NTA JEE Main 27th August 2021 Shift 2

NTA JEE Main 27th August 2021 Shift 2 - Question 1

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Match List-(I) with List-(II).
List-(I)                                                                List-(II)
(a) R$$_H$$ (Rydberg constant)                          (i) kg m$$^{-1}$$ s$$^{-1}$$
(b) $$h$$ (Planck's constant)                            (ii) kg m$$^2$$ s$$^{-1}$$
(c) $$\mu_B$$ (Magnetic field energy density)     (iii) m$$^{-1}$$
(d) $$\eta$$ (coefficient of viscosity)                     (iv) kg m$$^{-1}$$ s$$^{-2}$$
Choose the most appropriate answer from the options given below:

A

(a) - (iv), (b) - (ii), (c) - (i), (d) - (iii)

B

(a) - (ii), (b) - (iii), (c) - (iv), (d) - (i)

C

(a) - (iii), (b) - (ii), (c) - (iv), (d) - (i)

D

(a) - (iii), (b) - (ii), (c) - (i), (d) - (iv)

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NTA JEE Main 27th August 2021 Shift 2 - Question 2

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If force (F), length (L) and time (T) are taken as the fundamental quantities. Then what will be the dimension of density:

A

$$[FL^{-4} T^2]$$

B

$$[FL^{-3} T^3]$$

C

$$[FL^{-3} T^2]$$

D

$$[FL^{-5} T^2]$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 3

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Water drops are falling from a nozzle of a shower onto the floor from a height of 9.8 m. The drops fall at a regular interval of time. When the first drop strikes the floor, at that instant, the third drop begins to fall. Locate the position of second drop from the floor when the first drop strikes the floor.

A

2.94 m

B

4.18 m

C

2.45 m

D

7.35 m

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NTA JEE Main 27th August 2021 Shift 2 - Question 4

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A player kicks a football with an initial speed of 25 m s$$^{-1}$$ at an angle of 45° from the ground. What are the maximum height and the time taken by the football to reach at the highest point during motion? (Take g = 10 m s$$^{-2}$$)

A

$$h_{max}$$ = 15.625 m, T = 1.77 s

B

$$h_{max}$$ = 3.54 m, T = 0.125 s

C

$$h_{max}$$ = 10 m, T = 2.5 s

D

$$h_{max}$$ = 15.625 m, T = 3.54 s

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NTA JEE Main 27th August 2021 Shift 2 - Question 5

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The boxes of masses 2 kg and 8 kg are connected by a massless string passing over smooth pulleys. Calculate the time taken by box of mass 8 kg to strike the ground starting from rest. (g = 10 m s$$^{-2}$$)

A

0.25 s

B

0.34 s

C

0.2 s

D

0.4 s

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NTA JEE Main 27th August 2021 Shift 2 - Question 6

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The height of victoria's falls is 63 m. What is the difference in the temperature of water at the top and at the bottom of the fall? [Given 1 cal = 4.2 J and specific heat of water = 1 cal g$$^{-1}$$ °C$$^{-1}$$]

A

14.76 °C

B

1.476 °C

C

0.147 °C

D

0.014 °C

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NTA JEE Main 27th August 2021 Shift 2 - Question 7

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Two discs have moments of inertia $$I_1$$ and $$I_2$$ about their respective axes perpendicular to the plane and passing through the centre. They are rotating with angular speeds, $$\omega_1$$ and $$\omega_2$$ respectively and are brought into contact face to face with their axes of rotation coaxial. The loss in kinetic energy of the system in the process is given by:

A

$$\frac{I_1 I_2}{(I_1+I_2)}(\omega_1 - \omega_2)^2$$

B

$$\frac{(\omega_1-\omega_2)^2}{2(I_1+I_2)}$$

C

$$\frac{I_1 I_2}{2(I_1+I_2)}(\omega_1 - \omega_2)^2$$

D

$$\frac{(I_1-I_2)^2 \omega_1 \omega_2}{2(I_1+I_2)}$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 8

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A mass of 50 kg is placed at the center of a uniform spherical shell of mass 100 kg and radius 50 m. If the gravitational potential at a point, 25 m from the center is $$V$$ kg m$$^{-1}$$. The value of $$V$$ is:

A

+2$$G$$

B

-20$$G$$

C

-4$$G$$

D

-60$$G$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 9

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If the R.M.S. speed of oxygen molecules at 0°C is 160 m s$$^{-1}$$. Find the R.M.S. speed of hydrogen molecules at 0°C.

A

40 m s$$^{-1}$$

B

80 m s$$^{-1}$$

C

640 m s$$^{-1}$$

D

332 m s$$^{-1}$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 10

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Figure shows a rod $$AB$$, which is bent in a 120° circular arc of radius $$R$$. A charge $$(-Q)$$ is uniformly distributed over rod AB. What is the electric field $$\vec{E}$$ at the centre of curvature O?

A

$$\frac{3\sqrt{3}Q}{8\pi^2 \varepsilon_0 R^2} \hat{i}$$

B

$$\frac{3\sqrt{3}Q}{8\pi \varepsilon_0 R^2} \hat{i}$$

C

$$\frac{-3\sqrt{3}Q}{8\pi^2 \varepsilon_0 R^2} \hat{i}$$

D

$$\frac{3\sqrt{3}Q}{16\pi^2 \varepsilon_0 R^2} \hat{i}$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 11

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Three capacitors $$C_1 = 2 \mu$$F, $$C_2 = 6 \mu$$F and $$C_3 = 12 \mu$$F are connected as shown in the figure. Find the ratio of the charges on capacitors $$C_1$$, $$C_2$$ and $$C_3$$ respectively.

A

3 : 4 : 4

B

2 : 3 : 3

C

2 : 1 : 1

D

1 : 2 : 2

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NTA JEE Main 27th August 2021 Shift 2 - Question 12

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For full scale deflection of total 50 divisions, 50 mV voltage is required in galvanometer. The resistance of galvanometer if its current sensitivity is 2 div / mA will be:

A

1 $$\Omega$$

B

2 $$\Omega$$

C

5 $$\Omega$$

D

4 $$\Omega$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 13

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The colour coding on a carbon resistor is shown in the given figure. The resistance value of the given resistor is:

A

$$(7500 \pm 375)\Omega$$

B

$$(5700 \pm 285)\Omega$$

C

$$(7500 \pm 755)\Omega$$

D

$$(5700 \pm 375)\Omega$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 14

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A coaxial cable consists of an inner wire of radius $$a$$ surrounded by an outer shell of inner and outer radii $$b$$ and $$c$$ respectively. The inner wire carries an electric current $$i_0$$ which is distributed uniformly across cross-sectional area. The outer shell carries an equal current in opposite direction and distributed uniformly. What will be the ratio of the magnetic field at a distance $$x$$ from the axis when (i) $$x \lt a$$ and (ii) $$a \lt x \lt b$$?

A

$$\frac{a^2}{x^2}$$

B

$$\frac{b^2-a^2}{x^2}$$

C

$$\frac{x^2}{b^2-a^2}$$

D

$$\frac{x^2}{a^2}$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 15

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A constant magnetic field of 1 T is applied in the $$x > 0$$ region. A metallic circular ring of radius 1 m is moving with a constant velocity of 1 m s$$^{-1}$$ along the $$x$$-axis. At $$t = 0$$ s, the centre O of the ring is at $$x = -1$$ m. What will be the value of the induced emf in the ring at t = 1 s? (Assume the velocity of the ring does not change.)

A

0 V

B

2 V

C

$$2\pi$$ V

D

1 V

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NTA JEE Main 27th August 2021 Shift 2 - Question 16

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Curved surfaces of a plano-convex lens of refractive index $$\mu_1$$ and a plano-concave lens of refractive index $$\mu_2$$ have equal radius of curvature as shown in figure. Find the ratio of radius of curvature to the focal length of the combined lenses.

A

$$\frac{1}{\mu_2 - \mu_1}$$

B

$$\frac{1}{\mu_1 - \mu_2}$$

C

$$\mu_2 - \mu_1$$

D

$$\mu_1 - \mu_2$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 17

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The light waves from two coherent sources have same intensity $$I_1 = I_2 = I_0$$. In interference pattern the intensity of light at minima is zero. What will be the intensity of light at maxima?

A

$$I_0$$

B

$$2I_0$$

C

$$4I_0$$

D

$$5I_0$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 18

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A monochromatic neon lamp with wavelength of 670.5 nm illuminates a photo-sensitive material which has a stopping voltage of 0.48 V. What will be the stopping voltage if the source light is changed with another source of wavelength of 474.6 nm?

A

0.24 V

B

1.5 V

C

1.25 V

D

0.96 V

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NTA JEE Main 27th August 2021 Shift 2 - Question 19

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For a transistor $$\alpha$$ and $$\beta$$ are given as $$\alpha = \frac{I_c}{I_E}$$ and $$\beta = \frac{I_c}{I_B}$$. Then the correct relation between $$\alpha$$ and $$\beta$$ will be:

A

$$\alpha = \frac{1-\beta}{\beta}$$

B

$$\alpha\beta = 1$$

C

$$\beta = \frac{\alpha}{1-\alpha}$$

D

$$\alpha = \frac{\beta}{1-\beta}$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 20

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An antenna is mounted on a 400 m tall building. What will be the wavelength of signal that can be radiated effectively by the transmission tower upto a range of 44 km?

