NTA JEE Main 29th January 2023 Shift 1

NTA JEE Main 29th January 2023 Shift 1 - Question 1

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Match List I with List II:

List-I (Physical Quantity)	List-II (Dimensional Formula)
A. Pressure gradient	I. $$[M^0L^2T^{-2}]$$
B. Energy density	II. $$[M^1L^{-1}T^{-2}]$$
C. Electric Field	III. $$[M^1L^{-2}T^{-2}]$$
D. Latent heat	IV. $$[M^1L^1T^{-3}A^{-1}]$$

Choose the correct answer from the options given below:

A

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

B

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

C

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

D

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

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NTA JEE Main 29th January 2023 Shift 1 - Question 2

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A stone is projected at angle $$30°$$ to the horizontal. The ratio of kinetic energy of the stone at point of projection to its kinetic energy at the highest point of flight will be:

A

$$1 : 2$$

B

$$1 : 4$$

C

$$4 : 1$$

D

$$4 : 3$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 3

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A car is moving on a horizontal curved road with radius $$50$$ m. The approximate maximum speed of car will be, if friction between tyres and road is $$0.34$$. [Take $$g = 10$$ m s$$^{-2}$$]

A

$$3.4$$ m s$$^{-1}$$

B

$$22.4$$ m s$$^{-1}$$

C

$$13$$ m s$$^{-1}$$

D

$$17$$ m s$$^{-1}$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 4

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A block of mass $$m$$ slides down the plane inclined at angle $$30°$$ with an acceleration $$\frac{g}{4}$$. The value of coefficient of kinetic friction will be:

A

$$\frac{2\sqrt{3}+1}{2}$$

B

$$\frac{1}{2\sqrt{3}}$$

C

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

D

$$\frac{2\sqrt{3}-1}{2}$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 5

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Two particles of equal mass $$m$$ move in a circle of radius $$r$$ under the action of their mutual gravitational attraction. The speed of each particle will be:

A

$$\sqrt{\frac{Gm}{2r}}$$

B

$$\sqrt{\frac{4Gm}{r}}$$

C

$$\sqrt{\frac{Gm}{r}}$$

D

$$\sqrt{\frac{Gm}{4r}}$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 6

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Surface tension of a soap bubble is $$2.0 \times 10^{-2}$$ N m$$^{-1}$$. Work done to increase the radius of soap bubble from $$3.5$$ cm to $$7$$ cm will be: [Take $$\pi = \frac{22}{7}$$]

A

$$0.72 \times 10^{-4}$$ J

B

$$5.76 \times 10^{-4}$$ J

C

$$18.48 \times 10^{-4}$$ J

D

$$9.24 \times 10^{-4}$$ J

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NTA JEE Main 29th January 2023 Shift 1 - Question 7

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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: If $$dQ$$ and $$dW$$ represent the heat supplied to the system and the work done on the system respectively. Then according to the first law of thermodynamics $$dQ = dU - dW$$.
Reason R: First law of thermodynamics is based on law of conservation of energy.
In the light of the above statements, choose the correct answer from the option given below:

A

A is correct but R is not correct

B

A is not correct but R is correct

C

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

D

Both A and R are correct but R is not the correct explanation of A

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NTA JEE Main 29th January 2023 Shift 1 - Question 8

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A bicycle tyre is filled with air having pressure of $$270$$ kPa at $$27°$$C. The approximate pressure of the air in the tyre when the temperature increases to $$36°$$C is

A

$$270$$ kPa

B

$$262$$ kPa

C

$$278$$ kPa

D

$$360$$ kPa

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NTA JEE Main 29th January 2023 Shift 1 - Question 9

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A person observes two moving trains, $$A$$ reaching the station and $$B$$ leaving the station with equal speed of $$30$$ m s$$^{-1}$$. If both trains emit sounds with frequency $$300$$ Hz, (Speed of sound: $$330$$ m s$$^{-1}$$) approximate difference of frequencies heard by the person will be:

A

$$33$$ Hz

B

$$55$$ Hz

C

$$80$$ Hz

D

$$10$$ Hz

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NTA JEE Main 29th January 2023 Shift 1 - Question 10

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In a cuboid of dimension $$2L \times 2L \times L$$, a charge $$q$$ is placed at the centre of the surface $$S$$ having area of $$4L^2$$. The flux through the opposite surface to $$S$$ is given by

A

$$\frac{q}{12\epsilon_0}$$

B

$$\frac{q}{3\epsilon_0}$$

C

$$\frac{q}{2\epsilon_0}$$

D

$$\frac{q}{6\epsilon_0}$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 11

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Ratio of thermal energy released in two resistor $$R$$ and $$3R$$ connected in parallel in an electric circuit is:

