JEE Main 2024 (Online) 31st January Evening Shift

If for some \(m, n ;{ }^6 C_m+2\left({ }^6 C_{m+1}\right)+{ }^6 C_{m+2}>{ }^8 C_3\) and \({ }^{n-1} P_3:{ }^n P_4=1: 8\), then \({ }^n P_{m+1}+{ }^{\mathrm{n}+1} C_m\) is equal to

A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is

Let \(a, b, c\) be the lengths of three sides of a triangle satistying the condition \(\left(a^2+b^2\right) x^2-2 b(a+c) x+\left(b^2+c^2\right)=0\). If the set of all possible values of \(x\) is the interval \((\alpha, \beta)\), then $$12\l...

Let \(2^{\text {nd }}, 8^{\text {th }}\) and \(44^{\text {th }}\) terms of a non-constant A. P. be respectively the \(1^{\text {st }}, 2^{\text {nd }}\) and \(3^{\text {rd }}\) terms of a G. P. If the first term of the A. P. is 1, then the ...

Let \(A=\{1,2,3, \ldots \ldots \ldots \ldots, 100\}\). Let \(R\) be a relation on \(\mathrm{A}\) defined by \((x, y) \in R\) if and only if \(2 x=3 y\). Let \(R_1\) be a symmetric relation on \(A\) such that \(R \subset R_1\) and the number...

Let the mean and the variance of 6 observations \(a, b, 68,44,48,60\) be \(55\) and \(194\), respectively. If \(a>b\), then \(a+3 b\) is

Let \(A(a, b), B(3,4)\) and \(C(-6,-8)\) respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point \(P(2 a+3,7 b+5)\) from the line \(2 x+3 y-4=0\) measured parallel to the line $$x-2 y-1=...

Let \(A(-2,-1), B(1,0), C(\alpha, \beta)\) and \(D(\gamma, \delta)\) be the vertices of a parallelogram \(A B C D\). If the point \(C\) lies on \(2 x-y=5\) and the point \(D\) lies on \(3 x-2 y=6\), then the value of $$|\alpha+\beta+\gamma+...

The number of solutions, of the equation \(e^{\sin x}-2 e^{-\sin x}=2\), is :

Let \(\vec{a}=3 \hat{i}+2 \hat{j}+\hat{k}, \vec{b}=2 \hat{i}-\hat{j}+3 \hat{k}\) and \(\vec{c}\) be a vector such that \((\vec{a}+\vec{b}) \times \vec{c}=2(\vec{a} \times \vec{b})+24 \hat{j}-6 \hat{k}\) and $$(\vec{a}-\vec{b}+\hat{i}) \cdot...

An AC voltage \(V=20 \sin 200 \pi t\) is applied to a series LCR circuit which drives a current \(I=10 \sin \left(200 \pi t+\frac{\pi}{3}\right)\). The average power dissipated is:

The mass number of nucleus having radius equal to half of the radius of nucleus with mass number 192 is :

A nucleus has mass number \(A_1\) and volume \(V_1\). Another nucleus has mass number \(A_2\) and Volume \(V_2\). If relation between mass number is \(A_2=4 A_1\), then \(\frac{V_2}{V_1}=\) __________.

By what percentage will the illumination of the lamp decrease if the current drops by 20%?

The resistance per centimeter of a meter bridge wire is \(r\), with \(X \Omega\) resistance in left gap. Balancing length from left end is at \(40 \mathrm{~cm}\) with \(25 \Omega\) resistance in right gap. Now the wire is replaced by anothe...

In the following circuit, the battery has an emf of \(2 \mathrm{~V}\) and an internal resistance of \(\frac{2}{3} \Omega\). The power consumption in the entire circuit is _________ W.

In a photoelectric effect experiment a light of frequency 1.5 times the threshold frequency is made to fall on the surface of photosensitive material. Now if the frequency is halved and intensity is doubled, the number of photo electrons em...

The magnetic flux \(\phi\) (in weber) linked with a closed circuit of resistance \(8 \Omega\) varies with time (in seconds) as \(\phi=5 t^2-36 t+1\). The induced current in the circuit at \(t=2 \mathrm{~s}\) is __________ A.

Given below are two statements:
Statement I: Electromagnetic waves carry energy as they travel through space and this energy is equally shared by the electric and magnetic fields.
Statement II: When electromagnetic waves strike a surface, a...

