JEE Main 2024 (Online) 31st January Morning Shift

Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable \(x\) to be the number of rotten apples in a draw of two apples, the variance of \(x\) is

Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white...

Let \(\mathrm{S}\) be the set of positive integral values of \(a\) for which \(\frac{a x^2+2(a+1) x+9 a+4}{x^2-8 x+32} < 0, \forall x \in \mathbb{R}\). Then, the number of elements in \(\mathrm{S}\) is :

For \(0 < c < b < a\), let \((a+b-2 c) x^2+(b+c-2 a) x+(c+a-2 b)=0\) and \(\alpha \neq 1\) be one of its root. Then, among the two statements
(I) If \(\alpha \in(-1,0)\), then \(b\) cannot be the geometric mean of $a$ and \(c\)
(II) If $$\a...

The sum of the series \(\frac{1}{1-3 \cdot 1^2+1^4}+\frac{2}{1-3 \cdot 2^2+2^4}+\frac{3}{1-3 \cdot 3^2+3^4}+\ldots\) up to 10 -terms is

Let \(A=\{1,2,3,4\}\) and \(R=\{(1,2),(2,3),(1,4)\}\) be a relation on \(\mathrm{A}\). Let \(\mathrm{S}\) be the equivalence relation on \(\mathrm{A}\) such that \(R \subset S\) and the number of elements in \(\mathrm{S}\) is \(\mathrm{n}\)...

Let \(\alpha, \beta, \gamma, \delta \in \mathbb{Z}\) and let \(A(\alpha, \beta), B(1,0), C(\gamma, \delta)\) and \(D(1,2)\) be the vertices of a parallelogram \(\mathrm{ABCD}\). If \(A B=\sqrt{10}\) and the points \(\mathrm{A}\) and $$\math...

Let \(\vec{a}=3 \hat{i}+\hat{j}-2 \hat{k}, \vec{b}=4 \hat{i}+\hat{j}+7 \hat{k}\) and \(\vec{c}=\hat{i}-3 \hat{j}+4 \hat{k}\) be three vectors. If a vectors \(\vec{p}\) satisfies \(\vec{p} \times \vec{b}=\vec{c} \times \vec{b}\) and $$\vec{p...

The distance of the point \(Q(0,2,-2)\) form the line passing through the point \(P(5,-4, 3)\) and perpendicular to the lines \(\vec{r}=(-3 \hat{i}+2 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+5 \hat{k}), \lambda \in \mathbb{R}\) and $$\vec{r}=(\...

Let \(\vec{a}\) and \(\vec{b}\) be two vectors such that \(|\vec{a}|=1,|\vec{b}|=4\), and \(\vec{a} \cdot \vec{b}=2\). If \(\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}\) and the angle between \(\vec{b}\) and \(\vec{c}\) is \(\alpha\), then...

If the wavelength of the first member of Lyman series of hydrogen is \(\lambda\). The wavelength of the second member will be

The mass defect in a particular reaction is \(0.4 \mathrm{~g}\). The amount of energy liberated is \(n \times 10^7 \mathrm{~kWh}\), where \(n=\) __________. (speed of light \(\left.=3 \times 10^8 \mathrm{~m} / \mathrm{s}\right)\)

A parallel plate capacitor with plate separation \(5 \mathrm{~mm}\) is charged up by a battery. It is found that on introducing a dielectric sheet of thickness \(2 \mathrm{~mm}\), while keeping the battery connections intact, the capacitor ...

An artillery piece of mass \(M_1\) fires a shell of mass \(M_2\) horizontally. Instantaneously after the firing, the ratio of kinetic energy of the artillery and that of the shell is:

A solid circular disc of mass \(50 \mathrm{~kg}\) rolls along a horizontal floor so that its center of mass has a speed of \(0.4 \mathrm{~m} / \mathrm{s}\). The absolute value of work done on the disc to stop it is ________ J.

A body starts falling freely from height \(H\) hits an inclined plane in its path at height \(h\). As a result of this perfectly elastic impact, the direction of the velocity of the body becomes horizontal. The value of \(\frac{H}{h}\) for ...

A coin is placed on a disc. The coefficient of friction between the coin and the disc is \(\mu\). If the distance of the coin from the center of the disc is \(r\), the maximum angular velocity which can be given to the disc, so that the coi...

