rishi is not arrogant.
The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" is

Accepted Answer

(B) $(\sim B) \wedge (A \wedge C)$

Answer

(A) $B 	o (A \vee C)$

Answer

(C) $B 	o ((\sim A) \vee (\sim C))$

Answer

(D) $B 	o (A \wedge C)$

Question 18

The slope of normal at any point $(x, y)$, $x > 0$, $y > 0$ on the curve $y = y(x)$ is given by $\dfrac{x^2}{xy - x^2 y^2 - 1}$. If the curve passes through the point $(1, 1)$, then $e \cdot y(e)$ is equal to

Accepted Answer

(D) $\dfrac{1+	an(1)}{1-	an(1)}$

Answer

(A) $\dfrac{1-	an(1)}{1+	an(1)}$

Answer

(B) $	an(1)$

Answer

(C) $1$

Question 19

Let $\lambda^*$ be the largest value of $\lambda$ for which the function $f_{\lambda}(x) = 4\lambda x^3 - 36\lambda x^2 + 36x + 48$ is increasing for all $x \in R$. Then $f_{\lambda}*(1) + f_{\lambda}*(-1)$ is equal to :

Accepted Answer

(D) 72

Answer

(A) 36

Answer

(B) 48

Answer

(C) 64

Question 20

Let $S = \left\{z \in \mathbb{C} : |z-3| \le 1 	ext{ and } z(4+3i) + \bar{z}(4-3i) \le 24ight\}$. If $\alpha + i\beta$ is the point in S which is closest to $4i$, then $25(\alpha + \beta)$ is equal to ______.

Accepted Answer

80

Question 21

Let $S = \left\{\begin{pmatrix} -1 & a \ 0 & b \end{pmatrix}; a, b \in \{1, 2, 3, \ldots 100\}ight\}$ and let $T_n = \{A \in S : A^{n(n+1)} = I\}$. Then the number of elements in $\bigcap_{n=1}^{100} T_n$ is ______.

Accepted Answer

100

Question 22

The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is ______.

Accepted Answer

576

Question 23

The sum of all the elements of the set $\{\alpha \in \{1, 2, \ldots, 100\} : \mathrm{HCF}\,(\alpha, 24) = 1\}$ is ______.

Accepted Answer

1633

Question 24

The remainder on dividing $1 + 3 + 3^{2} + 3^{3} + ... + 3^{2021}$ by 50 is ______.

Accepted Answer

4

Question 25

The area (in sq. units) of the region enclosed between the parabola $y^{2} = 2x$ and the line $x + y = 4$ is ______.

Accepted Answer

18

Question 26

Let a circle $C : (x - h)^{2} + (y - k)^{2} = r^{2}$, $k > 0$, touch the x-axis at $(1, 0)$. If the line $x + y = 0$ intersects the circle C at P and Q such that the length of the chord PQ is 2, then the value of $h + k + r$ is equal to ______.

Accepted Answer

7

Question 27

In an examination, there are 10 true-false type questions. Out of 10, a student can guess the answer of 4 questions correctly with probability $\dfrac{3}{4}$ and the remaining 6 questions correctly with probability $\dfrac{1}{4}$. If the probability that the student guesses the answers of exactly 8 questions correctly out of 10 is $\dfrac{27k}{4^{10}}$, then k is equal to ______.

Accepted Answer

479

Question 28

Let the hyperbola $H : \dfrac{x^{2}}{a^{2}} - y^{2} = 1$ and the ellipse $E : 3x^{2} + 4y^{2} = 12$ be such that the length of latus rectum of H is equal to the length of latus rectum of E. If $e_{H}$ and $e_{E}$ are the eccentricities of H and E respectively, then the value of $12\left(e_{H}^{2} + e_{E}^{2}\right)$ is equal to ______.

Accepted Answer

42

Question 29

Let $P_{1}$ be a parabola with vertex $(3, 2)$ and focus $(4, 4)$ and $P_{2}$ be its mirror image with respect to the line $x + 2y = 6$. Then the directrix of $P_{2}$ is $x + 2y = $ ______.

Accepted Answer

10

Question 30

Identify the pair of physical quantities that have same dimensions :

Accepted Answer

(A) velocity gradient and decay constant

Answer

(B) wien's constant and Stefan constant

Answer

(C) angular frequency and angular momentum

Answer

(D) wave number and Avogadro number

Question 31

The distance between Sun and Earth is R. The duration of year if the distance between Sun and Earth becomes 3R will be :

Accepted Answer

(D) $3\sqrt{3}$ years

Answer

(A) $\sqrt{3}$ years

Answer

(B) 3 years

Answer

(C) 9 years

Question 32

A stone of mass m, tied to a string is being whirled in a vertical circle with a uniform speed. The tension in the string is :

Accepted Answer

(B) minimum at the highest position of the circular path

Answer

(A) the same throughout the motion

Answer

(C) minimum at the lowest position of the circular path

Answer

(D) minimum when the rope is in the horizontal position

Question 33

Two identical charged particles each having a mass 10 g and charge $2.0	imes10^{-7}$ C area placed on a horizontal table with a separation of L between then such that they stay in limited equilibrium. If the coefficient of friction between each particle and the table is 0.25, find the value of L. [Use g = 10 ms$^{-2}$]

Accepted Answer

(A) 12 cm

Answer

(B) 10 cm

Answer

(C) 8 cm

Answer

(D) 5 cm

Question 34

A Carnot engine take 5000 kcal of heat from a reservoir at $727^{\circ}$C and gives heat to a sink at $127^{\circ}$C. The work done by the engine is :

Accepted Answer

(C) $12.6	imes10^{6}$ J

Answer

(A) $3	imes10^{6}$ J

Answer

(B) Zero

Answer

(D) $8.4	imes10^{6}$ J

Question 35

What will be the most suitable combination of three resistors A = $2\Omega$, B = $4\Omega$, C = $6\ \Omega$ so that $\left(\dfrac{22}{3}\right)\Omega$ is equivalent resistance of combination?

Accepted Answer

(B) Parallel combination of A and B connected in series with C.

Answer

(A) Parallel combination of A and C connected in series with B.

Answer

(C) Series combination of A and C connected in parallel with B.

Answer

(D) Series combination of B and C connected in parallel with A.

Question 36

The soft-iron is a suitable material for making an electromagnet. This is because soft-iron has :

Accepted Answer

(C) high permeability and low retentively

Answer

(A) low coercively and high retentively

Answer

(B) low coercively and low permeability

Answer

(D) high permeability and high retentively

Question 37

A proton, a deuteron and an $\alpha$-particle with same kinetic energy enter into a uniform magnetic field at right angle to magnetic field. The ratio of the radii of their respective circular paths is :

Accepted Answer

(D) $1:\sqrt{2}:1$

Answer

(A) $1:\sqrt{2}:\sqrt{2}$

Answer

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

Answer

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

Question 38

Given below are two statements :
Statement-I : The reactance of an ac circuit is zero. It is possible that the circuit contains a capacitor and an inductor.
Statement-II : In ac circuit, the average poser delivered by the source never becomes zero.
In the light of the above statements, choose the correct answer from the options given below :

Accepted Answer

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

Answer

(A) Both Statement I and Statement II are true.

Answer

(B) Both Statement I and Statement II are false.

Answer

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

Question 39

Potential energy as a function of r is given by $U = \dfrac{A}{r^{10}} - \dfrac{B}{r^{5}}$ , where r is the interatomic distance, A and B are positive constants. The equilibrium distance between the two atoms will be :

Accepted Answer

(C) $\left(\dfrac{2A}{B}\right)^{\frac{1}{5}}$

Answer

(A) $\left(\dfrac{A}{B}\right)^{\frac{1}{5}}$

Answer

(B) $\left(\dfrac{B}{A}\right)^{\frac{1}{5}}$

Answer

(D) $\left(\dfrac{B}{2A}\right)^{\frac{1}{5}}$

Question 40

An object of mass 5 kg is thrown vertically upwards from the ground. The air resistance produces a constant retarding force of 10 N throughout the motion. The ratio of time of ascent to the time of descent will be equal to : [Use g = 10 ms$^{-2}$]

Accepted Answer

(B) $\sqrt{2}:\sqrt{3}$

Answer

(A) 1 : 1

Answer

(C) $\sqrt{3}:\sqrt{2}$

Answer

(D) 2 : 3

Question 41

A fly wheel is accelerated uniformly from rest and rotates through 5 rad in the first second. The angle rotated by the fly wheel in the next second, will be :

Accepted Answer

(B) 15 rad

Answer

(A) 7.5 rad

Answer

(C) 20 rad

Answer

(D) 30 rad

Question 42

If the charge on a capacitor is increased by 2 C, the energy stored in it increases by 44%. The original charge on the capacitor is (in C) :

Accepted Answer

(A) 10

Answer

(B) 20

Answer

(C) 30

Answer

(D) 40

Question 43

A long cylindrical volume contains a uniformly distributed charge of density $\rho$. The radius of cylindrical volume is R. A charge particle (q) revolves around the cylinder in a circular path. The kinetic of the particle is :

Accepted Answer

(A) $\dfrac{\rho qR^{2}}{4\varepsilon_{0}}$

Answer

(B) $\dfrac{\rho qR^{2}}{2\varepsilon_{0}}$

Answer

(C) $\dfrac{q\rho}{4\varepsilon_{0}R^{2}}$

Answer

(D) $\dfrac{4\varepsilon_{0}R^{2}}{q\rho}$

Question 44

An electric bulb is rated as 200 W. What will be the peak magnetic field at 4 m distance produced by the radiations coming from this bulb? Consider this bulb as a point source with 3.5% efficiency.

Accepted Answer

(B) $1.71 	imes 10^{-8}$ T

Answer

(A) $1.19 	imes 10^{-8}$ T

Answer

(C) $0.84 	imes 10^{-8}$ T

Answer

(D) $3.36 	imes 10^{-8}$ T

Question 45

The light of two different frequencies whose photons have energies 3.8 eV and 1.4 eV respectively, illuminate a metallic surface whose work function is 0.6 eV successively. The ratio of maximum speeds of emitted electrons for the two frequencies respectivly will be :

Accepted Answer

(B) 2 : 1

Answer

(A) 1 : 1

Answer

(C) 4 : 1

Answer

(D) 1 : 4

Question 46

Two light beams of intensities in the ratio of 9 : 4 are allowed to interfere. The .ratio of the intensity of maxima and minima will be :

Accepted Answer

(D) 25 : 1

Answer

(A) 2 : 3

Answer

(B) 16 : 81

Answer

(C) 25 : 169

Question 47

In Bohr's atomic model of hydrogen, let K. P and E are the kinetic energy, potential energy and total energy of the electron respectively. Choose the correct option when the electron undergoes transitions to a higher level :

Accepted Answer

(B) K decreases. P and E increase.

Answer

(A) All K. P and E increase.

Answer

(C) P decreases. K and E increase.

Answer

(D) K increases. P and E decrease.

Question 48

A body is projected from the ground at an angle of $45^\circ$ with the horizontal. Its velocity after 2s is $20$ ms$^{-1}$. The maximum height reached by the body during its motion is ________m. (use g $= 10$ms$^{-2}$)

Accepted Answer

20

Question 49

An antenna is placed in a dielectric medium of dielectric constant 6.25. If the maximum size of that antenna is 5.0 mm. it can radiate a signal of minimum frequency of ______GHz. (Given $\mu_{r} = 1$ for dielectric medium)

Accepted Answer

6

Question 50

A potentiometer wire of length 10 m and resistance $20\ \Omega$ is connected in series with a 25 V battery and an external resistance $30\ \Omega$ . A cell of emf E in secondary circuit is balanced by 250 cm long potentiometer wire. The value of E (in volt) is $\dfrac{x}{10}$ . The value of x is ________.

Accepted Answer

25

Question 51

Two travelling waves of equal amplitudes and equal frequencies move in opposite directions along a string. They interfere to produce a stationary wave whose equation is given by $y = (10 \cos \pi x \sin \dfrac{2\pi t}{T})$ cm The amplitude of the particle at $x = \dfrac{4}{3}$ cm will be ________ cm.

Accepted Answer

5

Question 52

In the given circuit- the value of current I$_L$ will be ______ mA.
(When R$_L$ = 1 k$\Omega$)

Accepted Answer

5

Question 53

A sample contains 10$^{-2}$ kg each of two substances A and B with half lives 4 s and 8 s respectively. The ratio of then atomic weights is 1 : 2. The ratio of the amounts of A and B after 16 s is $\dfrac{x}{100}$. the value of x is ______.

Accepted Answer

25

Question 54

A ray of ligh is incident at an angle of incidence $60^\circ$ on the glass slab of refractive index $\sqrt{3}$. After refraction, the light ray emerges out from other parallel faces and lateral shift between incident ray and emergent ray is $4\sqrt{3}$ cm. The thickness of the glass slab is ______ cm.

Accepted Answer

12

Question 55

A circular coil of 1000 turns each with area 1m$^2$ is rotated about its vertical diameter at the rate of one revolution per second in a uniform horizontal magnetic field of 0.07T. The maximum voltage generation will be ______V.

Accepted Answer

440

Question 56

A monoatomic gas performs a work of $\dfrac{Q}{4}$ where Q is the heat supplied to it. The molar heat capaticy of the gas will be ______R during this transformation.
Where R is the gas constant.

Accepted Answer

2

Question 57

In an experment ot verify Newton's law of cooling, a graph is plotted between, the temperature difference ($\Delta$T) of the water and surroundings and time as shown in figure. The initial temperature of water is taken as $80^\circ$C. The value of t$_2$ as mentioned in the graph will be ______.

Accepted Answer

16

Question 58

120 of an organic compound that contains only carbon and hydrogen gives 330g of CO$_2$ and 270g of water on complete combustion. The percentage of carbon and hydrogen, respectively are.

Accepted Answer

(D) 75 and 25

Answer

(A) 25 and 75

Answer

(B) 40 and 60

Answer

(C) 60 and 40

Question 59

The energy of one mole of photons of radiation of wavelength 300 nm is
(Given : h = $6.63 	imes 10^{-34}$ Js, N$_A$ = $6.02 	imes 10^{23}$ mol$^{-1}$, c = $3 	imes 10^{8}$ ms$^{-1}$)

Accepted Answer

(C) 399 kJ mol$^{-1}$

Answer

(A) 235 kJ mol$^{-1}$

Answer

(B) 325 kJ mol$^{-1}$

Answer

(D) 435 kJ mol$^{-1}$

Question 60

The correct order of bound orders of C$_2^{2-}$, N$_2^{2-}$ and O$_2^{2-}$ is, respectively.

Accepted Answer

(B) O$_2^{2-}$ < N$_2^{2-}$ < C$_2^{2-}$

Answer

(A) C$_2^{2-}$ < N$_2^{2-}$ < O$_2^{2-}$

Answer

(C) C$_2^{2-}$ < O$_2^{2-}$ < N$_2^{2-}$

Answer

(D) N$_2^{2-}$ < C$_2^{2-}$ < O$_2^{2-}$

Question 61

At $25^\circ$C and 1 atm pressure, the enthalpies of combustion are as given below:
The enthalpy of formation of ethane is

Accepted Answer

(C) $-$86.0 kJ mol$^{-1}$

Answer

(A) +54.0 kJ mol$^{-1}$

Answer

(B) $-$68.0 kJ mol$^{-1}$

Answer

(D) +97.0 kJ mol$^{-1}$

Question 62

For a first order reaction, the time required for completion of 90% reaction is 'x' times the half life of the reaction. The value of 'x' is
(Given: ln 10 = 2.303 and log 2 = 0.3010)

Accepted Answer

(C) 3.32

Answer

(A) 1.12

Answer

(B) 2.43

Answer

(D) 33.31

Question 63

Metals generally melt at very high temperature. Amongst the following, the metal with the highest melting point will be

Accepted Answer

(B) Ag

Answer

(A) Hg

Answer

(C) Ga

Answer

(D) Cs

Question 64

Which of the following chemical reactions represents Hall-Heroult Process?

Accepted Answer

(B) 2Al$_2$O$_3$ + 3C $\rightarrow$ 4Al + 3CO$_2$

Answer

(A) Cr$_2$O$_3$ + 2Al $\rightarrow$ Al$_2$O$_3$ + 2Cr

Answer

(C) FeO + CO $\rightarrow$ Fe + CO$_2$

Answer

(D) 2[Au(CN)$_2$]$^{-}_{(aq)}$ + Zn(s) $\rightarrow$ 2Au(s) + [Zn(CN)$_4$]$^{2-}$

JEE Main 24 June 2022 Shift 2 Question Paper with Answers

Physics — JEE Main 24 June 2022 Shift 2

Chemistry — JEE Main 24 June 2022 Shift 2

Mathematics — JEE Main 24 June 2022 Shift 2

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