A

605 m

B

302 m

C

37.8 m

D

75.6 m

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NTA JEE Main 27th August 2021 Shift 2 - Question 21

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A bullet of 10 g, moving with velocity $$v$$, collides head-on with the stationary bob of a pendulum and recoils with velocity 100 m s$$^{-1}$$. The length of the pendulum is 0.5 m and mass of the bob is 1 kg. The minimum value of $$v$$ in m s$$^{-1}$$, so that the pendulum describes a circle. (Assume the string to be inextensible and g = 10 m s$$^{-2}$$)

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NTA JEE Main 27th August 2021 Shift 2 - Question 22

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Wires $$W_1$$ and $$W_2$$ are made of same material having the breaking stress of $$1.25 \times 10^9$$ N m$$^{-2}$$. $$W_1$$ and $$W_2$$ have cross-sectional area of $$8 \times 10^{-7}$$ m$$^2$$ and $$4 \times 10^{-7}$$ m$$^2$$, respectively. Masses of 20 kg and 10 kg hang from them as shown in the figure. The maximum mass that can be placed in the pan without breaking the wires is _________ kg (Use g = 10 m s$$^{-2}$$)

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NTA JEE Main 27th August 2021 Shift 2 - Question 23

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A heat engine operates between a cold reservoir at temperature $$T_2 = 400$$ K and a hot reservoir at temperature $$T_1$$. It takes 300 J of heat from the hot reservoir and delivers 240 J of heat to the cold reservoir in a cycle. The minimum temperature of the hot reservoir has to be _________.

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NTA JEE Main 27th August 2021 Shift 2 - Question 24

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Two simple harmonic motion, are represented by the equations
$$y_1 = 10\sin\left(3\pi t + \frac{\pi}{3}\right)$$; $$y_2 = 5\left(\sin 3\pi t + \sqrt{3}\cos 3\pi t\right)$$
Ratio of amplitude of $$y_1$$ to $$y_2$$ = $$x$$ : 1. The value of $$x$$ is _________.

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NTA JEE Main 27th August 2021 Shift 2 - Question 25

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A tuning fork is vibrating at 250 Hz. The length of the shortest closed organ pipe that will resonate with the tuning fork will be _________ cm. (Take speed of sound in air as 340 m s$$^{-1}$$)

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NTA JEE Main 27th August 2021 Shift 2 - Question 26

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The ratio of the equivalent resistance of the network (shown in figure) between the points $$a$$ and $$b$$ when switch is open and switch is closed is $$x : 8$$. The value of $$x$$ is _________.

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NTA JEE Main 27th August 2021 Shift 2 - Question 27

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An AC circuit has an inductor and a resistor of resistance $$R$$ in series, such that $$X_L = 3R$$. Now, a capacitor is added in series such that $$X_C = 2R$$. The ratio of the new power factor with the old power factor of the circuit is $$\sqrt{5} : x$$. The value of $$x$$ is _________.

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NTA JEE Main 27th August 2021 Shift 2 - Question 28

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A plane electromagnetic wave with a frequency of 30 MHz travels in free space. At a particular point in space and time, the electric field is 6 V m$$^{-1}$$. The magnetic field at this point will be $$x \times 10^{-8}$$ T. The value of $$x$$ is _________.

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NTA JEE Main 27th August 2021 Shift 2 - Question 29

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$$X$$ different wavelength may be observed in the spectrum from a hydrogen sample if the atoms are excited to states with principal quantum number $$n = 6$$? The value of $$X$$ is _________.

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NTA JEE Main 27th August 2021 Shift 2 - Question 30

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A zener diode of power rating 2 W is to be used as a voltage regulator. If the zener diode has a breakdown of 10 V and it has to regulate voltage fluctuated between 6 V and 14 V, the value of $$R_s$$ for safe operation should be _________ $$\Omega$$.

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NTA JEE Main 27th August 2021 Shift 2 - Question 31

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The correct order of ionic radii for the ions, P$$^{3-}$$, S$$^{2-}$$, Ca$$^{2+}$$, K$$^+$$, Cl$$^-$$ is:

A

P$$^{3-}$$ > S$$^{2-}$$ > Cl$$^-$$ > K$$^+$$ > Ca$$^{2+}$$

B

P$$^{3-}$$ > S$$^{2-}$$ > Cl$$^-$$ > Ca$$^{2+}$$ > K$$^+$$

C

Cl$$^-$$ > S$$^{2-}$$ > P$$^{3-}$$ > Ca$$^{2+}$$ > K$$^+$$

D

K$$^+$$ > Ca$$^{2+}$$ > P$$^{3-}$$ > S$$^{2-}$$ > Cl$$^-$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 32

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The compound/s which will show significant intermolecular H-bonding is/are:

A

(a), (b) and (c)

B

(b) only

C

(c) only

D

(a) and (b) only

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NTA JEE Main 27th August 2021 Shift 2 - Question 33

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The oxide that gives H$$_2$$O$$_2$$ most readily on treatment with H$$_2$$O is:

A

Na$$_2$$O$$_2$$

B

BaO$$_2$$.8H$$_2$$O

C

SnO$$_2$$

D

PbO$$_2$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 34

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Choose the correct statement from the following:

A

LiF has least negative standard enthalpy of formation among alkali metal fluorides.

B

The standard enthalpy of formation for alkali metal bromides becomes less negative on descending the group.

C

The low solubility of CsI in water is due to its high lattice enthalpy.

D

Among the alkali metal halides, LiF is least soluble in water.

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NTA JEE Main 27th August 2021 Shift 2 - Question 35

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The major product (A) formed in the reaction given below is:

A

B

C

D

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NTA JEE Main 27th August 2021 Shift 2 - Question 36

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In stratosphere most of the ozone formation is assisted by:

A

$$\gamma$$-rays

B

cosmic rays

C

visible radiations

D

ultraviolet radiation

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NTA JEE Main 27th August 2021 Shift 2 - Question 37

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Lyophilic sols are more stable than lyophobic sols because:

A

the colloidal particles have no charge.

B

the colloidal particles have positive charge.

C

the colloidal particles are solvated.

D

there is a strong electrostatic repulsion between the negatively charged colloidal particles.

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NTA JEE Main 27th August 2021 Shift 2 - Question 38

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Match List-I with List-II:
List-I (Name of ore/mineral)     List-II (Chemical formula)
(a) Calamine                             (i) ZnS
(b) Malachite                            (ii) FeCO$$_3$$
(c) Siderite                              (iii) ZnCO$$_3$$
(d) Sphalerite                           (iv) CuCO$$_3$$.Cu(OH)$$_2$$
Choose the most appropriate answer from the options given below:

A

(a) - (iv), (b) - (iii), (c) - (i), (d) - (ii)

B

(a) - (iii), (b) - (ii), (c) - (iv), (d) - (i)

C

(a) - (iii), (b) - (iv), (c) - (i), (d) - (ii)

D

(a) - (iii), (b) - (iv), (c) - (ii), (d) - (i)

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NTA JEE Main 27th August 2021 Shift 2 - Question 39

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Which one of the following is formed (mainly) when red phosphorus is heated in a sealed tube at 803 K?

A

$$\alpha$$-Black phosphorus

B

White phosphorus

C

Yellow phosphorus

D

$$\beta$$-Black phosphorus

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NTA JEE Main 27th August 2021 Shift 2 - Question 40

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Potassium permanganate on heating at 513 K gives a product which is:

A

paramagnetic and colourless

B

diamagnetic and colourless

C

diamagnetic and green

D

paramagnetic and green

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NTA JEE Main 27th August 2021 Shift 2 - Question 41

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Which one of the following is used to remove most of plutonium from spent nuclear fuel?

A

BrO$$_3$$

B

I$$_2$$O$$_5$$

C

ClF$$_3$$

D

O$$_2$$F$$_2$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 42

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The addition of dilute NaOH to Cr$$^{3+}$$ salt solution will give:

A

a solution of [Cr(OH)$$_4$$]$$^-$$

B

precipitate of [Cr(OH)$$_6$$]$$^{3-}$$

C

precipitate of Cr$$_2$$O$$_3$$(H$$_2$$O)$$_n$$

D

precipitate of Cr(OH)$$_3$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 43

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The major product of the following reaction, if it occurs by S$$_N$$2 mechanism is:

A

B

C

D

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NTA JEE Main 27th August 2021 Shift 2 - Question 44

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Given below are two statements:
Statement I : Ethyl pent-4-yn-oate on reaction with CH$$_3$$MgBr gives a 3°-alcohol.
Statement II : In this reaction one mole of ethyl pent-4-yn-oate utilizes two moles of CH$$_3$$MgBr.
In the light of the above statements, choose the most appropriate answer from the options given below:

A

Both Statement I and Statement II are false

B

Statement I is true but Statement II is false

C

Statement I is false but Statement II is true

D

Both Statement I and Statement II are true

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NTA JEE Main 27th August 2021 Shift 2 - Question 45

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Which one of the following reactions will not yield propanoic acid?

A

CH$$_3$$CH$$_2$$CH$$_2$$Br + Mg, CO$$_2$$ dry ether /H$$_3$$O$$^+$$

B

CH$$_3$$CH$$_2$$CH$$_3$$ + KMnO$$_4$$(Heat), OH$$^-$$ /H$$_3$$O$$^+$$

C

CH$$_3$$CH$$_2$$CCl$$_3$$ + OH$$^-$$ /H$$_3$$O$$^+$$

D

CH$$_3$$CH$$_2$$COCH$$_3$$ + OI$$^-$$ /H$$_3$$O$$^+$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 46

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The correct structures of A and B formed in the following reactions are:

A

B

C

D

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NTA JEE Main 27th August 2021 Shift 2 - Question 47

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Which one of the following is the major product of the given reaction?

A

B

C

D

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NTA JEE Main 27th August 2021 Shift 2 - Question 48

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Which one of the following chemicals is responsible for the production of HCl in the stomach leading to irritation and pain?

A

B

C

D

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NTA JEE Main 27th August 2021 Shift 2 - Question 49

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Which one of the following tests used for the identification of functional groups in organic compounds does not use copper reagent?

A

Seliwanoff's test

B

Biuret test for peptide bond

C

Barfoed's test

D

Benedict's test

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NTA JEE Main 27th August 2021 Shift 2 - Question 50

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Hydrolysis of sucrose gives:

A

$$\alpha$$-D-(+)-Glucose and $$\alpha$$-D-(-)-Fructose

B

$$\alpha$$-D-(+)-Glucose and $$\beta$$-D-(-)-Fructose

C

$$\alpha$$-D-(-)-Glucose and $$\beta$$-D-(-)-Fructose

D

$$\alpha$$-D-(-)-Glucose and $$\alpha$$-D-(+)-Fructose

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NTA JEE Main 27th August 2021 Shift 2 - Question 51

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100 g of propane is completely reacted with 1000 g of oxygen. The mole fraction of carbon dioxide in the resulting mixture is $$x \times 10^{-2}$$. The value of $$x$$ is _________. (Nearest integer)

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NTA JEE Main 27th August 2021 Shift 2 - Question 52

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The number of photons emitted by a monochromatic (single frequency) infrared range finder of power 1 mW and wavelength of 1000 nm, in 0.1 second is $$x \times 10^{13}$$. The value of x is _________. (Nearest integer)
(h = $$6.63 \times 10^{-34}$$ Js, c = $$3.00 \times 10^8$$ ms$$^{-1}$$)

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NTA JEE Main 27th August 2021 Shift 2 - Question 53

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The number of species having non-pyramidal shape among the following is:
(A) SO$$_3$$
(B) NO$$_3^-$$
(C) PCl$$_3$$
(D) CO$$_3^{2-}$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 54

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Two flasks I and II shown below are connected by a valve of negligible volume.

When the valve is opened, the final pressure of the system in bar is $$x \times 10^{-2}$$. The value of x is _________. (Integer answer)
[Assume : Ideal gas; 1 bar = $$10^5$$ Pa; Molar mass of N$$_2$$ = 28.0 mol$$^{-1}$$; R = 8.31 J mol$$^{-1}$$ K$$^{-1}$$]

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NTA JEE Main 27th August 2021 Shift 2 - Question 55

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Data given for the following reaction is as follows:
FeO$$_{(s)}$$ + C$$_{(graphite)}$$ $$\rightarrow$$ Fe$$_{(s)}$$ + CO$$_{(g)}$$
Substance                $$\Delta_f H°$$(kJ mol$$^{-1}$$)                               $$\Delta S°$$(J mol$$^{-1}$$ K$$^{-1}$$)
FeO$$(s)$$                                          -266.3                                        57.49
C$$(graphite)$$                                  0                                              5.74
Fe$$(s)$$                                              0                                               27.28
CO$$(g)$$                                           -110.5                                         197.6
The minimum temperature in K at which the reaction becomes spontaneous is _________. (Integer answer)

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NTA JEE Main 27th August 2021 Shift 2 - Question 56

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When 5.1 g of solid NH$$_4$$HS is introduced into a two litre evacuated flask at 27°C, 20% of the solid decomposes into gaseous ammonia and hydrogen sulphide. The K$$_p$$ for the reaction at 27°C is $$x \times 10^{-2}$$. The value of x is _________. (Integer answer)
[Given 1R = 0.082 L atm K$$^{-1}$$ mol$$^{-1}$$]

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NTA JEE Main 27th August 2021 Shift 2 - Question 57

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40 g of glucose (Molar mass = 180) is mixed with 200 mL of water. The freezing point of solution is _________ K. (Nearest integer)
[Given : K$$_f$$ = 1.86 K kg mol$$^{-1}$$; Density of water = 1.00 g cm$$^{-3}$$; Freezing point of water = 273.15 K]

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NTA JEE Main 27th August 2021 Shift 2 - Question 58

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The resistance of conductivity cell with cell constant 1.14 cm$$^{-1}$$, containing 0.001M KCl at 298 K is 1500$$\Omega$$. The molar conductivity of 0.001M KCl solution at 298 K in S cm$$^2$$ mol$$^{-1}$$ is _________. (Integer answer)

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NTA JEE Main 27th August 2021 Shift 2 - Question 59

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The first order rate constant for the decomposition of CaCO$$_3$$ at 700 K is $$6.36 \times 10^{-3}$$ s$$^{-1}$$ and activation energy is 209 kJ mol$$^{-1}$$. Its rate constant (in s$$^{-1}$$) at 600 K is $$x \times 10^{-6}$$. The value of x is _________. (Nearest integer)
[Given R = 8.31 J K$$^{-1}$$ mol$$^{-1}$$; log $$6.36 \times 10^{-3}$$ = -2.19, $$10^{-4.79}$$ = $$1.62 \times 10^{-5}$$]

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NTA JEE Main 27th August 2021 Shift 2 - Question 60

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The number of optical isomers possible for [Cr(C$$_2$$O$$_4$$)$$_3$$]$$^{3-}$$ is:

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NTA JEE Main 27th August 2021 Shift 2 - Question 61

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The set of all values of $$k \gt -1$$, for which the equation $$(3x^2+4x+3)^2 - (k+1)(3x^2+4x+3)(3x^2+4x+2) + k(3x^2+4x+2)^2 = 0$$ has real roots, is:

A

$$\left[-\frac{1}{2}, 1\right)$$

B

$$\left(1, \frac{5}{2}\right]$$

C

$$\left(\frac{1}{2}, \frac{3}{2}\right] - \{1\}$$

D

$$[2, 3)$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 62

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If $$0 < x < 1$$ and $$y = \frac{1}{2}x^2 + \frac{2}{3}x^3 + \frac{3}{4}x^4 + \ldots$$, then the value of $$e^{1+y}$$ at $$x = \frac{1}{2}$$ is:

A

$$\frac{1}{2}e^2$$

B

$$2e$$

C

$$2e^2$$

D

$$\frac{1}{2}\sqrt{e}$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 63

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Let $$A(a, 0)$$, $$B(b, 2b+1)$$ and $$C(0, b)$$, $$b \neq 0$$, $$|b| \neq 1$$, be points such that the area of triangle $$ABC$$ is 1 sq. unit, then the sum of all possible values of $$a$$ is:

A

$$\frac{-2b}{b+1}$$

B

$$\frac{2b^2}{b+1}$$

C

$$\frac{-2b^2}{b+1}$$

D

$$\frac{2b}{b+1}$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 64

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If two tangents drawn from a point $$P$$ to the parabola $$y^2 = 16(x-3)$$ are at right angles, then the locus of point $$P$$ is:

A

$$x + 4 = 0$$

B

$$x + 2 = 0$$

C

$$x + 3 = 0$$

D

$$x + 1 = 0$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 65

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If $$\lim_{x \to \infty} \left(\sqrt{x^2 - x + 1} - ax\right) = b$$, then the ordered pair $$(a, b)$$ is:

A

$$\left(1, -\frac{1}{2}\right)$$

B

$$\left(-1, \frac{1}{2}\right)$$

C

$$\left(-1, -\frac{1}{2}\right)$$

D

$$\left(1, \frac{1}{2}\right)$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 66

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The Boolean expression $$(p \wedge q) \Rightarrow ((r \wedge q) \wedge p)$$ is equivalent to:

A

$$(p \wedge r) \Rightarrow (p \wedge q)$$

B

$$(q \wedge r) \Rightarrow (p \wedge q)$$

C

$$(p \wedge q) \Rightarrow (r \wedge q)$$

D

$$(p \wedge q) \Rightarrow (r \vee q)$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 67

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Two poles $$AB$$ of length $$a$$ metres and $$CD$$ of length $$a + b$$ $$(b \neq a)$$ metres are erected at the same horizontal level with bases at $$B$$ and $$D$$. If $$BD = x$$ and $$\tan \angle ACB = \frac{1}{2}$$, then:

A

$$x^2 + 2(a+2b)x - b(a+b) = 0$$

B

$$x^2 + 2(a+2b)x + a(a+b) = 0$$

C

$$x^2 - 2ax + b(a+b) = 0$$

D

$$x^2 - 2ax + a(a+b) = 0$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 68

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Let $$Z$$ be the set of all integers,
$$A = \{(x,y) \in Z \times Z : (x-2)^2 + y^2 \leq 4\}$$
$$B = \{(x,y) \in Z \times Z : x^2 + y^2 \leq 4\}$$ and
$$C = \{(x,y) \in Z \times Z : (x-2)^2 + (y-2)^2 \leq 4\}$$
If the total number of relations from $$A \cap B$$ to $$A \cap C$$ is $$2^p$$, then the value of $$p$$ is:

A

25

B

9

C

16

D

49

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NTA JEE Main 27th August 2021 Shift 2 - Question 69

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Let $$[\lambda]$$ be the greatest integer less than or equal to $$\lambda$$. The set of all values of $$\lambda$$ for which the system of linear equations $$x + y + z = 4$$, $$3x + 2y + 5z = 3$$, $$9x + 4y + (28 + [\lambda])z = [\lambda]$$ has a solution is:

A

$$R$$

B

$$(-\infty, -9) \cup [-8, \infty)$$

C

$$(-\infty, -9) \cup (-9, \infty)$$

D

$$[-9, -8)$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 70

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Let $$A = \begin{bmatrix} [x+1] & [x+2] & [x+3] \\ [x] & [x+3] & [x+3] \\ [x] & [x+2] & [x+4] \end{bmatrix}$$, where $$[x]$$ denotes the greatest integer less than or equal to $$x$$. If det$$(A) = 192$$, then the set of values of $$x$$ is in the interval:

A

$$[62, 63)$$

B

$$[65, 66)$$

C

$$[60, 61)$$

D

$$[68, 69)$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 71

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If $$y(x) = \cot^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right)$$, $$x \in \left(\frac{\pi}{2}, \pi\right)$$, then $$\frac{dy}{dx}$$ at $$x = \frac{5\pi}{6}$$ is:

A

0

B

-1

C

$$-\frac{1}{2}$$

D

$$\frac{1}{2}$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 72

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A box open from top is made from a rectangular sheet of dimension $$a \times b$$ by cutting squares each of side $$x$$ from each of the four corners and folding up the flaps. If the volume of the box is maximum, then $$x$$ is equal to:

A

$$\frac{a+b+\sqrt{a^2+b^2-ab}}{6}$$

B

$$\frac{a+b-\sqrt{a^2+b^2-ab}}{12}$$

C

$$\frac{a+b-\sqrt{a^2+b^2-ab}}{6}$$

D

$$\frac{a+b-\sqrt{a^2+b^2+ab}}{6}$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 73

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Let $$M$$ and $$m$$ respectively be the maximum and minimum values of the function $$f(x) = \tan^{-1}(\sin x + \cos x)$$ in $$\left[0, \frac{\pi}{2}\right]$$. Then the value of $$\tan(M - m)$$ is equal to:

A

$$2 - \sqrt{3}$$

B

$$3 - 2\sqrt{2}$$

C

$$3 + 2\sqrt{2}$$

D

$$2 + \sqrt{3}$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 74

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The value of the integral $$\int_0^1 \frac{\sqrt{x} dx}{(1+x)(1+3x)(3+x)}$$ is:

A

$$\frac{\pi}{4}\left(1 - \frac{\sqrt{3}}{2}\right)$$

B

$$\frac{\pi}{8}\left(1 - \frac{\sqrt{3}}{6}\right)$$

C

$$\frac{\pi}{8}\left(1 - \frac{\sqrt{3}}{2}\right)$$

D

$$\frac{\pi}{4}\left(1 - \frac{\sqrt{3}}{6}\right)$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 75

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The area of the region bounded by the parabola $$(y-2)^2 = (x-1)$$, the tangent to it at the point whose ordinate is 3 and the x-axis, is:

A

4

B

6

C

9

D

10

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NTA JEE Main 27th August 2021 Shift 2 - Question 76

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If the solution curve of the differential equation $$(2x - 10y^3)dy + ydx = 0$$, passes through the points $$(0, 1)$$ and $$(2, \beta)$$, then $$\beta$$ is a root of the equation?

A

$$y^5 - 2y - 2 = 0$$

B

$$y^5 - y^2 - 1 = 0$$

C

$$2y^5 - y^2 - 2 = 0$$

D

$$2y^5 - 2y - 1 = 0$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 77

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A differential equation representing the family of parabolas with axis parallel to y-axis and whose length of latus rectum is the distance of the point $$(2, -3)$$ from the line $$3x + 4y = 5$$, is given by:

A

$$11\frac{d^2x}{dy^2} = 10$$

B

$$11\frac{d^2y}{dx^2} = 10$$

C

$$10\frac{d^2y}{dx^2} = 11$$

D

$$10\frac{d^2x}{dy^2} = 11$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 78

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The equation of the plane passing through the line of intersection of the planes $$\vec{r} \cdot (2\hat{i} + 3\hat{j} - \hat{k}) + 4 = 0$$ and $$\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1$$ and parallel to the x-axis, is

A

$$\vec{r} \cdot (\hat{i} + 3\hat{k}) + 6 = 0$$

B

$$\vec{r} \cdot (\hat{i} - 3\hat{k}) + 6 = 0$$

C

$$\vec{r} \cdot (\hat{j} - 3\hat{k}) - 6 = 0$$

D

$$\vec{r} \cdot (\hat{j} - 3\hat{k}) + 6 = 0$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 79

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The angle between the straight lines, whose direction cosines $$l, m, n$$ are given by the equations $$2l + 2m - n = 0$$ and $$mn + nl + lm = 0$$, is:

A

$$\frac{\pi}{3}$$

B

$$\frac{\pi}{2}$$

C

$$\cos^{-1}\left(\frac{8}{9}\right)$$

D

$$\pi - \cos^{-1}\left(\frac{4}{9}\right)$$

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NTA JEE Main 27th August 2021 Shift 2 - Question 80

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Each of the persons $$A$$ and $$B$$ independently tosses three fair coins. The probability that both of them get the same number of heads is:

A

$$\frac{5}{8}$$

B

$$\frac{1}{8}$$

C

$$\frac{5}{16}$$

D

1

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NTA JEE Main 27th August 2021 Shift 2 - Question 81

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Let $$z_1$$ and $$z_2$$ be two complex numbers such that $$\arg(z_1 - z_2) = \frac{\pi}{4}$$ and $$z_1, z_2$$ satisfy the equation $$|z - 3| = \text{Re}(z)$$. Then the imaginary part $$z_1 + z_2$$ is equal to _________.

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NTA JEE Main 27th August 2021 Shift 2 - Question 82

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Let $$S = \{1, 2, 3, 4, 5, 6, 9\}$$. Then the number of elements in the set $$T = \{A \subseteq S : A \neq \phi$$ and the sum of all the elements of $$A$$ is not a multiple of 3$$\}$$ is _________.

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NTA JEE Main 27th August 2021 Shift 2 - Question 83

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$$3 \times 7^{22} + 2 \times 10^{22} - 44$$ when divided by 18 leaves the remainder _________.

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NTA JEE Main 27th August 2021 Shift 2 - Question 84

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Let $$S$$ be the sum of all solutions (in radians) of the equation $$\sin^4\theta + \cos^4\theta - \sin\theta\cos\theta = 0$$ in $$[0, 4\pi]$$ then $$\frac{8S}{\pi}$$ is equal to _________.

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NTA JEE Main 27th August 2021 Shift 2 - Question 85

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Two circles each of radius 5 units touch each other at the point $$(1, 2)$$. If the equation of their common tangent is $$4x + 3y = 10$$, and $$C_1(\alpha, \beta)$$ and $$C_2(\gamma, \delta)$$, $$C_1 \neq C_2$$ are their centres, then $$|(\alpha + \beta)(\gamma + \delta)|$$ is equal to _________.

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NTA JEE Main 27th August 2021 Shift 2 - Question 86

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Let $$P(a\sec\theta, b\tan\theta)$$ and $$Q(a\sec\phi, b\tan\phi)$$ where $$\theta + \phi = \frac{\pi}{2}$$, be two points on the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$. If the ordinate of the point of intersection of normals at $$P$$ and $$Q$$ is $$-k\left(\frac{a^2+b^2}{2b}\right)$$, then $$k$$ is equal to _________.

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NTA JEE Main 27th August 2021 Shift 2 - Question 87

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An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2. The variance of marks obtained by 30 girls is also 2. The average marks of all 50 candidates is 15. If $$\mu$$ is the average marks of girls and $$\sigma^2$$ is the variance of marks of 50 candidates, then $$\mu + \sigma^2$$ is equal to _________.

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NTA JEE Main 27th August 2021 Shift 2 - Question 88

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$$\int \frac{2e^x+3e^{-x}}{4e^x+7e^{-x}} dx = \frac{1}{14}(ux + v\log_e(4e^x + 7e^{-x})) + C$$, where $$C$$ is a constant of integration, then $$u + v$$ is equal to _________.

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NTA JEE Main 27th August 2021 Shift 2 - Question 89

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Let $$S$$ be the mirror image of the point $$Q(1, 3, 4)$$ with respect to the plane $$2x - y + z + 3 = 0$$ and let $$R(3, 5, \gamma)$$ be a point of this plane. Then the square of the length of the line segment $$SR$$ is _________.

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NTA JEE Main 27th August 2021 Shift 2 - Question 90

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The probability distribution of random variable $$X$$ is given by

Let $$p = P(1 < X < 4 | X < 3)$$. If $$5p = \lambda K$$, then $$\lambda$$ is equal to _________.

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