The output of the given circuit diagram is -

Force between two point charges \(q_1\) and \(q_2\) placed in vacuum at '\(r\)' cm apart is \(F\). Force between them when placed in a medium having dielectric constant \(K=5\) at '\(r / 5\)' \(\mathrm{cm}\) apart will be:

The distance between charges \(+q\) and \(-q\) is \(2 l\) and between \(+2 q\) and \(-2 q\) is \(4 l\). The electrostatic potential at point \(P\) at a distance \(r\) from center \(O\) is $$-\alpha\left[\frac{q l}{r^2}\right] \times 10^9 \m...

Light from a point source in air falls on a convex curved surface of radius \(20 \mathrm{~cm}\) and refractive index 1.5. If the source is located at \(100 \mathrm{~cm}\) from the convex surface, the image will be formed at ________ $$\math...

The mass of the moon is \(\frac{1}{144}\) times the mass of a planet and its diameter is \(\frac{1}{16}\) times the diameter of a planet. If the escape velocity on the planet is \(v\), the escape velocity on the moon will be :

The speed of sound in oxygen at S.T.P. will be approximately: (given, \(R=8.3 \mathrm{~JK}^{-1}, \gamma=1.4\))

A gas mixture consists of 8 moles of argon and 6 moles of oxygen at temperature T. Neglecting all vibrational modes, the total internal energy of the system is:

A block of mass \(5 \mathrm{~kg}\) is placed on a rough inclined surface as shown in the figure. If \(\overrightarrow{F_1}\) is the force required to just move the block up the inclined plane and \(\overrightarrow{F_2}\) is the force requi...

A light string passing over a smooth light fixed pulley connects two blocks of masses \(m_1\) and \(m_2\). If the acceleration of the system is \(g / 8\), then the ratio of masses is:

A uniform magnetic field of \(2 \times 10^{-3} \mathrm{~T}\) acts along positive \(Y\)-direction. A rectangular loop of sides \(20 \mathrm{~cm}\) and \(10 \mathrm{~cm}\) with current of \(5 \mathrm{~A}\) is in \(Y-Z\) plane. The current is ...

Two circular coils \(P\) and \(Q\) of 100 turns each have same radius of \(\pi \mathrm{~cm}\). The currents in \(P\) and \(R\) are \(1 A\) and \(2 A\) respectively. \(P\) and \(Q\) are placed with their planes mutually perpendicular with th...

A small spherical ball of radius \(r\), falling through a viscous medium of negligible density has terminal velocity '\(v\)'. Another ball of the same mass but of radius \(2 r\), falling through the same viscous medium will have terminal ve...

Two blocks of mass \(2 \mathrm{~kg}\) and \(4 \mathrm{~kg}\) are connected by a metal wire going over a smooth pulley as shown in figure. The radius of wire is \(4.0 \times 10^{-5} \mathrm{~m}\) and Young's modulus of the metal is $$2.0 \ti...

A body of mass '\(m\)' is projected with a speed '\(u\)' making an angle of \(45^{\circ}\) with the ground. The angular momentum of the body about the point of projection, at the highest point is expressed as \(\frac{\sqrt{2} m u^3}{X g}\)....

Two identical spheres each of mass \(2 \mathrm{~kg}\) and radius \(50 \mathrm{~cm}\) are fixed at the ends of a light rod so that the separation between the centers is \(150 \mathrm{~cm}\). Then, moment of inertia of the system about an axi...

The time period of simple harmonic motion of mass \(M\) in the given figure is \(\pi \sqrt{\frac{\alpha M}{5 k}}\), where the value of \(\alpha\) is _________.

Consider two physical quantities \(A\) and \(B\) related to each other as \(E=\frac{B-x^2}{A t}\) where \(E, x\) and \(t\) have dimensions of energy, length and time respectively. The dimension of \(A B\) is

The measured value of the length of a simple pendulum is \(20 \mathrm{~cm}\) with \(2 \mathrm{~mm}\) accuracy. The time for 50 oscillations was measured to be 40 seconds with 1 second resolution. From these measurements, the accuracy in the...

If two vectors \(\vec{A}\) and \(\vec{B}\) having equal magnitude \(R\) are inclined at angle \(\theta\), then

When unpolarized light is incident at an angle of \(60^{\circ}\) on a transparent medium from air, the reflected ray is completely polarized. The angle of refraction in the medium is:

A body of mass \(2 \mathrm{~kg}\) begins to move under the action of a time dependent force given by \(\vec{F}=\left(6 t \hat{i}+6 t^2 \hat{j}\right) N\). The power developed by the force at the time \(t\) is given by: