NTA JEE Mains 29th Jan 2024 Shift 2

NTA JEE Mains 29th Jan 2024 Shift 2 - Question 1

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A physical quantity $$Q$$ is found to depend on quantities $$a, b, c$$ by the relation $$Q = \frac{a^4 b^3}{c^2}$$. The percentage error in $$a, b$$ and $$c$$ are $$3\%, 4\%$$ and $$5\%$$ respectively. Then, the percentage error in $$Q$$ is:

A

$$66\%$$

B

$$43\%$$

C

$$34\%$$

D

$$14\%$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 2

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A particle is moving in a straight line. The variation of position $$x$$ as a function of time $$t$$ is given as $$x = (t^3 - 6t^2 + 20t + 15)$$ m. The velocity of the body when its acceleration becomes zero is:

A

$$4 \text{ m s}^{-1}$$

B

$$8 \text{ m s}^{-1}$$

C

$$10 \text{ m s}^{-1}$$

D

$$6 \text{ m s}^{-1}$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 3

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A stone of mass $$900$$ g is tied to a string and moved in a vertical circle of radius $$1$$ m making $$10$$ rpm. The tension in the string, when the stone is at the lowest point is (if $$\pi^2 = 9.8$$ and $$g = 9.8 \text{ m s}^{-2}$$)

A

$$97$$ N

B

$$9.8$$ N

C

$$8.82$$ N

D

$$17.8$$ N

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 4

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The bob of a pendulum was released from a horizontal position. The length of the pendulum is $$10$$ m. If it dissipates $$10\%$$ of its initial energy against air resistance, the speed with which the bob arrives at the lowest point is: [Use, $$g = 10 \text{ m s}^{-2}$$]

A

$$6\sqrt{5} \text{ m s}^{-1}$$

B

$$5\sqrt{6} \text{ m s}^{-1}$$

C

$$5\sqrt{5} \text{ m s}^{-1}$$

D

$$2\sqrt{5} \text{ m s}^{-1}$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 5

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A bob of mass $$m$$ is suspended by a light string of length $$L$$. It is imparted a minimum horizontal velocity at the lowest point $$A$$ such that it just completes half circle reaching the top most position $$B$$. The ratio of kinetic energies $$\frac{(K.E.)_A}{(K.E.)_B}$$ is:

A

$$3 : 2$$

B

$$5 : 1$$

C

$$2 : 5$$

D

$$1 : 5$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 6

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A planet takes $$200$$ days to complete one revolution around the Sun. If the distance of the planet from Sun is reduced to one fourth of the original distance, how many days will it take to complete one revolution?

A

$$25$$

B

$$50$$

C

$$100$$

D

$$20$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 7

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A wire of length $$L$$ and radius $$r$$ is clamped at one end. If its other end is pulled by a force $$F$$, its length increases by $$l$$. If the radius of the wire and the applied force both are reduced to half of their original values keeping original length constant, the increase in length will become:

A

$$3$$ times

B

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

C

$$4$$ times

D

$$2$$ times

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 8

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A small liquid drop of radius $$R$$ is divided into $$27$$ identical liquid drops. If the surface tension is $$T$$, then the work done in the process will be:

A

$$8\pi R^2 T$$

B

$$3\pi R^2 T$$

C

$$\frac{1}{8}\pi R^2 T$$

D

$$4\pi R^2 T$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 9

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The temperature of a gas having $$2.0 \times 10^{25}$$ molecules per cubic meter at $$1.38$$ atm (Given, $$k = 1.38 \times 10^{-23} \text{ J K}^{-1}$$) is:

A

$$500$$ K

B

$$200$$ K

C

$$100$$ K

D

$$300$$ K

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 10

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$$N$$ moles of a polyatomic gas $$(f = 6)$$ must be mixed with two moles of a monoatomic gas so that the mixture behaves as a diatomic gas. The value of $$N$$ is:

A

$$6$$

B

$$3$$

C

$$4$$

D

$$2$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 11

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An electric field is given by $$\left(6\hat{i} + 5\hat{j} + 3\hat{k}\right)$$ N C$$^{-1}$$. The electric flux through a surface area $$30\hat{i}$$ m$$^2$$ lying in YZ-plane (in SI unit) is:

A

$$90$$

B

$$150$$

C

$$180$$

D

$$60$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 12

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In the given circuit, the current in resistance $$R_3$$ is:

A

$$1$$ A

B

$$1.5$$ A

C

$$2$$ A

D

$$2.5$$ A

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 13

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Two particles $$X$$ and $$Y$$ having equal charges are being accelerated through the same potential difference. Thereafter, they enter normally in a region of uniform magnetic field and describes circular paths of radii $$R_1$$ and $$R_2$$ respectively. The mass ratio of $$X$$ and $$Y$$ is:

A

$$\left(\frac{R_2}{R_1}\right)^2$$

B

$$\left(\frac{R_1}{R_2}\right)^2$$

C

$$\frac{R_1}{R_2}$$

D

$$\frac{R_2}{R_1}$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 14

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In an a.c. circuit, voltage and current are given by: $$V = 100\sin(100t)$$ V and $$I = 100\sin\left(100t + \frac{\pi}{3}\right)$$ mA respectively. The average power dissipated in one cycle is:

A

$$5$$ W

B

$$10$$ W

C

$$2.5$$ W

D

$$25$$ W

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 15

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A plane electromagnetic wave of frequency $$35$$ MHz travels in free space along the $$X$$-direction. At a particular point (in space and time) $$\vec{E} = 9.6\hat{j}$$ V m$$^{-1}$$. The value of magnetic field at this point is:

A

$$3.2 \times 10^{-8}\hat{k}$$ T

B

$$3.2 \times 10^{-8}\hat{i}$$ T

C

$$9.6\hat{j}$$ T

D

$$9.6 \times 10^{-8}\hat{k}$$ T

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 16

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If the distance between object and its two times magnified virtual image produced by a curved mirror is $$15$$ cm, the focal length of the mirror must be:

A

$$15$$ cm

B

$$-12$$ cm

C

$$-10$$ cm

D

$$\frac{10}{3}$$ cm

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 17

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In Young's double slit experiment, light from two identical sources are superimposing on a screen. The path difference between the two lights reaching at a point on the screen is $$\frac{7\lambda}{4}$$. The ratio of intensity of fringe at this point with respect to the maximum intensity of the fringe is:

A

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

B

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

C

$$\frac{1}{3}$$

D

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

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 18

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Two sources of light emit with a power of $$200$$ W. The ratio of number of photons of visible light emitted by each source having wavelengths $$300$$ nm and $$500$$ nm respectively, will be:

A

$$1 : 5$$

B

$$1 : 3$$

C

$$5 : 3$$

D

$$3 : 5$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 19

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Given below are two statements:
Statement I: Most of the mass of the atom and all its positive charge are concentrated in a tiny nucleus and the electrons revolve around it, is Rutherford's model.
Statement II: An atom is a spherical cloud of positive charges with electrons embedded in it, is a special case of Rutherford's model.
In the light of the above statements, choose the most appropriate from the options given below.

A

Both statement I and statement II are false

B

Statement I is false but statement II is true

C

Statement I is true but statement II is false

D

Both statement I and statement II are true

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 20

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The truth table for this given circuit is:

A

B

C

D

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 21

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A particle is moving in a circle of radius $$50$$ cm in such a way that at any instant the normal and tangential components of its acceleration are equal. If its speed at $$t = 0$$ is $$4 \text{ m s}^{-1}$$, the time taken to complete the first revolution will be $$\frac{1}{\alpha}\left[1 - e^{-2\pi}\right]$$ s, where $$\alpha =$$ ______.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 22

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A body of mass $$5$$ kg moving with a uniform speed $$3\sqrt{2} \text{ m s}^{-1}$$ in $$X-Y$$ plane along the line $$y = x + 4$$. The angular momentum of the particle about the origin will be ______ kg m$$^2$$ s$$^{-1}$$.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 23

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Two metallic wires $$P$$ and $$Q$$ have same volume and are made up of same material. If their area of cross sections are in the ratio $$4 : 1$$ and force $$F_1$$ is applied to $$P$$, an extension of $$\Delta l$$ is produced. The force which is required to produce same extension in $$Q$$ is $$F_2$$. The value of $$\frac{F_1}{F_2}$$ is ______.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 24

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A simple harmonic oscillator has an amplitude $$A$$ and time period $$6\pi$$ second. Assuming the oscillation starts from its mean position, the time required by it to travel from $$x = A$$ to $$x = \frac{\sqrt{3}}{2}A$$ will be $$\frac{\pi}{x}$$ s, where $$x =$$ ______.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 25

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In the given circuit, the current flowing through the resistance $$20 \;\Omega$$ is $$0.3$$ A, while the ammeter reads $$0.9$$ A. The value of $$R_1$$ is ______ $$\Omega$$.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 26

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A charge of $$4.0 \;\mu$$C is moving with a velocity of $$4.0 \times 10^6 \text{ m s}^{-1}$$ along the positive $$y$$-axis under a magnetic field $$B$$ of strength $$\left(2\hat{k}\right)$$ T. The force acting on the charge is $$x\hat{i}$$ N. The value of $$x$$ is ______.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 27

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A horizontal straight wire $$5$$ m long extending from east to west falling freely at right angle to horizontal component of earth's magnetic field $$0.60 \times 10^{-4}$$ Wb m$$^{-2}$$. The instantaneous value of emf induced in the wire when its velocity is $$10 \text{ m s}^{-1}$$ is ______ $$\times 10^{-3}$$ V.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 28

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In the given figure, the charge stored in $$6\;\mu$$F capacitor, when points $$A$$ and $$B$$ are joined by a connecting wire is ______ $$\mu$$C.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 29

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In a single slit diffraction pattern, a light of wavelength $$6000$$ $$\mathring{A}$$ is used. The distance between the first and third minima in the diffraction pattern is found to be $$3$$ mm when the screen is placed $$50$$ cm away from slits. The width of the slit is ______ $$\times 10^{-4}$$ m.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 30

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Hydrogen atom is bombarded with electrons accelerated through a potential different of $$V$$, which causes excitation of hydrogen atoms. If the experiment is being formed at $$T = 0$$ K. The minimum potential difference needed to observe any Balmer series lines in the emission spectra will be $$\frac{\alpha}{10}$$ V, where $$\alpha =$$ ______.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 31

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Match List I with List II

Choose the correct answer from the options given below:

A

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

B

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

C

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

D

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

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 32

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The element having the highest first ionization enthalpy is

A

Si

B

Al

C

N

D

C

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 33

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Given below are two statements:
Statement I: Fluorine has most negative electron gain enthalpy in its group.
Statement II: Oxygen has least negative electron gain enthalpy in its group.
In the light of the above statements, choose the most appropriate from the options given below.

A

Both Statement I and Statement II are true

B

Statement I is true but Statement II is false

C

Both Statement I and Statement II are false

D

Statement I is false but Statement II is true

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 34

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According to IUPAC system, the compound is named as:

A

Cyclohex-1-en-2-ol

B

1-Hydroxyhex-2-ene

C

Cyclohex-1-en-3-ol

D

Cyclohex-2-en-1-ol

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 35

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The ascending acidity order of the following H atoms is

A

C < D < B < A

B

A < B < C < D

C

A < B < D < C

D

D < C < B < A

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 36

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Match List I with List II 

Choose the correct answer from the options given below:

A

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

B

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

C

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

D

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

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 37

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Which one of the following will show geometrical isomerism?

A

B

C

D

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 38

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Chromatographic technique/s based on the principle of differential adsorption is/are
A. Column chromatography B. Thin layer chromatography C. Paper chromatography
Choose the most appropriate answer from the options given below:

A

B only

B

A only

C

A & B only

D

C only

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 39

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Anomalous behaviour of oxygen is due to its

A

Large size and high electronegativity

B

Small size and low electronegativity

C

Small size and high electronegativity

D

Large size and low electronegativity

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 40

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Which of the following acts as a strong reducing agent? (Atomic number: Ce = 58, Eu = 63, Gd = 64, Lu = 71)

A

Lu$$^{3+}$$

B

Gd$$^{3+}$$

C

Eu$$^{2+}$$

D

Ce$$^{4+}$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 41

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Which of the following statements are correct about Zn, Cd and Hg?
A. They exhibit high enthalpy of atomization as the d-subshell is full.
B. Zn and Cd do not show variable oxidation state while Hg shows +I and +II.
C. Compounds of Zn, Cd and Hg are paramagnetic in nature.
D. Zn, Cd and Hg are called soft metals.
Choose the most appropriate from the options given below:

A

B, D only

B

B, C only

C

A, D only

D

C, D only

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 42

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The correct IUPAC name of $$K_2MnO_4$$ is:

A

Potassium tetraoxopermanganate (VI)

B

Potassium tetraoxidomanganate (VI)

C

Dipotassium tetraoxidomanganate (VII)

D

Potassium tetraoxidomanganese (VI)

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 43

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Alkyl halide is converted into alkyl isocyanide by reaction with

A

NaCN

B

NH$$_4$$CN

C

KCN

D

AgCN

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 44

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Phenol treated with chloroform in presence of sodium hydroxide, which further hydrolysed in presence of an acid results

A

Salicyclic acid

B

Benzene-1,2-diol

C

Benzene-1,3-diol

D

2-Hydroxybenzaldehyde

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 45

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Identify the reagents used for the following conversion

A

$$A = LiAlH_4, B = NaOH_{(aq)}, C = NH_2-NH_2/KOH$$ ethylene glycol

B

$$A = LiAlH_4, B = NaOH_{(alc)}, C = Zn/HCl$$

C

$$A = DIBAL-H, B = NaOH_{(aq)}, C = NH_2-NH_2/KOH$$ ethylene glycol

D

$$A = DIBAL-H, B = NaOH_{(alc)}, C = Zn/HCl$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 46

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Which of the following reaction is correct?

A

B

C

D

$$C_2H_5CONH_2 + Br_2 + NaOH \rightarrow C_2H_5CH_2NH_2 + Na_2CO_3 + NaBr + H_2O$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 47

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The product A formed in the following reaction is:

A

B

C

D

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 48

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On passing a gas, '$$X$$', through Nessler's reagent, a brown precipitate is obtained. The gas '$$X$$' is

A

H$$_2$$S

B

CO$$_2$$

C

NH$$_3$$

D

Cl$$_2$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 49

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A reagent which gives brilliant red precipitate with Nickel ions in basic medium is

A

sodium nitroprusside

B

neutral FeCl$$_3$$

C

meta-dinitrobenzene

D

dimethyl glyoxime

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 50

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Match List I with List II

Choose the correct answer from the options given below:

A

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

B

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

C

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

D

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

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 51

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The total number of molecules with zero dipole moment among $$CH_4, BF_3, H_2O, HF, NH_3, CO_2$$ and $$SO_2$$ is ______.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 52

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The total number of 'Sigma' and Pi bonds in 2-formylhex-4-enoic acid is ______.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 53

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The total number of anti bonding molecular orbitals, formed from $$2s$$ and $$2p$$ atomic orbitals in a diatomic molecule is ______.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 54

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Standard enthalpy of vapourisation for $$CCl_4$$ is $$30.5$$ kJ mol$$^{-1}$$. Heat required for vapourisation of $$284$$ g of $$CCl_4$$ at constant temperature is ______ kJ. (Given molar mass in g mol$$^{-1}$$; C = 12, Cl = 35.5)

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 55

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The following concentrations were observed at $$500$$ K for the formation of $$NH_3$$ from $$N_2$$ and $$H_2$$. At equilibrium: $$[N_2] = 2 \times 10^{-2}$$ M, $$[H_2] = 3 \times 10^{-2}$$ M and $$[NH_3] = 1.5 \times 10^{-2}$$ M. Equilibrium constant for the reaction is ______.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 56

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If $$50$$ mL of $$0.5$$ M oxalic acid is required to neutralise $$25$$ mL of NaOH solution, the amount of NaOH in $$50$$ mL of given NaOH solution is ______ g.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 57

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Molality of $$0.8$$ M $$H_2SO_4$$ solution (density $$1.06$$ g cm$$^{-3}$$) is ______ $$\times 10^{-3}$$ m.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 58

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A constant current was passed through a solution of $$AuCl_4^-$$ ion between gold electrodes. After a period of $$10.0$$ minutes, the increase in mass of cathode was $$1.314$$ g. The total charge passed through the solution is ______ $$\times 10^{-2}$$ F. (Given atomic mass of Au = 197)

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 59

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The half-life of radioisotopic bromine-82 is $$36$$ hours. The fraction which remains after one day is ______ $$\times 10^{-2}$$.
(Given antilog $$0.2006 = 1.587$$)

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 60

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Oxidation state of Fe (Iron) in complex formed in Brown ring test.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 61

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Let $$r$$ and $$\theta$$ respectively be the modulus and amplitude of the complex number $$z = 2 - i\left(2\tan\frac{5\pi}{8}\right)$$, then $$(r, \theta)$$ is equal to

A

$$\left(2\sec\frac{3\pi}{8}, \frac{3\pi}{8}\right)$$

B

$$\left(2\sec\frac{3\pi}{8}, \frac{5\pi}{8}\right)$$

C

$$\left(2\sec\frac{5\pi}{8}, \frac{3\pi}{8}\right)$$

D

$$\left(2\sec\frac{11\pi}{8}, \frac{11\pi}{8}\right)$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 62

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Number of ways of arranging $$8$$ identical books into $$4$$ identical shelves where any number of shelves may remain empty is equal to

A

$$18$$

B

$$16$$

C

$$12$$

D

$$15$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 63

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If $$\log_e a, \log_e b, \log_e c$$ are in an A.P. and $$\log_e a - \log_e 2b, \log_e 2b - \log_e 3c, \log_e 3c - \log_e a$$ are also in an A.P., then $$a : b : c$$ is equal to

A

$$9 : 6 : 4$$

B

$$16 : 4 : 1$$

C

$$25 : 10 : 4$$

D

$$6 : 3 : 2$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 64

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If each term of a geometric progression $$a_1, a_2, a_3, \ldots$$ with $$a_1 = \frac{1}{8}$$ and $$a_2 \neq a_1$$, is the arithmetic mean of the next two terms and $$S_n = a_1 + a_2 + \ldots + a_n$$, then $$S_{20} - S_{18}$$ is equal to

A

$$2^{15}$$

B

$$-2^{18}$$

C

$$2^{18}$$

D

$$-2^{15}$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 65

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The sum of the solutions $$x \in R$$ of the equation $$\frac{3\cos 2x + \cos^3 2x}{\cos^6 x - \sin^6 x} = x^3 - x^2 + 6$$ is

A

$$0$$

B

$$1$$

C

$$-1$$

D

$$3$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 66

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Let $$A$$ be the point of intersection of the lines $$3x + 2y = 14, 5x - y = 6$$ and $$B$$ be the point of intersection of the lines $$4x + 3y = 8, 6x + y = 5$$. The distance of the point $$P(5, -2)$$ from the line $$AB$$ is

A

$$\frac{13}{2}$$

B

$$8$$

C

$$\frac{5}{2}$$

D

$$6$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 67

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The distance of the point $$(2, 3)$$ from the line $$2x - 3y + 28 = 0$$, measured parallel to the line $$\sqrt{3}x - y + 1 = 0$$, is equal to

A

$$4\sqrt{2}$$

B

$$6\sqrt{3}$$

C

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

D

$$4 + 6\sqrt{3}$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 68

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If the mean and variance of five observations are $$\frac{24}{5}$$ and $$\frac{194}{25}$$ respectively and the mean of first four observations is $$\frac{7}{2}$$, then the variance of the first four observations is equal to

A

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

B

$$\frac{77}{12}$$

C

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

D

$$\frac{105}{4}$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 69

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If $$R$$ is the smallest equivalence relation on the set $$\{1, 2, 3, 4\}$$ such that $$\{(1, 2), (1, 3)\} \subset R$$, then the number of elements in $$R$$ is ______.

