JEE Main 17 March 2021 Shift 1 Question Paper

Question 1

Let $\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$ and $\vec{b} = 7\hat{i} + \hat{j} - 6\hat{k}$. If $\vec{r} 	imes \vec{a} = \vec{r} 	imes \vec{b}, \vec{r} \cdot \left(\hat{i} + 2\hat{j} + \hat{k}ight) = -3$, then $\vec{r} \cdot \left(2\hat{i} - 3\hat{j} + \hat{k}ight)$ is equal to :

Accepted Answer

(A) 12

Answer

(B) 8

Answer

(C) 13

Answer

(D) 10

Question 2

The system of equations $kx + y + z = 1$, $x + ky + z = k$ and $x + y + zk = k^2$ has no solution if k is equal to :

Accepted Answer

(D) $-2$

Answer

(A) 0

Answer

(B) 1

Answer

(C) $-1$

Question 3

If $\cot^{-1}(\alpha) = \cot^{-1} 2 + \cot^{-1} 8 + \cot^{-1} 18 + \cot^{-1} 32 + .....$ upto 100 terms, then $\alpha$ is :

Accepted Answer

(A) 1.01

Answer

(B) 1.00

Answer

(C) 1.02

Answer

(D) 1.03

Question 4

The equation of the plane which contains the y-axis and passes through the point $(1, 2, 3)$ is :

Accepted Answer

(D) $3x - z = 0$

Answer

(A) $x + 3z = 10$

Answer

(B) $x + 3z = 0$

Answer

(C) $3x + z = 6$

Question 5

If $A = \begin{pmatrix} 0 & \sin\alpha \ \sin\alpha & 0 \end{pmatrix}$ and $\det\left(A^2 - \frac{1}{2}Iight) = 0$, then a possible value of $\alpha$ is

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 6

If the Boolean expression $(p \Rightarrow q) \Leftrightarrow (q * (\sim p))$ is a tautology, then the Boolean expression $p * (\sim q)$ is equivalent to :

Accepted Answer

(A) $q \Rightarrow p$

Answer

(B) $\sim q \Rightarrow p$

Answer

(C) $p \Rightarrow \sim q$

Answer

(D) $p \Rightarrow q$

Question 7

Two dices are rolled. If both dices have six faces numbered 1,2,3,5,7 and 11, then the probability that the sum of the numbers on the top faces is less than or equal to 8 is :

Accepted Answer

(B) $\frac{17}{36}$

Answer

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

Answer

(C) $\frac{5}{12}$

Answer

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

Question 8

If the fourth term in the expansion of $\left(x + x^{\log_2 x}\right)^7$ is 4480, then the value of x where $x \in N$ is equal to :

Accepted Answer

(A) 2

Answer

(B) 4

Answer

(C) 3

Answer

(D) 1

Question 9

In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement?

Accepted Answer

(C) None of these

Answer

(A) P and Q

Answer

(B) P and R

Answer

(D) Q and R

Question 10

The sum of possible values of x for $	an^{-1}(x + 1) + \cot^{-1}\left(\frac{1}{x - 1}ight) = 	an^{-1}\left(\frac{8}{31}ight)$ is :

Accepted Answer

(A) $-\frac{32}{4}$

Answer

(B) $-\frac{31}{4}$

Answer

(C) $-\frac{30}{4}$

Answer

(D) $-\frac{33}{4}$

Question 11

The area of the triangle with vertices $A(z)$, $B(iz)$ and $C(z + iz)$ is :

Accepted Answer

(B) $\frac{1}{2}|z|^2$

Answer

(A) 1

Answer

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

Answer

(D) $\frac{1}{2}|z + iz|^2$

Question 12

The line $2x - y + 1 = 0$ is a tangent to the circle at the point $(2, 5)$ and the centre of the circle lies on $x - 2y = 4$. Then, the radius of the circle is:

Accepted Answer

(A) $3\sqrt{5}$

Answer

(B) $5\sqrt{3}$

Answer

(C) $5\sqrt{4}$

Answer

(D) $4\sqrt{5}$

Question 13

Team 'A' consists of 7 boys and n girls and Team 'B' has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then n is equal to :

Accepted Answer

(C) 4

Answer

(A) 5

Answer

(B) 2

Answer

(D) 6

Question 14

The value of $4 + \cfrac{1}{5 + \cfrac{1}{4 + \cfrac{1}{5 + \cfrac{1}{4 + .......\infty}}}}$ is :

Accepted Answer

(A) $2 + \frac{2}{5}\sqrt{30}$

Answer

(B) $2 + \frac{4}{\sqrt{5}}\sqrt{30}$

Answer

(C) $4 + \frac{4}{\sqrt{5}}\sqrt{30}$

Answer

(D) $5 + \frac{2}{5}\sqrt{30}$

Question 15

Choose the incorrect statement about the two circles whose equations are given below :
$x^2 + y^2 - 10x - 10y + 41 = 0$ and
$x^2 + y^2 - 16x - 10y + 80 = 0$

Accepted Answer

(B) Both circles' centres lie inside region of one another.

Answer

(A) Distance between two centres is the average of radii of both the circles.

Answer

(C) Both circles pass through the centre of each other.

Answer

(D) Circles have two intersection points.

Question 16

Which of the following is true for $y(x)$ that satisfies the differential equation $\frac{dy}{dx} = xy - 1 + x - y$ ; $y(0) = 0$ :

Accepted Answer

(A) $y(1) = e^{-\frac{1}{2}} - 1$

Answer

(B) $y(1) = e^{\frac{1}{2}} - e^{-\frac{1}{2}}$

Answer

(C) $y(1) = 1$

Answer

(D) $y(1) = e^{\frac{1}{2}} - 1$

Question 17

The value of $\lim_{x 	o 0^{+}} \frac{\cos^{-1}\left(x - [x]^2ight) \cdot \sin^{-1}\left(x - [x]^2ight)}{x - x^3}$, where $[x]$ denotes the greatest integer $\leq x$ is :

Accepted Answer

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

Answer

(A) $\pi$

Answer

(B) 0

Answer

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

Question 18

The maximum value of z in the following equation $z = 6xy + y^2$, where $3x + 4y \leq 100$ and $4x + 3y \leq 75$ for $x \geq 0$ and $y \geq 0$ is _______ .

Accepted Answer

904

Question 19

If the function $f(x) = \frac{\cos(\sin x) - \cos x}{x^4}$ is continuous at each point in its domain and $f(0) = \frac{1}{k}$, then k is _________ .

Accepted Answer

6

Question 20

If $f(x) = \sin\left(\cos^{-1}\left(\frac{1 - 2^{2x}}{1 + 2^{2x}}\right)\right)$ and its first derivative with respect to x is $-\frac{b}{a}\log_e 2$ when $x = 1$, where a and b are integers, then the minimum value of $|a^2 - b^2|$ is _______ .

Accepted Answer

481

Question 21

Let there be three independent events $E_1$, $E_2$ and $E_3$. The probability that only $E_1$ occurs is $\alpha$, only $E_2$ occurs is $\beta$ and only $E_3$ occurs is $\gamma$. Let 'p' denote the probability of none of events occurs that satisfies the equations $(\alpha - 2\beta)p = \alpha\beta$ and $(\beta - 3\gamma)p = 2\beta\gamma$. All the given probabilities are assumed to lie in the interval (0, 1).
Then, $\frac{	ext{Probability of occurrence of } E_1}{	ext{Probability of occurrence of } E_3}$ is equal to _____.

Accepted Answer

6

Question 22

If $\vec{a} = \alpha\hat{i} + \beta\hat{j} + 3\hat{k}$, $\vec{b} = -\beta\hat{i} - \alpha\hat{j} - \hat{k}$ and $\vec{c} = \hat{i} - 2\hat{j} - \hat{k}$ such that $\vec{a} \cdot \vec{b} = 1$ and $\vec{b} \cdot \vec{c} = -3$, then $\frac{1}{3}\left(\left(\vec{a} 	imes \vec{b}ight) \cdot \vec{c}ight)$ is equal to _______.

Accepted Answer

2

Question 23

If $A = \begin{bmatrix} 2 & 3 \ 0 & -1 \end{bmatrix}$, then the value of $\det(A^4) + \det\left(A^{10} - (	ext{Adj}(2A))^{10}ight)$ is _________ .

Accepted Answer

16

Question 24

If $[\cdot]$ represents the greatest integer function, then the value of $\left| \int_{0}^{\sqrt{\frac{\pi}{2}}} \left[ \left[x^2\right] - \cos x \right] dx \right|$ is _________ .

Accepted Answer

1

Question 25

The minimum distance between any two points $P_1$ and $P_2$ while considering point $P_1$ on one circle and point $P_2$ on the other circle for the given circles' equations
$x^2 + y^2 - 10x - 10y + 41 = 0$
$x^2 + y^2 - 24x - 10y + 160 = 0$ is ______ .

Accepted Answer

1

Question 26

If the equation of the plane passing through the line of intersection of the planes $2x - 7y + 4z - 3 = 0$, $3x - 5y + 4z + 11 = 0$ and the point $(-2, 1, 3)$ is $ax + by + cz - 7 = 0$, then the value of $2a + b + c - 7$ is ______ .

Accepted Answer

4

Question 27

If $(2021)^{3762}$ is divided by 17, then the remainder is _______ .

Accepted Answer

4

Question 28

A triangular plate is shown. A force $\vec{F} = 4\hat{i} - 3\hat{j}$ is applied at point P. The torque at point P with respect to point 'O' and 'Q' are :

Accepted Answer

(A) $-15 - 20\sqrt{3}$ , $15 - 20\sqrt{3}$

Answer

(B) $15 + 20\sqrt{3}$ , $15 - 20\sqrt{3}$

Answer

(C) $15 - 20\sqrt{3}$ , $15 + 20\sqrt{3}$

Answer

(D) $-15 + 20\sqrt{3}$ , $15 + 20\sqrt{3}$

Question 29

When two soap bubbles of radii a and b (b > a) coalesce, the radius of curvature of common surface is :

Accepted Answer

(A) $\frac{ab}{b-a}$

Answer

(B) $\frac{a+b}{ab}$

Answer

(C) $\frac{b-a}{ab}$

Answer

(D) $\frac{ab}{a+b}$

Question 30

A polyatomic ideal gas has 24 vibrational modes. What is the value of $\gamma$ ?

Accepted Answer

(A) 1.03

Answer

(B) 1.30

Answer

(C) 1.37

Answer

(D) 10.3

Question 31

If an electron is moving in the $n^{th}$ orbit of the hydrogen atom, then its velocity $(v_n)$ for the $n^{th}$ orbit is given as :

Accepted Answer

(B) $v_n \propto \frac{1}{n}$

Answer

(A) $v_n \propto n$

Answer

(C) $v_n \propto n^2$

Answer

(D) $v_n \propto \frac{1}{n^2}$

Question 32

An electron of mass m and a photon have same energy E. The ratio of wavelength of electron to that of photon is : (c being the velocity of light)

Accepted Answer

(B) $\frac{1}{c}\left(\frac{E}{2m}\right)^{1/2}$

Answer

(A) $\frac{1}{c}\left(\frac{2m}{E}\right)^{1/2}$

Answer

(C) $\left(\frac{E}{2m}\right)^{1/2}$

Answer

(D) $c\,(2mE)^{1/2}$

Question 33

Two identical metal wires of thermal conductivities $K_1$ and $K_2$ respectively are connected in series. The effective thermal conductivity of the combination is :

Accepted Answer

(A) $\frac{2K_1 K_2}{K_1 + K_2}$

Answer

(B) $\frac{K_1 + K_2}{2K_1 K_2}$

Answer

(C) $\frac{K_1 + K_2}{K_1 K_2}$

Answer

(D) $\frac{K_1 K_2}{K_1 + K_2}$

Question 34

The vernier scale used for measurement has a positive zero error of 0.2 mm. If while taking a measurement it was noted that '0' on the vernier scale lies between 8.5 cm and 8.6 cm, vernier coincidence is 6, then the correct value of measurement is_____ cm.
(least count = 0.01 cm)

Accepted Answer

(B) 8.54 cm

Answer

(A) 8.36 cm

Answer

(C) 8.58 cm

Answer

(D) 8.56 cm

Question 35

An AC current is given by $I = I_1 \sin\omega t + I_2 \cos\omega t$. A hot wire ammeter will give a reading :

Accepted Answer

(B) $\sqrt{\frac{I_1^2 + I_2^2}{2}}$

Answer

(A) $\sqrt{\frac{I_1^2 - I_2^2}{2}}$

Answer

(C) $\frac{I_1 + I_2}{\sqrt{2}}$

Answer

(D) $\frac{I_1 + I_2}{2\sqrt{2}}$

Question 36

A modern grand-prix racing car of mass m is travelling on a flat track in a circular arc of radius R with a speed v. If the coefficient of static friction between the tyres and the track is $\mu_s$, then the magnitude of negative lift $F_L$ acting downwards on the car is :
(Assume forces on the four tyres are identical and g = acceleration due to gravity)

Accepted Answer

(B) $m\left(\frac{v^2}{\mu_s R} - g\right)$

Answer

(A) $m\left(\frac{v^2}{\mu_s R} + g\right)$

Answer

(C) $m\left(g - \frac{v^2}{\mu_s R}\right)$

Answer

(D) $-m\left(g + \frac{v^2}{\mu_s R}\right)$

Question 37

A car accelerates from rest at a constant rate $\alpha$ for some time after which it decelerates at a constant rate $\beta$ to come to rest. If the total time elapsed is t seconds, the total distance travelled is :

Accepted Answer

(C) $\frac{\alpha\beta}{2(\alpha+\beta)}t^2$

Answer

(A) $\frac{4\alpha\beta}{(\alpha+\beta)}t^2$

Answer

(B) $\frac{2\alpha\beta}{(\alpha+\beta)}t^2$

Answer

(D) $\frac{\alpha\beta}{4(\alpha+\beta)}t^2$

Question 38

A solenoid of 1000 turns per metre has a core with relative permeability 500. Insulated windings of the solenoid carry an electric current of 5A. The magnetic flux density produced by the solenoid is :
(permeability of free space = $4\pi 	imes 10^{-7}$ H/m)

Accepted Answer

(A) $\pi$T

Answer

(B) $2 	imes 10^{-3}\ \pi$T

Answer

(C) $\frac{\pi}{5}$ T

Answer

(D) $10^{-4}\pi$T

Question 39

A mass M hangs on a massless rod of length $l$ which rotates at a constant angular frequency. The mass M moves with steady speed in a circular path of constant radius. Assume that the system is in steady circular motion with constant angular velocity $\omega$. The angular momentum of M about point A is $L_A$ which lies in the positive z direction and the angular momentum of M about B is $L_B$. The correct statement for this system is :

Accepted Answer

(D) $L_A$ is constant, both in magnitude and direction

Answer

(A) $L_A$ and $L_B$ are both constant in magnitude and direction

Answer

(B) $L_B$ is constant in direction with varying magnitude

Answer

(C) $L_B$ is constant, both in magnitude and direction

Question 40

For what value of displacement the kinetic energy and potential energy of a simple harmonic oscillation become equal ?

Accepted Answer

(C) $x = \pm \frac{A}{\sqrt{2}}$

Answer

(A) $x = 0$

Answer

(B) $x = \pm A$

Answer

(D) $x = \frac{A}{2}$

Question 41

A Carnot's engine working between 400 K and 800 K has a work output of 1200 J per cycle. The amount of heat energy supplied to the engine from the source in each cycle is :

Accepted Answer

(D) 2400 J

Answer

(A) 3200 J

Answer

(B) 1800 J

Answer

(C) 1600 J

Question 42

The thickness at the centre of a plano convex lens is 3 mm and the diameter is 6 cm. If the speed of light in the material of the lens is $2 	imes 10^8$ $ms^{-1}$. The focal length of the lens is _____.

Accepted Answer

(D) 30 cm

Answer

(A) 0.30 cm

Answer

(B) 15 cm

Answer

(C) 1.5 cm

Question 43

The output of the given combination gates represents :

Accepted Answer

(B) NAND Gate

Answer

(A) XOR Gate

Answer

(C) AND Gate

Answer

(D) NOR Gate

Question 44

A boy is rolling a 0.5 kg ball on the frictionless floor with the speed of 20 $ms^{-1}$. The ball gets deflected by an obstacle on the way. After deflection it moves with 5% of its initial kinetic energy. What is the speed of the ball now ?

Accepted Answer

(B) 4.47 $ms^{-1}$

Answer

(A) 19.0 $ms^{-1}$

Answer

(C) 14.41 $ms^{-1}$

Answer

(D) 1.00 $ms^{-1}$

Question 45

Which level of the single ionized carbon has the same energy as the ground state energy of hydrogen atom?

Accepted Answer

(B) 6

Answer

(A) 1

Answer

(C) 4

Answer

(D) 8

Question 46

Two ideal polyatomic gases at temperatures $T_1$ and $T_2$ are mixed so that there is no loss of energy. If $F_1$ and $F_2$, $m_1$ and $m_2$, $n_1$ and $n_2$ be the degrees of freedom, masses, number of molecules of the first and second gas respectively, the temperature of mixture of these two gases is :

Accepted Answer

(B) $\frac{n_1 F_1 T_1 + n_2 F_2 T_2}{n_1 F_1 + n_2 F_2}$

Answer

(A) $\frac{n_1 T_1 + n_2 T_2}{n_1 + n_2}$

Answer

(C) $\frac{n_1 F_1 T_1 + n_2 F_2 T_2}{F_1 + F_2}$

Answer

(D) $\frac{n_1 F_1 T_1 + n_2 F_2 T_2}{n_1 + n_2}$

Question 47

A current of 10A exists in a wire of crosssectional area of 5 $mm^2$ with a drift velocity of $2 	imes 10^{-3}$ $ms^{-1}$. The number of free electrons in each cubic meter of the wire is ___.

Accepted Answer

(B) $625 	imes 10^{25}$

Answer

(A) $2 	imes 10^6$

Answer

(C) $2 	imes 10^{25}$

Answer

(D) $1 	imes 10^{23}$

Question 48

For VHF signal broadcasting, ____ km$^2$ of maximum service area will be covered by an antenna tower of height 30m, if the receiving antenna is placed at ground. Let radius of the earth be 6400 km. (Round off to the Nearest Integer) (Take $\pi$ as 3.14)

Accepted Answer

1206

Question 49

The angular speed of truck wheel is increased from 900 rpm to 2460 rpm in 26 seconds. The number of revolutions by the truck engine during this time is _______.
(Assuming the acceleration to be uniform).

Accepted Answer

728

Question 50

The equivalent resistance of series combination of two resistors is 's'. When they are connected in parallel, the equivalent resistance is 'p'. If s = np, then the minimum value for n is ___.
(Round off to the Nearest Integer)

Accepted Answer

4

Question 51

Four identical rectangular plates with length, $l$ = 2 cm and breadth, b = $\frac{3}{2}$ cm are arranged as shown in figure. The equivalent capacitance between A and C is $\frac{\mathrm{x}\,\varepsilon_0}{\mathrm{d}}$. The value of x is ___.
(Round off to the Nearest Integer)

Accepted Answer

2

Question 52

The radius in kilometer to which the present radius of earth (R = 6400 km) to be compressed so that the escape velocity is increased 10 time is ________.

Accepted Answer

64

Question 53

Consider two identical springs each of spring constant k and negligible mass compared to the mass M as shown. Fig.1 shows one of them and Fig.2 shows their series combination. The ratios of time period of oscillation of the two SHM is $\frac{\mathrm{T}_b}{\mathrm{T}_a} = \sqrt{\mathrm{x}}$, where value of x is _______.
(Round off to the Nearest Integer)

Accepted Answer

2

Question 54

The following bodies,
(1) a ring
(2) a disc
(3) a solid cylinder
(4) a solid sphere,
of same mass 'm' and radius 'R' are allowed to roll down without slipping simultaneously from the top of the inclined plane. The body which will reach first at the bottom of the inclined plane is ______.
[Mark the body as per their respective numbering given in the question]

Accepted Answer

4

Question 55

A parallel plate capacitor whose capacitance C is 14 pF is charged by a battery to a potential difference V = 12V between its plates. The charging battery is now disconnected and a porcelin plate with k = 7 is inserted between the plates, then the plate would oscillate back and forth between the plates with a constant mechanical energy of ______ pJ.
(Assume no friction)

Accepted Answer

864

Question 56

Two blocks (m = 0.5 kg and M = 4.5 kg) are arranged on a horizontal frictionless table as shown in figure. The coefficient of static friction between the two blocks is $\frac{3}{7}$. Then the maximum horizontal force that can be applied on the larger block so that the blocks move together is ______ N. (Round off to the Nearest Integer) [Take g as 9.8 ms$^{-2}$]

Accepted Answer

21

Question 57

If $2.5 	imes 10^{-6}$ N average force is exerted by a light wave on a non-reflecting surface of 30 cm$^2$ area during 40 minutes of time span, the energy flux of light just before it falls on the surface is ____ W/cm$^2$.
(Round off to the Nearest Integer)
(Assume complete absorption and normal incidence conditions are there)

Accepted Answer

25

Question 58

With respect to drug-enzyme interaction, identify the wrong statement:

Accepted Answer

(C) Allosteric inhibitor competes with the enzyme's active site

Answer

(A) Non-Competitive inhibitor binds to the allosteric site

Answer

(B) Allosteric inhibitor changes the enzyme's active site

Answer

(D) Competitive inhibitor binds to the enzyme's active site

Question 59

Which of the following is an aromatic compound?

Accepted Answer

(A) (1)

Answer

(B) (2)

Answer

(C) (3)

Answer

(D) (4)

Question 60

The product "A" in the above reaction is:

Accepted Answer

(B) (2)

Answer

(A) (1)

Answer

(C) (3)

Answer

(D) (4)

Question 61

A central atom in a molecule has two lone pairs of electrons and forms three single bonds. The shape of this molecule is:

Accepted Answer

(C) T-shaped

Answer

(A) see-saw

Answer

(B) planar triangular

Answer

(D) trigonal pyramidal

Question 62

Given below are two statements:
Statement I : Potassium permanganate on heating at 573 K forms potassium manganate.
Statement II : Both potassium permanganate and potassium manganate are tetrahedral and paramagnetic in nature.
In the light of the above statements, choose the most appropriate answer from the options given below:

Accepted Answer

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

Answer

(B) Both statement I and statement II are true

Answer

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

Answer

(D) Both statement I and statement II are false

Question 63

Which of the following is correct structure of tyrosine?

Accepted Answer

(D) (4)

Answer

(A) (1)

Answer

(B) (2)

Answer

(C) (3)

Question 64

The above reaction requires which of the following reaction conditions?

Accepted Answer

(D) 623 K, 300 atm

Answer

(A) 573 K, Cu, 300 atm

Answer

(B) 623 K, Cu, 300 atm

Answer

(C) 573 K, 300 atm

JEE Main 17 March 2021 Shift 1 Question Paper with Answers

Physics — JEE Main 17 March 2021 Shift 1

Chemistry — JEE Main 17 March 2021 Shift 1

Mathematics — JEE Main 17 March 2021 Shift 1

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