A

$$3 : 1$$

B

$$1 : 1$$

C

$$1 : 3$$

D

$$1 : 27$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 12

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A single current carrying loop of wire carrying current $$I$$ flowing in anticlockwise direction seen from $$+ve$$ z direction and lying in $$xy$$ plane is shown in figure. The plot of $$\hat{j}$$ component of magnetic field $$(B_y)$$ at a distance $$a$$ (less than radius of the coil) and on $$yz$$ plane vs $$z$$ coordinate looks like

A

B

C

D

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NTA JEE Main 29th January 2023 Shift 1 - Question 13

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The magnitude of magnetic induction at mid-point $$O$$ due to current arrangement as shown in figure will be

A

$$\frac{\mu_0 I}{2\pi a}$$

B

$$0$$

C

$$\frac{\mu_0 I}{4\pi a}$$

D

$$\frac{\mu_0 I}{\pi a}$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 14

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Find the mutual inductance in the arrangement, when a small circular loop of wire of radius $$R$$ is placed inside a large square loop of wire of side $$L(L \gg R)$$. The loops are coplanar and their centres coincide: 

A

$$M = \frac{\sqrt{2}\mu_0 R^2}{L}$$

B

$$M = \frac{2\sqrt{2}\mu_0 R}{L^2}$$

C

$$M = \frac{2\sqrt{2}\mu_0 R^2}{L}$$

D

$$M = \frac{\sqrt{2}\mu_0 R}{L^2}$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 15

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Which of the following are true?
A. Speed of light in vacuum is dependent on the direction of propagation.
B. Speed of light in a medium is independent of the wavelength of light.
C. The speed of light is independent of the motion of the source.
D. The speed of light in a medium is independent of intensity.
Choose the correct answer from the question given below:

A

A and C only

B

B and D only

C

B and C only

D

C and D only

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NTA JEE Main 29th January 2023 Shift 1 - Question 16

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The threshold wavelength for photoelectric emission from a material is $$5500$$ $$\text{\AA}$$. Photoelectrons will be emitted, when this material is illuminated with monochromatic radiation from a
A. 75 W infra-red lamp
B. 10 W infra-red lamp
C. 75 W ultra-violet lamp
D. 10 W ultra-violet lamp
Choose the correct answer from the options given below:

A

B and C only

B

A and D only

C

C only

D

C and D only

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NTA JEE Main 29th January 2023 Shift 1 - Question 17

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If a radioactive element having half-life of $$30$$ min is undergoing beta decay, the fraction of radioactive element remains undecayed after $$90$$ min will be:

A

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

B

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

C

$$\frac{1}{4}$$

D

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

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NTA JEE Main 29th January 2023 Shift 1 - Question 18

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Which of the following statement is not correct in the case of light emitting diodes?
A. It is a heavily doped p-n junction.
B. It emits light only when it is forward biased.
C. It emits light only when it is reverse biased.
D. The energy of the light emitted is equal to or slightly less than the energy gap of the semiconductor used.
Choose the correct answer from the options given below:

A

C and D

B

A

C

C

D

B

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NTA JEE Main 29th January 2023 Shift 1 - Question 19

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If the height of transmitting and receiving antennas are $$80$$ m each, the maximum line of sight distance will be:
Given: Earth's radius $$= 6.4 \times 10^6$$ m.

A

$$32$$ km

B

$$28$$ km

C

$$36$$ km

D

$$64$$ km

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NTA JEE Main 29th January 2023 Shift 1 - Question 20

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A tennis ball is dropped on to the floor from a height of $$9.8$$ m. It rebounds to a height $$5.0$$ m. Ball comes in contact with the floor for $$0.2$$ s. The average acceleration during contact is ______ m s$$^{-2}$$. [Given $$g = 10$$ m s$$^{-2}$$]

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NTA JEE Main 29th January 2023 Shift 1 - Question 21

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A $$0.4$$ kg mass takes $$8$$ s to reach ground when dropped from a certain height $$P$$ above surface of earth. The loss of potential energy in the last second of fall is ______ J. [Take $$g = 10$$ m s$$^{-2}$$]

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NTA JEE Main 29th January 2023 Shift 1 - Question 22

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A solid sphere of mass $$2$$ kg is making pure rolling on a horizontal surface with kinetic energy $$2240$$ J. The velocity of centre of mass of the sphere will be ______ m s$$^{-1}$$.

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NTA JEE Main 29th January 2023 Shift 1 - Question 23

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A body cools from $$60°$$C to $$40°$$C in $$6$$ minutes. If, temperature of surroundings is $$10°$$C. Then, after the next 6 minutes, its temperature will be ______ °C.

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NTA JEE Main 29th January 2023 Shift 1 - Question 24

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Two simple harmonic waves having equal amplitudes of $$8$$ cm and equal frequency of $$10$$ Hz are moving along the same direction. The resultant amplitude is also $$8$$ cm. The phase difference between the individual waves is ______ degree.

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NTA JEE Main 29th January 2023 Shift 1 - Question 25

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A point charge $$q_1 = 4q_0$$ is placed at origin. Another point charge $$q_2 = -q_0$$ is placed at $$x = 12$$ cm. Charge of proton is $$q_0$$. The proton is placed on $$x$$-axis so that the electrostatic force on the proton in zero. In this situation, the position of the proton from the origin is ______ cm.

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NTA JEE Main 29th January 2023 Shift 1 - Question 26

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A certain elastic conducting material is stretched into a circular loop. It is placed with its plane perpendicular to a uniform magnetic field $$B = 0.8$$ T. When released the radius of the loop starts shrinking at a constant rate of $$2$$ cm s$$^{-1}$$. The induced emf in the loop at an instant when the radius of the loop is $$10$$ cm will be ______ mV.

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NTA JEE Main 29th January 2023 Shift 1 - Question 27

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In a Young's double slit experiment, two slits are illuminated with a light of wavelength $$800$$ nm. The line joining $$A_1P$$ is perpendicular to $$A_1A_2$$ as shown in the figure. If the first minimum is detected at $$P$$, the value of slits separation $$a$$ will be:

The distance of screen from slits $$D = 5$$ cm. Answer in mm.

A

0.4 mm

B

0.5 mm

C

0.2 mm

D

0.1 mm

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NTA JEE Main 29th January 2023 Shift 1 - Question 28

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As shown in figures, three identical polaroids $$P_1$$, $$P_2$$ and $$P_3$$ are placed one after another. The pass axis of $$P_2$$ and $$P_3$$ are inclined at angle of $$60°$$ and $$90°$$ with respect to axis of $$P_1$$. The source $$S$$ has an intensity of $$256$$ W m$$^{-2}$$. The intensity of light at point $$O$$ is ______ W m$$^{-2}$$.

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NTA JEE Main 29th January 2023 Shift 1 - Question 29

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A radioactive element $$^{242}_{92}$$X emits two $$\alpha$$-particles, one electron and two positrons. The product nucleus is represented by $$^{234}_{P}$$Y. The value of $$P$$ is ______.

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NTA JEE Main 29th January 2023 Shift 1 - Question 30

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In a metre bridge experiment the balance point is obtained if the gaps are closed by $$2$$ $$\Omega$$ and $$3$$ $$\Omega$$. A shunt of $$X$$ $$\Omega$$ is added to $$3$$ $$\Omega$$ resistor to shift the balancing point by $$22.5$$ cm. The value of $$X$$ is ______.

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NTA JEE Main 29th January 2023 Shift 1 - Question 31

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The shortest wavelength of hydrogen atom in Lyman series is $$\lambda$$. The longest wavelength in Balmer series of He$$^+$$ is

A

$$\frac{5}{9}\lambda$$

B

$$\frac{9\lambda}{5}$$

C

$$\frac{36\lambda}{5}$$

D

$$\frac{5\lambda}{9}$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 32

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The bond dissociation energy is highest for

A

Cl$$_2$$

B

I$$_2$$

C

Br$$_2$$

D

F$$_2$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 33

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For 1 mol of gas, the plot of pV vs p is shown below. p is the pressure and V is the volume of the gas.

What is the value of compressibility factor at point A?

A

$$1 - \frac{a}{RTV}$$

B

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

C

$$1 - \frac{b}{V}$$

D

$$1 + \frac{a}{RTV}$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 34

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Which of the given compounds can enhance the efficiency of hydrogen storage tank?

A

Li/P$$_4$$

B

SiH$$_4$$

C

NaNi$$_5$$

D

Di-isobutylaluminium hydride

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NTA JEE Main 29th January 2023 Shift 1 - Question 35

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The magnetic behaviour of Li$$_2$$O, Na$$_2$$O$$_2$$ and KO$$_2$$, respectively, are

A

diamagnetic, paramagnetic and diamagnetic

B

paramagnetic, paramagnetic and diamagnetic

C

paramagnetic, diamagnetic and paramagnetic

D

diamagnetic, diamagnetic and paramagnetic

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NTA JEE Main 29th January 2023 Shift 1 - Question 36

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The correct order of hydration enthalpies is
(A) K$$^+$$
(B) Rb$$^+$$
(C) Mg$$^{2+}$$
(D) Cs$$^+$$
(E) Ca$$^{2+}$$
Choose the correct answer from the options below:

A

C $$>$$ A $$>$$ E $$>$$ B $$>$$ D

B

E $$>$$ C $$>$$ A $$>$$ B $$>$$ D

C

C $$>$$ E $$>$$ A $$>$$ D $$>$$ B

D

C $$>$$ E $$>$$ A $$>$$ B $$>$$ D

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NTA JEE Main 29th January 2023 Shift 1 - Question 37

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During the borax bead test with CuSO$$_4$$, a blue green colour of the bead was observed in oxidising flame due to the formation of

A

Cu$$_3$$B$$_2$$

B

Cu

C

Cu(BO$$_2$$)$$_2$$

D

CuO

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NTA JEE Main 29th January 2023 Shift 1 - Question 38

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Compound that will give positive Lassaigne's test for both nitrogen and halogen is

A

N$$_2$$H$$_4$$ $$\cdot$$ HCl

B

CH$$_3$$NH$$_2$$. HCl

C

NH$$_4$$Cl

D

NH$$_2$$OH. HCl

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NTA JEE Main 29th January 2023 Shift 1 - Question 39

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Correct statement about smog is

A

NO$$_2$$ is present in classical smog

B

Both NO$$_2$$ and SO$$_2$$ are present in classical smog

C

Photochemical smog has high concentration of oxidizing agents

D

Classical smog also has high concentration of oxidizing agents

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NTA JEE Main 29th January 2023 Shift 1 - Question 40

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The standard electrode potential (M$$^{3+}$$/M$$^{2+}$$) for V, Cr, Mn & Co are $$-0.26$$ V, $$-0.41$$ V, $$+1.57$$ V and $$+1.97$$ V, respectively. The metal ions which can liberate H$$_2$$ from a dilute acid are

A

V$$^{2+}$$ and Mn$$^{2+}$$

B

Cr$$^{2+}$$ and Co$$^{2+}$$

C

V$$^{2+}$$ and Cr$$^{2+}$$

D

Mn$$^{2+}$$ and Co$$^{2+}$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 41

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Following figure shows dependence of molar conductance of two electrolytes on concentration. $$\Lambda^0_m$$ is the limiting molar conductivity.

The number of Incorrect statement(s) from the following is ______
(A) $$\Lambda^0_m$$ for electrolyte A is obtained by extrapolation
(B) For electrolyte B, $$\Lambda_m$$ Vs $$\sqrt{c}$$ graph is a straight line with intercept equal to $$\Lambda^0_m$$
(C) At infinite dilution, the value of degree of dissociation approach zero for electrolyte B.
(D) $$\Lambda_m$$ for any electrolyte A or B can be calculated using $$\lambda°$$ for individual ions.

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NTA JEE Main 29th January 2023 Shift 1 - Question 42

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Which of the following salt solutions would coagulate the colloid solution formed when FeCl$$_3$$ is added to NaOH solution, at the fastest rate?

A

10 mL of 0.2 mol dm$$^{-3}$$ AlCl$$_3$$

B

10 mL of 0.1 mol dm$$^{-3}$$ Na$$_2$$SO$$_4$$

C

10 mL of 0.1 mol dm$$^{-3}$$ Ca$$_3$$(PO$$_4$$)$$_2$$

D

10 mL of 0.15 mol dm$$^{-3}$$ CaCl$$_2$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 43

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The reaction representing the Mond process for metal refining is ______

A

$$\text{Ni} + 4\text{CO} \xrightarrow{\Delta} \text{Ni(CO)}_4$$

B

$$2\text{K[Au(CN)}_2] + \text{Zn} \xrightarrow{\Delta} \text{K}_2[\text{Zn(CN)}_4] + 2\text{Au}$$

C

$$\text{Zr} + 2\text{I}_2 \xrightarrow{\Delta} \text{ZrI}_4$$

D

$$\text{ZnO} + \text{C} \xrightarrow{\Delta} \text{Zn} + \text{CO}$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 44

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"A" obtained by Ostwald's method involving air oxidation of NH$$_3$$, upon further air oxidation produces "B". "B" on hydration forms an oxoacid of Nitrogen along with evolution of "A". The oxoacid also produces "A" and gives positive brown ring test

A

NO$$_2$$, N$$_2$$O$$_5$$

B

NO$$_2$$, N$$_2$$O$$_4$$

C

NO, NO$$_2$$

D

N$$_2$$O$$_3$$, NO$$_2$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 45

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Chiral complex from the following is:
Here en = ethylene diamine

A

cis $$-$$ [PtCl$$_2$$(en)$$_2$$]$$^{2+}$$

B

trans $$-$$ [PtCl$$_2$$(en)$$_2$$]$$^{2+}$$

C

cis $$-$$ [PtCl$$_2$$(NH$$_3$$)$$_2$$]

D

trans $$-$$ [Co(NH$$_3$$)$$_4$$Cl$$_2$$]$$^+$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 46

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Identify the correct order for the given property for following compounds

Choose the correct answer from the option below:

A

(B), (C) and (D) only

B

(A), (C) and (E) only

C

(A), (C) and (D) only

D

(A), (B) and (E) only

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NTA JEE Main 29th January 2023 Shift 1 - Question 47

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The increasing order of pK$$_a$$ for the following phenols is
(A) 2,4-Dinitrophenol
(B) 4-Nitrophenol
(C) 2,4,5-Trimethylphenol
(D) Phenol
(E) 3-Chlorophenol
Choose the correct answer from the option given below:

A

(A), (E), (B), (D), (C)

B

(A), (B), (E), (D), (C)

C

(C), (D), (E), (B), (A)

D

(C), (E), (D), (B), (A)

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NTA JEE Main 29th January 2023 Shift 1 - Question 48

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Match List I with List II.

List I (Reaction)	List II (Reagents)
(A) Hoffmann Degradation	(I) Conc. KOH, $$\Delta$$
(B) Clemmensen reduction	(II) CHCl$$_3$$, NaOH/H$$_3$$O$$^+$$
(C) Cannizzaro reaction	(III) Br$$_2$$, NaOH
(D) Reimer-Tiemann reaction	(IV) Zn $$-$$ Hg/HCl

A

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

B

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

C

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

D

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

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NTA JEE Main 29th January 2023 Shift 1 - Question 49

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The major product 'P' for the following sequence of reactions is:

A

B

C

D

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NTA JEE Main 29th January 2023 Shift 1 - Question 50

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Match List I with List II

List I (Antimicrobials)	List II (Names)
(A) Narrow Spectrum Antibiotic	(I) Furacin
(B) Antiseptic	(II) Sulphur dioxide
(C) Disinfectants	(III) Penicillin-G
(D) Broad spectrum antibiotic	(IV) Chloramphenicol

A

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

B

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

C

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

D

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

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NTA JEE Main 29th January 2023 Shift 1 - Question 51

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Number of cyclic tripeptides formed with 2 amino acids A and B is:

A

$$2$$

B

$$3$$

C

$$5$$

D

$$4$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 52

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The number of molecules or ions from the following, which do not have odd number of electrons are ______.
(A) NO$$_2$$
(B) ICl$$_4^-$$
(C) BrF$$_3$$
(D) ClO$$_2$$
(E) NO$$_2^+$$
(F) NO

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NTA JEE Main 29th January 2023 Shift 1 - Question 53

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Consider the following reaction approaching equilibrium at $$27°$$C and 1 atm pressure
$$\text{A} + \text{B} \underset{K_r=10^2}{\overset{K_f=10^3}{\rightleftharpoons}} \text{C} + \text{D}$$
The standard Gibb's energy change $$(\Delta_r G°)$$ at $$27°$$C is $$(-)$$ ______ kJ mol$$^{-1}$$. (Nearest integer).
(Given: R $$= 8.3$$ J K$$^{-1}$$ mol$$^{-1}$$ and $$\ln 10 = 2.3$$)

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NTA JEE Main 29th January 2023 Shift 1 - Question 54

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Water decomposes at 2300 K
$$\text{H}_2\text{O}(g) \to \text{H}_2(g) + \frac{1}{2}\text{O}_2(g)$$
The percent of water decomposing at 2300 K and 1 bar is ______ (Nearest integer). Equilibrium constant for the reaction is $$2 \times 10^{-3}$$ at 2300 K

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NTA JEE Main 29th January 2023 Shift 1 - Question 55

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Millimoles of calcium hydroxide required to produce 100 mL of the aqueous solution of pH 12 is $$x \times 10^{-1}$$. The value of $$x$$ is ______ (Nearest integer). Assume complete dissociation.

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NTA JEE Main 29th January 2023 Shift 1 - Question 56

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Following chromatogram was developed by adsorption of compound 'A' on a 6 cm TLC glass plate. Retardation factor of the compound 'A' is ______ $$\times 10^{-1}$$.

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NTA JEE Main 29th January 2023 Shift 1 - Question 57

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17 mg of a hydrocarbon (M.F. C$$_{10}$$H$$_{16}$$) takes up 8.40 mL of the H$$_2$$ gas measured at 0°C and 760 mm of Hg. Ozonolysis of the same hydrocarbon yields

The number of double bond/s present in the hydrocarbon is

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NTA JEE Main 29th January 2023 Shift 1 - Question 58

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Solid Lead nitrate is dissolved in 1 litre of water. The solution was found to boil at $$100.15°$$C. When $$0.2$$ mol of NaCl is added to the resulting solution, it was observed that the solution froze at $$-0.8°$$C. The solutibility product of PbCl$$_2$$ formed is ______ $$\times 10^{-6}$$ at 298 K. (Nearest integer)
Given: K$$_b = 0.5$$ K kg mol$$^{-1}$$ and K$$_f = 1.8$$ kg mol$$^{-1}$$. Assume molality to be equal to molarity in all cases.

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NTA JEE Main 29th January 2023 Shift 1 - Question 59

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For certain chemical reaction X $$\to$$ Y, the rate of formation of product is plotted against the time as shown in the figure. The number of Correct statement/s from the following is ______

(A) Over all order of this reaction is one
(B) Order of this reaction can't be determined
(C) In region-I and III, the reaction is of first and zero order respectively
(D) In region-II, the reaction is of first order
(E) In region-II, the order of reaction is in the range of 0.1 to 0.9.

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NTA JEE Main 29th January 2023 Shift 1 - Question 60

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The sum of bridging carbonyls in W(CO)$$_6$$ and Mn$$_2$$(CO)$$_{10}$$ is ______.

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NTA JEE Main 29th January 2023 Shift 1 - Question 61

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Let $$\lambda \neq 0$$ be a real number. Let $$\alpha, \beta$$ be the roots of the equation $$14x^2 - 31x + 3\lambda = 0$$ and $$\alpha, \gamma$$ be the roots of the equation $$35x^2 - 53x + 4\lambda = 0$$. Then $$\frac{3\alpha}{\beta}$$ and $$\frac{4\alpha}{\gamma}$$ are the roots of the equation:

A

$$7x^2 + 245x - 250 = 0$$

B

$$7x^2 - 245x + 250 = 0$$

C

$$49x^2 - 245x + 250 = 0$$

D

$$49x^2 + 245x + 250 = 0$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 62

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For two non-zero complex numbers $$z_1$$ and $$z_2$$, if $$\text{Re}(z_1 z_2) = 0$$ and $$\text{Re}(z_1 + z_2) = 0$$, then which of the following are possible?
(A) Im $$(z_1) > 0$$ and Im $$(z_2) > 0$$
(B) Im $$(z_1) < 0$$ and Im $$(z_2) > 0$$
(C) Im $$(z_1) > 0$$ and Im $$(z_2) < 0$$
(D) Im $$(z_1) < 0$$ and Im $$(z_2) < 0$$
Choose the correct answer from the options given below:

A

B and D

B

B and C

C

A and B

D

A and C

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NTA JEE Main 29th January 2023 Shift 1 - Question 63

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Let $$f(\theta) = 3\left(\sin^4\left(\frac{3\pi}{2} - \theta\right) + \sin^4(3\pi + \theta)\right) - 2\left(1 - \sin^2 2\theta\right)$$ and $$S = \left\{\theta \in [0, \pi] : f'(\theta) = -\frac{\sqrt{3}}{2}\right\}$$. If $$4\beta = \sum_{\theta \in S} \theta$$ then $$f(\beta)$$ is equal to

A

$$\frac{11}{8}$$

B

$$\frac{5}{4}$$

C

$$\frac{9}{8}$$

D

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

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NTA JEE Main 29th January 2023 Shift 1 - Question 64

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A light ray emits from the origin making angle $$30°$$ with the positive $$x$$-axis. After getting reflected by the line $$x + y = 1$$, if this ray intersects x-axis at Q, then the abscissa of Q is

A

$$\frac{2}{(\sqrt{3}-1)}$$

B

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

C

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

D

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

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NTA JEE Main 29th January 2023 Shift 1 - Question 65

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Let $$B$$ and $$C$$ be the two points on the line $$y + x = 0$$ such that $$B$$ and $$C$$ are symmetric with respect to the origin. Suppose $$A$$ is a point on $$y - 2x = 2$$ such that $$\triangle ABC$$ is an equilateral triangle. Then, the area of the $$\triangle ABC$$ is

A

$$3\sqrt{3}$$

B

$$2\sqrt{3}$$

C

$$\frac{8}{\sqrt{3}}$$

D

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

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NTA JEE Main 29th January 2023 Shift 1 - Question 66

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Let the tangents at the points $$A(4, -11)$$ and $$B(8, -5)$$ on the circle $$x^2 + y^2 - 3x + 10y - 15 = 0$$, intersect at the point $$C$$. Then the radius of the circle, whose centre is $$C$$ and the line joining $$A$$ and $$B$$ is its tangent, is equal to

A

$$\frac{3\sqrt{3}}{4}$$

B

$$2\sqrt{13}$$

C

$$\sqrt{13}$$

D

$$\frac{2\sqrt{13}}{3}$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 67

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Let $$x = 2$$ be a root of the equation $$x^2 + px + q = 0$$ and $$f(x) = \begin{cases} \frac{1-\cos(x^2-4px+q^2+8q+16)}{(x-2p)^4}, & x \neq 2p \\ 0, & x = 2p \end{cases}$$. Then $$\lim_{x \to 2p^+} [f(x)]$$
where $$[.]$$ denotes greatest integer function, is

A

$$2$$

B

$$1$$

C

$$0$$

D

$$-1$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 68

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If $$p, q$$ and $$r$$ are three propositions, then which of the following combination of truth values of $$p$$, $$q$$ and $$r$$ makes the logical expression $$\{(p \vee q) \wedge ((\neg p) \vee r)\} \to ((\neg q) \vee r)$$ false?

A

$$p = T, q = F, r = T$$

B

$$p = T, q = T, r = F$$

C

$$p = F, q = T, r = F$$

D

$$p = T, q = F, r = F$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 69

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Let $$\alpha$$ and $$\beta$$ be real numbers. Consider a $$3 \times 3$$ matrix $$A$$ such that $$A^2 = 3A + \alpha I$$. If $$A^4 = 21A + \beta I$$, then

A

$$\alpha = 1$$

B

$$\alpha = 4$$

C

$$\beta = 8$$

D

$$\beta = -8$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 70

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Consider the following system of equations
$$\alpha x + 2y + z = 1$$
$$2\alpha x + 3y + z = 1$$
$$3x + \alpha y + 2z = \beta$$
For some $$\alpha, \beta \in \mathbb{R}$$. Then which of the following is NOT correct.

A

It has no solution if $$\alpha = -1$$ and $$\beta \neq 2$$

B

It has no solution for $$\alpha = -1$$ and for all $$\beta \in \mathbb{R}$$

C

It has no solution for $$\alpha = 3$$ and for all $$\beta \neq 2$$

D

It has a solution for all $$\alpha \neq -1$$ and $$\beta = 2$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 71

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The domain of $$f(x) = \dfrac{\log_{(x+1)}(x-2)}{e^{2\log_e x} - (2x+3)}$$, $$x \in R$$ is

A

$$\mathbb{R} - \{-1, 3\}$$

B

$$(2, \infty) - \{3\}$$

C

$$(-1, \infty) - \{3\}$$

D

$$\mathbb{R} - \{3\}$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 72

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Let $$f : R \to R$$ be a function such that $$f(x) = \frac{x^2+2x+1}{x^2+1}$$. Then

A

$$f(x)$$ is many-one in $$(-\infty, -1)$$

B

$$f(x)$$ is many-one in $$(1, \infty)$$

C

$$f(x)$$ is one-one in $$[1, \infty)$$ but not in $$(-\infty, \infty)$$

D

$$f(x)$$ is one-one in $$(-\infty, \infty)$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 73

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Let $$f(x) = x + \frac{a}{\pi^2-4}\sin x + \frac{b}{\pi^2-4}\cos x$$, $$x \in \mathbb{R}$$ be a function which satisfies $$f(x) = x + \int_0^{\pi/2} \sin(x+y)f(y)dy$$. Then $$(a+b)$$ is equal to

A

$$-\pi(\pi+2)$$

B

$$-2\pi(\pi+2)$$

C

$$-2\pi(\pi-2)$$

D

$$-\pi(\pi-2)$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 74

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Let $$[x]$$ denote the greatest integer $$\leq x$$. Consider the function $$f(x) = \max\{x^2, 1 + [x]\}$$. Then the value of the integral $$\int_0^2 f(x) dx$$ is:

A

$$\frac{5+4\sqrt{2}}{3}$$

B

$$\frac{8+4\sqrt{2}}{3}$$

C

$$\frac{1+5\sqrt{2}}{3}$$

D

$$\frac{4+5\sqrt{2}}{3}$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 75

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Let $$A = \left\{(x,y) \in \mathbb{R}^2 : y \geq 0, 2x \leq y \leq \sqrt{4-(x-1)^2}\right\}$$ and
$$B = \left\{(x,y) \in \mathbb{R} \times \mathbb{R} : 0 \leq y \leq \min\left\{2x, \sqrt{4-(x-1)^2}\right\}\right\}$$. Then the ratio of the area of $$A$$ to the area of $$B$$ is

A

$$\frac{\pi-1}{\pi+1}$$

B

$$\frac{\pi}{\pi-1}$$

C

$$\frac{\pi}{\pi+1}$$

D

$$\frac{\pi+1}{\pi-1}$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 76

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Let $$\Delta$$ be the area of the region $$\{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \leq 21, y^2 \leq 4x, x \geq 1\}$$. Then $$\frac{1}{2}\left(\Delta - 21\sin^{-1}\frac{2}{\sqrt{7}}\right)$$ is equal to

A

$$2\sqrt{3} - \frac{1}{3}$$

B

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

C

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

D

$$\sqrt{3} - \frac{4}{3}$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 77

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Let $$y = f(x)$$ be the solution of the differential equation $$y(x+1)dx - x^2 dy = 0$$, $$y(1) = e$$. Then $$\lim_{x \to 0^+} f(x)$$ is equal to

A

$$0$$

B

$$\frac{1}{e}$$

C

$$e^2$$

D

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

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NTA JEE Main 29th January 2023 Shift 1 - Question 78

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If the vectors $$\vec{a} = \lambda\hat{i} + \mu\hat{j} + 4\hat{k}$$, $$\vec{b} = -2\hat{i} + 4\hat{j} - 2\hat{k}$$ and $$\vec{c} = 2\hat{i} + 3\hat{j} + \hat{k}$$ are coplanar and the projection of $$\vec{a}$$ on the vector $$\vec{b}$$ is $$\sqrt{54}$$ units, then the sum of all possible values of $$\lambda + \mu$$ is equal to

A

$$0$$

B

$$6$$

C

$$24$$

D

$$18$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 79

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Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is

A

$$\frac{5}{24}$$

B

$$\frac{2}{15}$$

C

$$\frac{1}{6}$$

D

$$\frac{5}{36}$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 80

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Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If $$\mu$$ and $$\sigma^2$$ represent mean and variance of X, respectively, then $$10(\mu^2 + \sigma^2)$$ is equal to

A

$$20$$

B

$$250$$

C

$$25$$

D

$$30$$

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NTA JEE Main 29th January 2023 Shift 1 - Question 81

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If all the six digit numbers $$x_1x_2x_3x_4x_5x_6$$ with $$0 < x_1 < x_2 < x_3 < x_4 < x_5 < x_6$$ are arranged in the increasing order, then the sum of the digits in the $$72^{th}$$ number is ______.

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NTA JEE Main 29th January 2023 Shift 1 - Question 82

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Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is

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NTA JEE Main 29th January 2023 Shift 1 - Question 83

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Let $$a_1, a_2, a_3, \ldots$$ be a GP of increasing positive numbers. If the product of fourth and sixth terms is $$9$$ and the sum of fifth and seventh terms is $$24$$, then $$a_1a_9 + a_2a_4a_9 + a_5 + a_7$$ is equal to

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NTA JEE Main 29th January 2023 Shift 1 - Question 84

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Let the coefficients of three consecutive terms in the binomial expansion of $$(1 + 2x)^n$$ be in the ratio $$2 : 5 : 8$$. Then the coefficient of the term, which is in the middle of these three terms, is

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NTA JEE Main 29th January 2023 Shift 1 - Question 85

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If the co-efficient of $$x^9$$ in $$\left(\alpha x^3 + \frac{1}{\beta x}\right)^{11}$$ and the co-efficient of $$x^{-9}$$ in $$\left(\alpha x - \frac{1}{\beta x^3}\right)^{11}$$ are equal, then $$(\alpha\beta)^2$$ is equal to

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NTA JEE Main 29th January 2023 Shift 1 - Question 86

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Let $$f : \mathbb{R} \to \mathbb{R}$$ be a differentiable function that satisfies the relation $$f(x+y) = f(x) + f(y) - 1, \forall x, y \in \mathbb{R}$$. If $$f'(0) = 2$$, then $$|f(-2)|$$ is equal to

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NTA JEE Main 29th January 2023 Shift 1 - Question 87

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Suppose f is a function satisfying $$f(x + y) = f(x) + f(y)$$ for all $$x, y \in \mathbb{N}$$ and $$f(1) = \frac{1}{5}$$. If $$\sum_{n=1}^{m} \frac{f(n)}{n(n+1)(n+2)} = \frac{1}{12}$$ then m is equal to ______.

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NTA JEE Main 29th January 2023 Shift 1 - Question 88

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Let $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be three non-zero non-coplanar vectors. Let the position vectors of four points $$A$$, $$B$$, $$C$$ and $$D$$ be $$\vec{a} - \vec{b} + \vec{c}$$, $$\lambda\vec{a} - 3\vec{b} + 4\vec{c}$$, $$-\vec{a} + 2\vec{b} - 3\vec{c}$$ and $$2\vec{a} - 4\vec{b} + 6\vec{c}$$ respectively. If $$\vec{AB}$$, $$\vec{AC}$$ and $$\vec{AD}$$ are coplanar, then $$\lambda$$ is:

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NTA JEE Main 29th January 2023 Shift 1 - Question 89

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Let the equation of the plane P containing the line $$x + 10 = \frac{8-y}{2} = z$$ be $$ax + by + 3z = 2(a+b)$$ and the distance of the plane P from the point $$(1, 27, 7)$$ be $$c$$. Then $$a^2 + b^2 + c^2$$ is equal to

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NTA JEE Main 29th January 2023 Shift 1 - Question 90

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Let the co-ordinates of one vertex of $$\triangle ABC$$ be $$A(0, 2, \alpha)$$ and the other two vertices lie on the line $$\frac{x+\alpha}{5} = \frac{y-1}{2} = \frac{z+4}{3}$$. For $$\alpha \in \mathbb{Z}$$, if the area of $$\triangle ABC$$ is $$21$$ sq. units and the line segment $$BC$$ has length $$2\sqrt{21}$$ units, then $$\alpha^2$$ is equal to ______.

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