A

$$10$$

B

$$12$$

C

$$8$$

D

$$15$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 70

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Let $$A = \begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix}$$ and $$P = \begin{bmatrix} 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{bmatrix}$$. The sum of the prime factors of $$|P^{-1}AP - 2I|$$ is equal to

A

$$26$$

B

$$27$$

C

$$66$$

D

$$23$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 71

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Let $$x = \frac{m}{n}$$ ($$m, n$$ are co-prime natural numbers) be a solution of the equation $$\cos\left(2\sin^{-1}x\right) = \frac{1}{9}$$ and let $$\alpha, \beta (\alpha > \beta)$$ be the roots of the equation $$mx^2 - nx - m + n = 0$$. Then the point $$(\alpha, \beta)$$ lies on the line

A

$$3x + 2y = 2$$

B

$$5x - 8y = -9$$

C

$$3x - 2y = -2$$

D

$$5x + 8y = 9$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 72

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Let $$y = \log_e\left(\frac{1 - x^2}{1 + x^2}\right), -1 < x < 1$$. Then at $$x = \frac{1}{2}$$, the value of $$225(y' - y'')$$ is equal to

A

$$732$$

B

$$746$$

C

$$742$$

D

$$736$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 73

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The function $$f(x) = 2x + 3x^{\frac{2}{3}}, x \in R$$, has

A

exactly one point of local minima and no point of local maxima

B

exactly one point of local maxima and no point of local minima

C

exactly one point of local maxima and exactly one point of local minima

D

exactly two points of local maxima and exactly one point of local minima

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 74

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The function $$f(x) = \frac{x}{x^2 - 6x - 16}, x \in \mathbb{R} - \{-2, 8\}$$

A

decreases in $$(-2, 8)$$ and increases in $$(-\infty, -2) \cup (8, \infty)$$

B

decreases in $$(-\infty, -2) \cup (-2, 8) \cup (8, \infty)$$

C

decreases in $$(-\infty, -2)$$ and increases in $$(8, \infty)$$

D

increases in $$(-\infty, -2) \cup (-2, 8) \cup (8, \infty)$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 75

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If $$\int \frac{\sin^{\frac{3}{2}}x + \cos^{\frac{3}{2}}x}{\sqrt{\sin^3 x \cos^3 x \sin(x - \theta)}} dx = A\sqrt{\cos\theta\tan x - \sin\theta} + B\sqrt{\cos\theta - \sin\theta\cot x} + C$$, where $$C$$ is the integration constant, then $$AB$$ is equal to

A

$$4\text{ cosec}(2\theta)$$

B

$$4\sec\theta$$

C

$$2\sec\theta$$

D

$$8\text{ cosec}(2\theta)$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 76

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If $$\sin\left(\frac{y}{x}\right) = \log_e|x| + \frac{\alpha}{2}$$ is the solution of the differential equation $$x\cos\left(\frac{y}{x}\right)\frac{dy}{dx} = y\cos\left(\frac{y}{x}\right) + x$$ and $$y(1) = \frac{\pi}{3}$$, then $$\alpha^2$$ is equal to

A

$$3$$

B

$$12$$

C

$$4$$

D

$$9$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 77

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Let $$\vec{OA} = \vec{a}, \vec{OB} = 12\vec{a} + 4\vec{b}$$ and $$\vec{OC} = \vec{b}$$, where $$O$$ is the origin. If $$S$$ is the parallelogram with adjacent sides $$OA$$ and $$OC$$, then $$\frac{\text{area of the quadrilateral } OABC}{\text{area of } S}$$ is equal to _____

A

$$6$$

B

$$10$$

C

$$7$$

D

$$8$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 78

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Let a unit vector $$\hat{u} = x\hat{i} + y\hat{j} + z\hat{k}$$ make angles $$\frac{\pi}{2}, \frac{\pi}{3}$$ and $$\frac{2\pi}{3}$$ with the vectors $$\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{k}, \frac{1}{\sqrt{2}}\hat{j} + \frac{1}{\sqrt{2}}\hat{k}$$ and $$\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j}$$ respectively. If $$\vec{v} = \frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j} + \frac{1}{\sqrt{2}}\hat{k}$$, then $$|\hat{u} - \vec{v}|^2$$ is equal to

A

$$\frac{11}{2}$$

B

$$\frac{5}{2}$$

C

$$9$$

D

$$7$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 79

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Let $$P(3, 2, 3), Q(4, 6, 2)$$ and $$R(7, 3, 2)$$ be the vertices of $$\triangle PQR$$. Then, the angle $$\angle QPR$$ is

A

$$\frac{\pi}{6}$$

B

$$\cos^{-1}\left(\frac{7}{18}\right)$$

C

$$\cos^{-1}\left(\frac{1}{18}\right)$$

D

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

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 80

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An integer is chosen at random from the integers $$1, 2, 3, \ldots, 50$$. The probability that the chosen integer is a multiple of at least one of $$4, 6$$ and $$7$$ is

A

$$\frac{8}{25}$$

B

$$\frac{21}{50}$$

C

$$\frac{9}{50}$$

D

$$\frac{14}{25}$$

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 81

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Let the set $$C = \{(x, y) \mid x^2 - 2^y = 2023, x, y \in \mathbb{N}\}$$. Then $$\sum_{(x,y) \in C}(x + y)$$ is equal to ______.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 82

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Let $$\alpha, \beta$$ be the roots of the equation $$x^2 - \sqrt{6}x + 3 = 0$$ such that $$\text{Im}(\alpha) > \text{Im}(\beta)$$. Let $$a, b$$ be integers not divisible by $$3$$ and $$n$$ be a natural number such that $$\frac{\alpha^{99}}{\beta} + \alpha^{98} = 3^n(a + ib), i = \sqrt{-1}$$. Then $$n + a + b$$ is equal to ______.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 83

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Remainder when $$64^{32^{32}}$$ is divided by $$9$$ is equal to ______.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 84

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Let $$P(\alpha, \beta)$$ be a point on the parabola $$y^2 = 4x$$. If $$P$$ also lies on the chord of the parabola $$x^2 = 8y$$ whose mid point is $$\left(1, \frac{5}{4}\right)$$, then $$(\alpha - 28)(\beta - 8)$$ is equal to ______.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 85

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Let the slope of the line $$45x + 5y + 3 = 0$$ be $$27r_1 + \frac{9r_2}{2}$$ for some $$r_1, r_2 \in R$$. Then $$\lim_{x \to 3}\left(\int_3^x \frac{8t^2}{\frac{3r_2 x}{2} - r_2 x^2 - r_1 x^3 - 3x} dt\right)$$ is equal to ______.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 86

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Let for any three distinct consecutive terms $$a, b, c$$ of an A.P, the lines $$ax + by + c = 0$$ be concurrent at the point $$P$$ and $$Q(\alpha, \beta)$$ be a point such that the system of equations $$x + y + z = 6, 2x + 5y + \alpha z = \beta$$ and $$x + 2y + 3z = 4$$, has infinitely many solutions. Then $$(PQ)^2$$ is equal to ______.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 87

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Let $$f(x) = \sqrt{\lim_{r \to x}\left\{\frac{2r^2[(f(r))^2 - f(x)f(r)]}{r^2 - x^2} - r^3 e^{\frac{f(r)}{r}}\right\}}$$ be differentiable in $$(-\infty, 0) \cup (0, \infty)$$ and $$f(1) = 1$$. Then the value of $$ae$$, such that $$f(a) = 0$$, is equal to ______.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 88

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If $$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\sqrt{1 - \sin 2x} \, dx = \alpha + \beta\sqrt{2} + \gamma\sqrt{3}$$, where $$\alpha, \beta$$ and $$\gamma$$ are rational numbers, then $$3\alpha + 4\beta - \gamma$$ is equal to ______.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 89

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Let the area of the region $$\{(x, y) : 0 \leq x \leq 3, 0 \leq y \leq \min\{x^2 + 2, 2x + 2\}\}$$ be $$A$$. Then $$12A$$ is equal to ______.

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NTA JEE Mains 29th Jan 2024 Shift 2 - Question 90

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Let $$O$$ be the origin, and $$M$$ and $$N$$ be the points on the lines $$\frac{x-5}{4} = \frac{y-4}{1} = \frac{z-5}{3}$$ and $$\frac{x+8}{12} = \frac{y+2}{5} = \frac{z+11}{9}$$ respectively such that $$MN$$ is the shortest distance between the given lines. Then $$\vec{OM} \cdot \vec{ON}$$ is equal to ______.

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