Two conductors have the same resistances at \(0^{\circ} \mathrm{C}\) but their temperature coefficients of resistance are \(\alpha_1\) and \(\alpha_2\). The respective temperature coefficients for their series and parallel combinations are ...

Equivalent resistance of the following network is __________ \(\Omega\).

When a metal surface is illuminated by light of wavelength \(\lambda\), the stopping potential is \(8 \mathrm{~V}\). When the same surface is illuminated by light of wavelength \(3 \lambda\), stopping potential is \(2 \mathrm{~V}\). The thr...

A coil is places perpendicular to a magnetic field of \(5000 \mathrm{~T}\). When the field is changed to \(3000 \mathrm{~T}\) in \(2 \mathrm{~s}\), an induced emf of \(22 \mathrm{~V}\) is produced in the coil. If the diameter of the coil is...

A small square loop of wire of side \(l\) is placed inside a large square loop of wire of side \(L\left(L=l^2\right)\). The loops are coplanar and their centers coincide. The value of the mutual inductance of the system is $$\sqrt{x} \times...

In a plane EM wave, the electric field oscillates sinusoidally at a frequency of \(5 \times 10^{10} \mathrm{~Hz}\) and an amplitude of \(50 \mathrm{~Vm}^{-1}\). The total average energy density of the electromagnetic field of the wave is : ...

Identify the logic operation performed by the given circuit.

Two charges \(q\) and \(3 q\) are separated by a distance '\(r\)' in air. At a distance \(x\) from charge \(q\), the resultant electric field is zero. The value of \(x\) is :

The refractive index of a prism with apex angle \(A\) is \(\cot A / 2\). The angle of minimum deviation is :

Four identical particles of mass \(m\) are kept at the four corners of a square. If the gravitational force exerted on one of the masses by the other masses is \(\left(\frac{2 \sqrt{2}+1}{32}\right) \frac{\mathrm{Gm}^2}{L^2}\), the length o...

The given figure represents two isobaric processes for the same mass of an ideal gas, then

The parameter that remains the same for molecules of all gases at a given temperature is :

In the given arrangement of a doubly inclined plane two blocks of masses \(M\) and \(m\) are placed. The blocks are connected by a light string passing over an ideal pulley as shown. The coefficient of friction between the surface of the pl...

A rigid wire consists of a semicircular portion of radius \(R\) and two straight sections. The wire is partially immerged in a perpendicular magnetic field \(B=B_0 \hat{k}\) as shown in figure. The magnetic force on the wire if it has a cur...

An electron moves through a uniform magnetic field \(\vec{B}=B_0 \hat{i}+2 B_0 \hat{j} T\). At a particular instant of time, the velocity of electron is \(\vec{u}=3 \hat{i}+5 \hat{j} \mathrm{~m} / \mathrm{s}\). If the magnetic force acting ...

The relation between time '\(t\)' and distance '\(x\)' is \(t=\alpha x^2+\beta x\), where \(\alpha\) and \(\beta\) are constants. The relation between acceleration \((a)\) and velocity \((v)\) is :

A small steel ball is dropped into a long cylinder containing glycerine. Which one of the following is the correct representation of the velocity time graph for the transit of the ball?

The depth below the surface of sea to which a rubber ball be taken so as to decrease its volume by \(0.02 \%\) is _______ \(m\).
(Take density of sea water \(=10^3 \mathrm{kgm}^{-3}\), Bulk modulus of rubber $$=9 \times 10^8 \mathrm{~Nm}^{-...

A particle performs simple harmonic motion with amplitude \(A\). Its speed is increased to three times at an instant when its displacement is \(\frac{2 A}{3}\). The new amplitude of motion is \(\frac{n A}{3}\). The value of \(n\) is _______...

If the percentage errors in measuring the length and the diameter of a wire are \(0.1 \%\) each. The percentage error in measuring its resistance will be:

A force is represented by \(F=a x^2+b t^{\frac{1}{2}}\)
where \(x=\) distance and \(t=\) time. The dimensions of \(b^2 / a\) are:

Two waves of intensity ratio \(1: 9\) cross each other at a point. The resultant intensities at that point, when (a) Waves are incoherent is \(I_1\) (b) Waves are coherent is \(I_2\) and differ in phase by \(60^{\circ}\). If $$\frac{I_1}{I_...

The fundamental frequency of a closed organ pipe is equal to the first overtone frequency of an open organ pipe. If length of the open pipe is \(60 \mathrm{~cm}\), the length of the closed pipe will be: