NTA JEE Main 27th August 2021 Shift 1

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

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The resultant of these forces $$\vec{OP}, \vec{OQ}, \vec{OR}, \vec{OS}$$ and $$\vec{OT}$$ is approximately ______ N.
[Take $$\sqrt{3} = 1.7, \sqrt{2} = 1.4$$. Given $$\hat{i}$$ and $$\hat{j}$$ unit vectors along $$x, y$$ axis]

A

$$-1.5\hat{i} - 15.5\hat{j}$$

B

$$9.25\hat{i} + 5\hat{j}$$

C

$$3\hat{i} + 15\hat{j}$$

D

$$2.5\hat{i} - 14.5\hat{j}$$

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

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If $$E$$ and $$H$$ represents the intensity of electric field and magnetizing field respectively, then the unit of $$\frac{E}{H}$$ will be:

A

joule

B

ohm

C

newton

D

mho

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NTA JEE Main 27th August 2021 Shift 1 - Question 3

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Which of the following is not a dimensionless quantity?

A

Power factor

B

Quality factor

C

Permeability of free space ($$\mu_0$$)

D

Relative magnetic permeability ($$\mu_r$$)

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NTA JEE Main 27th August 2021 Shift 1 - Question 4

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A huge circular arc of length 4.4 ly subtends an angle 4s at the centre of the circle. How long it would take for a body to complete 4 revolution if its speed is 8 AU per second?
Given : 1 ly = $$9.46 \times 10^{15}$$ m
1 AU = $$1.5 \times 10^{11}$$ m

A

$$3.5 \times 10^6$$ s

B

$$4.5 \times 10^{10}$$ s

C

$$4.1 \times 10^8$$ s

D

$$7.2 \times 10^8$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 5

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Moment of inertia of a square plate of side $$l$$ about the axis passing through one of the corner and perpendicular to the plane of square plate is given by:

A

$$\frac{Ml^2}{6}$$

B

$$\frac{2}{3}Ml^2$$

C

$$Ml^2$$

D

$$\frac{Ml^2}{12}$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 6

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In Millikan's oil drop experiment, what is viscous force acting on an uncharged drop of radius $$2.0 \times 10^{-5}$$ m and density $$1.2 \times 10^3$$ kg m$$^{-3}$$? Take viscosity of liquid = $$1.8 \times 10^{-5}$$ N s m$$^{-2}$$. (Neglect buoyancy due to air).

A

$$5.8 \times 10^{-10}$$ N

B

$$3.9 \times 10^{-10}$$ N

C

$$1.8 \times 10^{-10}$$ N

D

$$3.8 \times 10^{-11}$$ N

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NTA JEE Main 27th August 2021 Shift 1 - Question 7

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An ideal gas is expanding such that $$PT^3$$ = constant. The coefficient of volume expansion of the gas is:

A

$$\frac{2}{T}$$

B

$$\frac{3}{T}$$

C

$$\frac{1}{T}$$

D

$$\frac{4}{T}$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 8

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A balloon carries a total load of 185 kg at normal pressure and temperature of 27°C. What load will the balloon carry on rising to a height at which the barometric pressure is 45 cm of Hg and the temperature is -7°C. Assuming the volume constant?

A

214.15 kg

B

123.54 kg

C

219.07 kg

D

181.46 kg

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NTA JEE Main 27th August 2021 Shift 1 - Question 9

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The variation of displacement with time of a particle executing free simple harmonic motion is shown in the figure.

The potential energy $$U(x)$$ versus time $$(t)$$ plot of the particle is correctly shown in figure:

A

B

C

D

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NTA JEE Main 27th August 2021 Shift 1 - Question 10

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A uniformly charged disc of radius $$R$$ having surface charge density $$\sigma$$ is placed in the $$xy$$ plane with its center at the origin. Find the electric field intensity along the $$z$$-axis at a distance $$Z$$ from origin:

A

$$E = \frac{2\varepsilon_0}{\sigma}\left(\frac{1}{(Z^2+R^2)^{1/2}} + Z\right)$$

B

$$E = \frac{\sigma}{2\varepsilon_0}\left(1 + \frac{Z}{(Z^2+R^2)^{1/2}}\right)$$

C

$$E = \frac{\sigma}{2\varepsilon_0}\left(1 - \frac{Z}{(Z^2+R^2)^{1/2}}\right)$$

D

$$E = \frac{\sigma}{2\varepsilon_0}\left(\frac{1}{(Z^2+R^2)} + \frac{1}{Z^2}\right)$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 11

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Calculate the amount of charge on capacitor of 4 $$\mu$$F. The internal resistance of battery is 1$$\Omega$$:

A

4 $$\mu$$C

B

8 $$\mu$$C

C

16 $$\mu$$C

D

zero

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NTA JEE Main 27th August 2021 Shift 1 - Question 12

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Five identical cells each of internal resistance 1 $$\Omega$$ and emf 5 V are connected in series and in parallel with an external resistance $$R$$. For what value of $$R$$, current in series and parallel combination will remain the same?

A

1 $$\Omega$$

B

5 $$\Omega$$

C

25 $$\Omega$$

D

10 $$\Omega$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 13

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Two ions of masses 4 amu and 16 amu have charges +2e and +3e respectively. These ions pass through the region of the constant perpendicular magnetic field. The kinetic energy of both ions is the same. Then:

A

lighter ion will be deflected less than heavier ion

B

lighter ion will be deflected more than heavier ion

C

both ions will be deflected equally

D

no ion will be deflected

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NTA JEE Main 27th August 2021 Shift 1 - Question 14

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A bar magnet is passing through a conducting loop of radius $$R$$ with velocity $$v$$. The radius of the bar magnet is such that it just passes through the loop. The induced e.m.f. in the loop can be represented by the approximate curve:

A

B

C

D

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NTA JEE Main 27th August 2021 Shift 1 - Question 15

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The electric field in a plane electromagnetic wave is given by, $$E = 50\sin(500x - 10 \times 10^{10}t)$$ V m$$^{-1}$$. The velocity of an electromagnetic wave in this medium is: (Given $$c$$ = the speed of light in vacuum).

A

$$\frac{3}{2}c$$

B

$$c$$

C

$$\frac{c}{2}$$

D

$$\frac{2}{3}c$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 16

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An object is placed beyond the centre of curvature $$C$$ of the given concave mirror. If the distance of the object is $$d_1$$ from $$C$$ and the distance of the image formed is $$d_2$$ from $$C$$, the radius of curvature of this mirror is:

A

$$\dfrac{2d_1 d_2}{d_1 + d_2}$$

B

$$\dfrac{2d_1 d_2}{d_1 - d_2}$$

C

$$\dfrac{d_1 d_2}{d_1 + d_2}$$

D

$$\dfrac{d_1 d_2}{d_1 - d_2}$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 17

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Find the distance of the image from object $$O$$, formed by the combination of lenses in the figure:

A

75 cm

B

10 cm

C

infinity

D

20 cm

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NTA JEE Main 27th August 2021 Shift 1 - Question 18

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In a photoelectric experiment, increasing the intensity of incident light:

A

increases the number of photons incident and also increases the K.E. of the ejected electrons.

B

increases the frequency of photons incident and increases the K.E. of the ejected electrons.

C

increases the number of photons incident and the K.E. of the ejected electrons remains unchanged.

D

increases the frequency of photons incident and the K.E. of the ejected electrons remains unchanged.

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NTA JEE Main 27th August 2021 Shift 1 - Question 19

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There are $$10^{10}$$ radioactive nuclei in a given radioactive element. Its half-life time is 1 min. How many nuclei will remain after 30 s? $$(\sqrt{2} = 1.414)$$

A

$$7 \times 10^9$$

B

$$10^5$$

C

$$2 \times 10^{10}$$

D

$$4 \times 10^{10}$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 20

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For a transistor in CE mode to be used as an amplifier, it must be operated in:

A

Cut-off region only

B

Saturation region only

C

The active region only

D

Both cut-off and Saturation

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NTA JEE Main 27th August 2021 Shift 1 - Question 21

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If the velocity of a body related to displacement $$x$$ is given by $$v = \sqrt{5000 + 24x}$$ m s$$^{-1}$$, then the acceleration of the body is _________ m s$$^{-2}$$.

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NTA JEE Main 27th August 2021 Shift 1 - Question 22

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Two persons $$A$$ and $$B$$ perform same amount of work in moving a body through a certain distance $$d$$ with application of forces acting at angles 45° and 60° with the direction of displacement respectively. The ratio of force applied by person $$A$$ to the force applied by person $$B$$ is $$\frac{1}{\sqrt{x}}$$. The value of $$x$$ is _________.

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NTA JEE Main 27th August 2021 Shift 1 - Question 23

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A body of mass $$(2M)$$ splits into four masses $$\{m, M-m, m, M-m\}$$, which are rearranged to form a square as shown in the figure. The ratio of $$\frac{M}{m}$$ for which, the gravitational potential energy of the system becomes maximum is $$x : 1$$. The value of $$x$$ is _________.

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NTA JEE Main 27th August 2021 Shift 1 - Question 24

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A rod $$CD$$ of thermal resistance $$10.0$$ KW$$^{-1}$$ is joined at the middle of an identical rod $$AB$$ as shown in figure. The ends $$A$$, $$B$$ and $$D$$ are maintained at 200°C, 100°C and 125°C respectively. The heat current in $$CD$$ is _________ P W. The value of $$P$$ is _________.

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NTA JEE Main 27th August 2021 Shift 1 - Question 25

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Two cars $$X$$ and $$Y$$ are approaching each other with velocities 36 km h$$^{-1}$$ and 72 km h$$^{-1}$$ respectively. The frequency of a whistle sound as emitted by a passenger in car $$X$$, heard by the passenger in car $$Y$$ is 1320 Hz. If the velocity of sound in air is 340 ms$$^{-1}$$, the actual frequency of the whistle sound produced is _________ Hz.

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NTA JEE Main 27th August 2021 Shift 1 - Question 26

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First, a set of $$n$$ equal resistors of 10 $$\Omega$$ each are connected in series to a battery of E.M.F. 20 V and internal resistance 10 $$\Omega$$. A current $$I$$ is observed to flow. Then, the $$n$$ resistors are connected in parallel to the same battery. It is observed that the current is increased 20 times, then the value of $$n$$ is _________.

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

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A uniform conducting wire of length is 24$$a$$, and resistance $$R$$ is wound up as a current carrying coil in the shape of an equilateral triangle of side $$a$$ and then in the form of a square of side $$a$$. The coil is connected to a voltage source $$V_0$$. The ratio of magnetic moment of the coils in case of equilateral triangle to that for square is $$1 : \sqrt{y}$$ where $$y$$ is _________.

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NTA JEE Main 27th August 2021 Shift 1 - Question 28

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The alternating current is given by, $$i = \left\{\sqrt{42}\sin\left(\frac{2\pi}{T}t\right) + 10\right\}$$ A. The R.M.S. value of this current is _________ A.

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NTA JEE Main 27th August 2021 Shift 1 - Question 29

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A circuit is arranged as shown in figure. The output voltage $$V_0$$ is equal to _________ V.

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NTA JEE Main 27th August 2021 Shift 1 - Question 30

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A transmitting antenna has a height of 320 m and that of receiving antenna is 2000 m. The maximum distance between them for satisfactory communication in line of sight mode is $$d$$. The value of $$d$$ is _________ km.

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NTA JEE Main 27th August 2021 Shift 1 - Question 31

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The unit of the van der Waals gas equation parameter 'a' in $$\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$$ is:

A

dm$$^3$$ mol$$^{-1}$$

B

kg ms$$^{-1}$$

C

atm dm$$^6$$ mol$$^{-2}$$

D

kg ms$$^{-2}$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 32

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In polythionic acid, H$$_2$$S$$_x$$O$$_6$$ ($$x$$ = 3 to 5) the oxidation state(s) of sulphur is/are:

A

+6 only

B

+5 only

C

0 and +5 only

D

+3 and +5 only

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NTA JEE Main 27th August 2021 Shift 1 - Question 33

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Deuterium resembles hydrogen in properties but:

A

reacts vigorously than hydrogen

B

reacts just as hydrogen

C

emits $$\beta^+$$ particles

D

reacts slower than hydrogen

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NTA JEE Main 27th August 2021 Shift 1 - Question 34

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The number of water molecules in gypsum, dead burnt plaster and plaster of Paris, respectively are:

A

2, 0 and 0.5

B

0.5, 0 and 2

C

2, 0 and 1

D

5, 0 and 0.5

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NTA JEE Main 27th August 2021 Shift 1 - Question 35

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In which one of the following molecules strongest back donation of an electron pair from halide to boron is expected?

A

BI$$_3$$

B

BBr$$_3$$

C

BCl$$_3$$

D

BF$$_3$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 36

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The gas 'A' is having very low reactivity reaches to stratosphere. It is non-toxic and non-flammable but dissociated by UV-radiations in stratosphere. The intermediates formed initially from the gas 'A' are:

A

$$\dot{C}H_3 + \dot{C}F_2Cl$$

B

$$\dot{C}l + \dot{C}F_2Cl$$

C

$$Cl\dot{O} + \dot{C}H_3$$

D

$$Cl\dot{O} + \dot{C}F_2Cl$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 37

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Match List - I with List - II:
List-I (Property)                  List-II (Example)
(a) Diamagnetism               (i) MnO
(b) Ferrimagnetism            (ii) O$$_2$$
(c) Paramagnetism            (iii) NaCl
(d) Antiferromagnetism     (iv) Fe$$_3$$O$$_4$$
Choose the most appropriate answer from the options given below:

A

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

B

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

C

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

D

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

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NTA JEE Main 27th August 2021 Shift 1 - Question 38

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Tyndall effect is more effectively shown by:

A

lyophilic colloid

B

suspension

C

lyophobic colloid

D

true solution

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NTA JEE Main 27th August 2021 Shift 1 - Question 39

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Which refining process is generally used in the purification of low melting metals?

A

Chromatographic method

B

Electrolysis

C

Zone refining

D

Liquation

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NTA JEE Main 27th August 2021 Shift 1 - Question 40

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Match List - I with List - II:
List-I (Species)   List-II (Number of lone pairs of electrons on the central atom)
(a) XeF$$_2$$       (i) 0
(b) XeO$$_2$$F$$_2$$   (ii) 1
(c) XeO$$_3$$F$$_2$$   (iii) 2
(d) XeF$$_4$$       (iv) 3
Choose the most appropriate answer from the options given below:

A

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

B

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

C

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

D

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

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NTA JEE Main 27th August 2021 Shift 1 - Question 41

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The nature of oxides V$$_2$$O$$_3$$ and CrO is indexed as 'X' and 'Y' type respectively. The correct set of X and Y is:

A

X = acidic, Y = acidic

B

X = basic, Y = amphoteric

C

X = basic, Y = basic

D

X = amphoteric, Y = basic

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NTA JEE Main 27th August 2021 Shift 1 - Question 42

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Acidic ferric chloride solution on treatment with excess of potassium ferrocyanide gives a Prussian blue coloured colloidal species. It is:

A

HFe[Fe(CN)$$_6$$]

B

K$$_5$$Fe[Fe(CN)$$_6$$]$$_2$$

C

Fe$$_4$$[Fe(CN)$$_6$$]$$_3$$

D

KFe[Fe(CN)$$_6$$]

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NTA JEE Main 27th August 2021 Shift 1 - Question 43

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In the following sequence of reactions the P is:

A

B

C

D

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NTA JEE Main 27th August 2021 Shift 1 - Question 44

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Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Synthesis of ethyl phenyl ether may be achieved by Williamson synthesis.
Reason (R): Reaction of bromobenzene with sodium ethoxide yields ethyl phenyl ether.
In the light of the above statements, choose the most appropriate answer from the options given below.

A

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

B

(A) is correct but (R) is not correct

C

(A) is not correct but (R) is correct

D

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

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NTA JEE Main 27th August 2021 Shift 1 - Question 45

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The structure of the starting compound P used in the reaction given below is:

A

B

C

D

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NTA JEE Main 27th August 2021 Shift 1 - Question 46

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The major product of the following reaction is:

A

B

C

D

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NTA JEE Main 27th August 2021 Shift 1 - Question 47

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In the following sequence of reactions, the final product D is:

A

B

C

D

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NTA JEE Main 27th August 2021 Shift 1 - Question 48

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Which of the following is not a correct statement for primary aliphatic amines?

A

Primary amines can be prepared by the Gabriel phthalimide synthesis.

B

Primary amines are less basic than the secondary amines.

C

The intermolecular association in primary amines is less than the intermolecular association in secondary amines.

D

Primary amines on treating with nitrous acid solution form corresponding alcohols except methyl amine.

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NTA JEE Main 27th August 2021 Shift 1 - Question 49

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A

B

C

D
The correct statement about (A), (B), (C) and (D) is:

A

(B) and (C) are tranquillizers

B

(B), (C) and (D) are tranquillizers

C

(A) and (D) are tranquillizers

D

(A), (B) and (C) are narcotic analgesics

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NTA JEE Main 27th August 2021 Shift 1 - Question 50

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Out of following isomeric forms of uracil, which one is present in RNA?

A

B

C

D

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NTA JEE Main 27th August 2021 Shift 1 - Question 51

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The reaction that occurs in a breath analyser, a device used to determine the alcohol level in a person's blood stream is
$$2K_2Cr_2O_7 + 8H_2SO_4 + 3C_2H_6O \rightarrow 2Cr_2(SO_4)_3 + 3C_2H_4O_2 + 2K_2SO_4 + 11H_2O$$
If the rate of appearance of $$Cr_2(SO_4)_3$$ is 2.67 mol min$$^{-1}$$ at a particular time, the rate of disappearance of $$C_2H_6O$$ at the same time is _________ mol min$$^{-1}$$. (Nearest integer)

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NTA JEE Main 27th August 2021 Shift 1 - Question 52

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1 kg of 0.75 molal aqueous solution of sucrose can be cooled up to $$-4°$$C before freezing. The amount of ice (in g) that will be separated out is _________. (Nearest integer) [Given : $$K_f(H_2O) = 1.86$$ K kg mol$$^{-1}$$]

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NTA JEE Main 27th August 2021 Shift 1 - Question 53

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The number of $$f$$ electrons in the ground state electronic configuration of Np (Z = 93) is _________. (Integer answer)

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NTA JEE Main 27th August 2021 Shift 1 - Question 54

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The number of moles of NH$$_3$$, that must be added to 2 L of 0.80 M AgNO$$_3$$ in order to reduce the concentration of Ag$$^+$$ ions to $$5.0 \times 10^{-8}$$ M ($$K_{formation}$$ for $$[Ag(NH_3)_2]^+ = 1.0 \times 10^8$$) is _________. (Nearest integer)
[Assume no volume change on adding NH$$_3$$]

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NTA JEE Main 27th August 2021 Shift 1 - Question 55

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When 10 mL of an aqueous solution of KMnO$$_4$$ was titrated in acidic medium, equal volume of 0.1M of an aqueous solution of ferrous sulphate was required for complete discharge of colour. The strength of KMnO$$_4$$ in grams per litre is _________ $$\times 10^{-2}$$. (Nearest integer) [Atomic mass of K = 39, Mn = 55, O = 16]

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NTA JEE Main 27th August 2021 Shift 1 - Question 56

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The number of moles of CuO, that will be utilized in Dumas method for estimating nitrogen in a sample of 57.5 g of N, N-dimethylaminopentane is _________ $$\times 10^{-2}$$. (Nearest integer)

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NTA JEE Main 27th August 2021 Shift 1 - Question 57

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200 mL of 0.2 M HCl is mixed with 300 mL of 0.1 M NaOH. The molar heat of neutralization of this reaction is -57.1 kJ. The increase in temperature in °C of the system on mixing is $$x \times 10^{-2}$$. The value of x is _________. (Nearest integer)
[Given: Specific heat of water = 4.18 J g$$^{-1}$$ K$$^{-1}$$
Density of water = 1.00 g cm$$^{-3}$$]
(Assume no volume change on mixing)

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NTA JEE Main 27th August 2021 Shift 1 - Question 58

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The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is equal to $$\frac{h^2}{x \cdot m a_0^2}$$. The value of 10x is _________. ($$a_0$$ is radius of Bohr's orbit)
(Nearest integer)
[Given: $$\pi = 3.14$$]

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NTA JEE Main 27th August 2021 Shift 1 - Question 59

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In Carius method for estimation of halogens, 0.2 g of an organic compound gave 0.188 g of AgBr. The percentage of bromine in the compound is _________. (Nearest integer)
[Atomic mass: Ag = 108, Br = 80]

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NTA JEE Main 27th August 2021 Shift 1 - Question 60

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1 mol of an octahedral metal complex with formula MCl$$_3$$ . 2L on reaction with excess of AgNO$$_3$$ gives 1 mol of AgCl. The denticity of Ligand L is _________. (Integer answer)

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NTA JEE Main 27th August 2021 Shift 1 - Question 61

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If $$x^2 + 9y^2 - 4x + 3 = 0$$, $$x, y \in R$$, then $$x$$ and $$y$$ respectively lie in the intervals

A

$$\left[-\frac{1}{3}, \frac{1}{3}\right]$$ and $$\left[-\frac{1}{3}, \frac{1}{3}\right]$$

B

$$[1, 3]$$ and $$\left[-\frac{1}{3}, \frac{1}{3}\right]$$

C

$$\left[-\frac{1}{3}, \frac{1}{3}\right]$$ and $$[1, 3]$$

D

$$[1, 3]$$ and $$[1, 3]$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 62

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If $$S = \left\{z \in C : \frac{z-i}{z+2i} \in R\right\}$$, then

A

$$S$$ is a circle in the complex plane

B

$$S$$ contains exactly two elements

C

$$S$$ contains only one element

D

$$S$$ is a straight line in the complex plane

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NTA JEE Main 27th August 2021 Shift 1 - Question 63

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If for $$x, y \in R$$, $$x > 0$$, $$y = \log_{10} x + \log_{10} x^{1/3} + \log_{10} x^{1/9} + \ldots$$ upto $$\infty$$ terms and $$\frac{2+4+6+\ldots+2y}{3+6+9+\ldots+3y} = \frac{4}{\log_{10} x}$$, then the ordered pair $$(x, y)$$ is equal to

A

$$(10^6, 6)$$

B

$$(10^6, 9)$$

C

$$(10^2, 3)$$

D

$$(10^4, 6)$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 64

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If $$0 < x < 1$$, then $$\frac{3}{2}x^2 + \frac{5}{3}x^3 + \frac{7}{4}x^4 + \ldots$$, is equal to

A

$$x\left(\frac{x+1}{1-x}\right) + \log_e(1-x)$$

B

$$x\left(\frac{1-x}{1+x}\right) + \log_e(1-x)$$

C

$$\frac{1+x}{1-x} + \log_e(1-x)$$

D

$$\frac{1-x}{1+x} + \log_e(1-x)$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 65

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$$\sum_{k=0}^{20} \left({}^{20}C_k\right)^2$$ is equal to

A

$${}^{40}C_{21}$$

B

$${}^{41}C_{20}$$

C

$${}^{40}C_{20}$$

D

$${}^{40}C_{19}$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 66

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Let $$A$$ be a fixed point $$(0, 6)$$ and $$B$$ be a moving point $$(2t, 0)$$. Let $$M$$ be the mid-point of $$AB$$ and the perpendicular bisector of $$AB$$ meets the y-axis at $$C$$. The locus of the mid-point $$P$$ of MC is

A

$$3x^2 + 2y - 6 = 0$$

B

$$2x^2 - 3y + 9 = 0$$

C

$$3x^2 - 2y - 6 = 0$$

D

$$2x^2 + 3y - 9 = 0$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 67

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A tangent and a normal are drawn at the point $$P(2, -4)$$ on the parabola $$y^2 = 8x$$, which meet the directrix of the parabola at the points $$A$$ and $$B$$ respectively. If $$Q(a, b)$$ is a point such that $$AQBP$$ is a square, then $$2a + b$$ is equal to

A

-12

B

-20

C

-16

D

-18

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NTA JEE Main 27th August 2021 Shift 1 - Question 68

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If $$\alpha, \beta$$ are the distinct roots of $$x^2 + bx + c = 0$$, then $$\lim_{x \to \beta} \frac{e^{2(x^2+bx+c)} - 1 - 2(x^2+bx+c)}{(x-\beta)^2}$$ is equal to

A

$$2(b^2 + 4c)$$

B

$$b^2 - 4c$$

C

$$2(b^2 - 4c)$$

D

$$b^2 + 4c$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 69

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The statement $$(p \wedge (p \rightarrow q) \wedge (q \rightarrow r)) \rightarrow r$$ is

A

a tautology

B

equivalent to $$q \rightarrow \sim r$$

C

a fallacy

D

equivalent to $$p \rightarrow \sim r$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 70

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Let $$\frac{\sin A}{\sin B} = \frac{\sin(A-C)}{\sin(C-B)}$$, where $$A, B, C$$ are angles of a triangle $$ABC$$. If the lengths of the sides opposite these angles are $$a, b, c$$ respectively, then

A

$$b^2, c^2, a^2$$ are in A.P.

B

$$c^2, a^2, b^2$$ are in A.P.

C

$$b^2 - a^2 = a^2 + c^2$$

D

$$a^2, b^2, c^2$$ are in A.P.

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NTA JEE Main 27th August 2021 Shift 1 - Question 71

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If the matrix $$A = \begin{bmatrix} 0 & 2 \\ K & -1 \end{bmatrix}$$ satisfies $$A(A^3 + 3I) = 2I$$, then the value of $$K$$ is

A

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

B

1

C

-1

D

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

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NTA JEE Main 27th August 2021 Shift 1 - Question 72

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If $$(\sin^{-1} x)^2 - (\cos^{-1} x)^2 = a$$; $$0 < x < 1$$, $$a \neq 0$$, then the value of $$2x^2 - 1$$ is

A

$$\cos\left(\frac{2a}{\pi}\right)$$

B

$$\sin\left(\frac{2a}{\pi}\right)$$

C

$$\cos\left(\frac{4a}{\pi}\right)$$

D

$$\sin\left(\frac{4a}{\pi}\right)$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 73

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A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is

A

$$\frac{10}{2+3\sqrt{3}}$$

B

$$\frac{5}{3+\sqrt{3}}$$

C

$$\frac{10}{3+2\sqrt{3}}$$

D

$$\frac{5}{2+\sqrt{3}}$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 74

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If $$U_n = \left(1 + \frac{1}{n^2}\right)\left(1 + \frac{2^2}{n^2}\right)^2 \cdots \left(1 + \frac{n^2}{n^2}\right)^n$$, then $$\lim_{n \to \infty} (U_n)^{\frac{-4}{n^2}}$$ is equal to

A

$$\frac{16}{e^2}$$

B

$$\frac{4}{e}$$

C

$$\frac{4}{e^2}$$

D

$$\frac{e^2}{16}$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 75

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$$\int_6^{16} \frac{\log_e x^2}{\log_e x^2 + \log_e(x^2 - 44x + 484)} dx$$ is equal to

A

5

B

10

C

8

D

6

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NTA JEE Main 27th August 2021 Shift 1 - Question 76

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Let us consider a curve, $$y = f(x)$$ passing through the point $$(-2, 2)$$ and the slope of the tangent to the curve at any point $$(x, f(x))$$ is given by $$f(x) + xf'(x) = x^2$$. Then

A

$$x^3 - 3xf(x) - 4 = 0$$

B

$$x^2 + 2xf(x) - 12 = 0$$

C

$$x^3 + xf(x) + 12 = 0$$

D

$$x^2 + 2xf(x) + 4 = 0$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 77

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Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} = 2(y + 2\sin x - 5)x - 2\cos x$$ such that $$y(0) = 7$$. Then $$y(\pi)$$ is equal to

A

$$7e^{\pi^2} + 5$$

B

$$e^{\pi^2} + 5$$

C

$$2e^{\pi^2} + 5$$

D

$$3e^{\pi^2} + 5$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 78

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The distance of the point $$(1, -2, 3)$$ from the plane $$x - y + z = 5$$ measured parallel to a line, whose direction ratios are $$2, 3, -6$$, is

A

2

B

5

C

3

D

1

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NTA JEE Main 27th August 2021 Shift 1 - Question 79

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Equation of a plane at a distance $$\sqrt{\frac{2}{21}}$$ units from the origin, which contains the line of intersection of the planes $$x - y - z - 1 = 0$$ and $$2x + y - 3z + 4 = 0$$, is

A

$$-x + 2y + 2z - 3 = 0$$

B

$$3x - 4z + 3 = 0$$

C

$$3x - 1y - 5z + 2 = 0$$

D

$$4x - y - 5z + 2 = 0$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 80

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When a certain biased die is rolled, a particular face occurs with probability $$\frac{1}{6} - x$$ and its opposite face occurs with probability $$\frac{1}{6} + x$$. All other faces occur with probability $$\frac{1}{6}$$. Note that opposite faces sum to 7 in any die. If $$0 < x < \frac{1}{6}$$, and the probability of obtaining total sum = 7, when such a die is rolled twice, is $$\frac{13}{96}$$, then the value of $$x$$ is

A

$$\frac{1}{16}$$

B

$$\frac{1}{12}$$

C

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

D

$$\frac{1}{9}$$

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NTA JEE Main 27th August 2021 Shift 1 - Question 81

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If $$A = \{x \in R : |x-2| > 1\}$$, $$B = \{x \in R : \sqrt{x^2 - 3} > 1\}$$, $$C = \{x \in R : |x-4| \geq 2\}$$ and $$Z$$ is the set of all integers, then the number of subsets of the set $$(A \cap B \cap C)^c \cap Z$$ is _________.

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NTA JEE Main 27th August 2021 Shift 1 - Question 82

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A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55, is _________.

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NTA JEE Main 27th August 2021 Shift 1 - Question 83

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Let the equation $$x^2 + y^2 + px + (1-p)y + 5 = 0$$ represent circles of varying radius $$r \in (0, 5]$$. Then the number of elements in the set $$S = \{q : q = p^2$$ and $$q$$ is an integer$$\}$$ is _________.

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NTA JEE Main 27th August 2021 Shift 1 - Question 84

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If the minimum area of the triangle formed by a tangent to the ellipse $$\frac{x^2}{b^2} + \frac{y^2}{4a^2} = 1$$ and the co-ordinate axis is $$kab$$, then $$k$$ is equal to _________.

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NTA JEE Main 27th August 2021 Shift 1 - Question 85

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Let $$n$$ be an odd natural number such that the variance of 1, 2, 3, 4, ..., n is 14. Then $$n$$ is equal to _________.

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NTA JEE Main 27th August 2021 Shift 1 - Question 86

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If the system of linear equations
$$2x + y - z = 3$$
$$x - y - z = \alpha$$
$$3x + 3y + \beta z = 3$$
has infinitely many solutions, then $$|\alpha + \beta - \alpha\beta|$$ is equal to _________.

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NTA JEE Main 27th August 2021 Shift 1 - Question 87

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If $$y^{1/4} + y^{-1/4} = 2x$$, and $$(x^2 - 1)\frac{d^2y}{dx^2} + \alpha x\frac{dy}{dx} + \beta y = 0$$, then $$|\alpha - \beta|$$ is equal to _________.

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NTA JEE Main 27th August 2021 Shift 1 - Question 88

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The number of distinct real roots of the equation $$3x^4 + 4x^3 - 12x^2 + 4 = 0$$ is _________.

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NTA JEE Main 27th August 2021 Shift 1 - Question 89

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If $$\int \frac{dx}{(x^2+x+1)^2} = a\tan^{-1}\left(\frac{2x+1}{\sqrt{3}}\right) + b\left(\frac{2x+1}{x^2+x+1}\right) + C$$,$$x > 0$$ where $$C$$ is the constant of integration, then the value of $$9\left(\sqrt{3}a + b\right)$$ is equal to _________.

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NTA JEE Main 27th August 2021 Shift 1 - Question 90

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Let $$\vec{a} = \hat{i} + 5\hat{j} + \alpha\hat{k}$$, $$\vec{b} = \hat{i} + 3\hat{j} + \beta\hat{k}$$ and $$\vec{c} = -\hat{i} + 2\hat{j} - 3\hat{k}$$ be three vectors such that, $$|\vec{b} \times \vec{c}| = 5\sqrt{3}$$ and $$\vec{a}$$ is perpendicular to $$\vec{b}$$. Then the greatest amongst the values of $$|\vec{a}|^2$$ is _________.

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