JEE Main 2024 (Online) 9th April Morning Shift

Let \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked \(1,2,3,4\). If the probability that \(a x^2+b x+c=0\) has all real roots is $$\frac{m}{n...

Let \(\alpha, \beta\) be the roots of the equation \(x^2+2 \sqrt{2} x-1=0\). The quadratic equation, whose roots are \(\alpha^4+\beta^4\) and \(\frac{1}{10}(\alpha^6+\beta^6)\), is:

If the sum of the series \(\frac{1}{1 \cdot(1+\mathrm{d})}+\frac{1}{(1+\mathrm{d})(1+2 \mathrm{~d})}+\ldots+\frac{1}{(1+9 \mathrm{~d})(1+10 \mathrm{~d})}\) is equal to 5, then \(50 \mathrm{~d}\) is equal to :

Let \(A=\{2,3,6,7\}\) and \(B=\{4,5,6,8\}\). Let \(R\) be a relation defined on \(A \times B\) by \((a_1, b_1) R(a_2, b_2)\) if and only if \(a_1+a_2=b_1+b_2\). Then the number of elements in \(R\) is __________.

The frequency distribution of the age of students in a class of 40 students is given below.

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A variable line \(\mathrm{L}\) passes through the point \((3,5)\) and intersects the positive coordinate axes at the points \(\mathrm{A}\) and \(\mathrm{B}\). The minimum area of the triangle \(\mathrm{OAB}\), where \(\mathrm{O}\) is the or...

A ray of light coming from the point \(\mathrm{P}(1,2)\) gets reflected from the point \(\mathrm{Q}\) on the \(x\)-axis and then passes through the point \(R(4,3)\). If the point \(S(h, k)\) is such that \(P Q R S\) is a parallelogram, then...

Let \(|\cos \theta \cos (60-\theta) \cos (60+\theta)| \leq \frac{1}{8}, \theta \epsilon[0,2 \pi]\). Then, the sum of all \(\theta \in[0,2 \pi]\), where \(\cos 3 \theta\) attains its maximum value, is :

Let three vectors ,\(\overrightarrow{\mathrm{a}}=\alpha \hat{i}+4 \hat{j}+2 \hat{k}, \overrightarrow{\mathrm{b}}=5 \hat{i}+3 \hat{j}+4 \hat{k}, \overrightarrow{\mathrm{c}}=x \hat{i}+y \hat{j}+z \hat{k}\) form a triangle such that $$\vec{c}=...

Let \(\overrightarrow{O A}=2 \vec{a}, \overrightarrow{O B}=6 \vec{a}+5 \vec{b}\) and \(\overrightarrow{O C}=3 \vec{b}\), where \(O\) is the origin. If the area of the parallelogram with adjacent sides \(\overrightarrow{O A}\) and $$\overrig...

A bulb and a capacitor are connected in series across an ac supply. A dielectric is then placed between the plates of the capacitor. The glow of the bulb :

When a coil is connected across a \(20 \mathrm{~V}\) dc supply, it draws a current of \(5 \mathrm{~A}\). When it is connected across \(20 \mathrm{~V}, 50 \mathrm{~Hz}\) ac supply, it draws a current of \(4 \mathrm{~A}\). The self inductance...

The energy equivalent of \(1 \mathrm{~g}\) of substance is :

A star has \(100 \%\) helium composition. It starts to convert three \({ }^4 \mathrm{He}\) into one \({ }^{12} \mathrm{C}\) via triple alpha process as $${ }^4 \mathrm{He}+{ }^4 \mathrm{He}+{ }^4 \mathrm{He} \rightarrow{ }^{12} \mathrm{C}+\...

A capacitor is made of a flat plate of area A and a second plate having a stair-like structure as shown in figure. If the area of each stair is \(\frac{A}{3}\) and the height is \(d\), the capacitance of the arrangement is :

A galvanometer has a coil of resistance \(200 \Omega\) with a full scale deflection at \(20 \mu \mathrm{A}\). The value of resistance to be added to use it as an ammeter of range \((0-20) \mathrm{mA}\) is :

The equivalent resistance between A and B is :

The current flowing through the \(1 \Omega\) resistor is \(\frac{n}{10}\) A. The value of \(n\) is _______.

A proton, an electron and an alpha particle have the same energies. Their de-Broglie wavelengths will be compared as :

A plane EM wave is propagating along \(x\) direction. It has a wavelength of \(4 \mathrm{~mm}\). If electric field is in \(y\) direction with the maximum magnitude of \(60 \mathrm{~Vm}^{-1}\), the equation for magnetic field is :

A light emitting diode (LED) is fabricated using GaAs semiconducting material whose band gap is \(1.42 \mathrm{~eV}\). The wavelength of light emitted from the LED is :

At the centre of a half ring of radius \(\mathrm{R}=10 \mathrm{~cm}\) and linear charge density \(4 \mathrm{~nC} \mathrm{~m}^{-1}\), the potential is \(x \pi \mathrm{V}\). The value of \(x\) is _________.

Given below are two statements :
Statement (I) : When an object is placed at the centre of curvature of a concave lens, image is formed at the centre of curvature of the lens on the other side.
Statement (II) : Concave lens always forms a v...

An astronaut takes a ball of mass \(m\) from earth to space. He throws the ball into a circular orbit about earth at an altitude of \(318.5 \mathrm{~km}\). From earth's surface to the orbit, the change in total mechanical energy of the ball...

The volume of an ideal gas \((\gamma=1.5)\) is changed adiabatically from 5 litres to 4 litres. The ratio of initial pressure to final pressure is :

A sample of 1 mole gas at temperature \(T\) is adiabatically expanded to double its volume. If adiab constant for the gas is \(\gamma=\frac{3}{2}\), then the work done by the gas in the process is :

A heavy iron bar, of weight \(W\) is having its one end on the ground and the other on the shoulder of a person. The bar makes an angle \(\theta\) with the horizontal. The weight experienced by the person is :

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

Given below are two statements :
Statement (I) : When currents vary with time, Newton's third law is valid only if momentum carried by the electromagnetic field is taken into account
Statement (II) : Ampere's circuital law does not depend o...

A square loop of edge length \(2 \mathrm{~m}\) carrying current of \(2 \mathrm{~A}\) is placed with its edges parallel to the \(x\)-\(y\) axis. A magnetic field is passing through the \(x\)-\(y\) plane and expressed as $$\vec{B}=B_0(1+4 x) ...

A particle moving in a straight line covers half the distance with speed \(6 \mathrm{~m} / \mathrm{s}\). The other half is covered in two equal time intervals with speeds \(9 \mathrm{~m} / \mathrm{s}\) and \(15 \mathrm{~m} / \mathrm{s}\) re...

A sphere of relative density \(\sigma\) and diameter \(D\) has concentric cavity of diameter \(d\). The ratio of \(\frac{D}{d}\), if it just floats on water in a tank is :

Two persons pull a wire towards themselves. Each person exerts a force of \(200 \mathrm{~N}\) on the wire. Young's modulus of the material of wire is \(1 \times 10^{11} \mathrm{~N} \mathrm{~m}^{-2}\). Original length of the wire is $$2 \mat...

A string is wrapped around the rim of a wheel of moment of inertia \(0.40 \mathrm{~kgm}^2\) and radius \(10 \mathrm{~cm}\). The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of $...

The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes of \(4 \mathrm{~m}, 2 \mathrm{~ms}^{-1}\) and \(16 \mathrm{~ms}^{-2}\) at a certain instant. The amplitude of the motion is ...

The dimensional formula of latent heat is :

One main scale division of a vernier caliper is equal to \(\mathrm{m}\) units. If \(\mathrm{n}^{\text {th }}\) division of main scale coincides with \((n+1)^{\text {th }}\) division of vernier scale, the least count of the vernier caliper i...

If \(\vec{a}\) and \(\vec{b}\) makes an angle \(\cos ^{-1}\left(\frac{5}{9}\right)\) with each other, then \(|\vec{a}+\vec{b}|=\sqrt{2}|\vec{a}-\vec{b}|\) for \(|\vec{a}|=n|\vec{b}|\) The integer value of \(\mathrm{n}\) is _________.

In a Young's double slit experiment, the intensity at a point is \(\left(\frac{1}{4}\right)^{\text {th }}\) of the maximum intensity, the minimum distance of the point from the central maximum is _________ \(\mu \mathrm{m}\). (Given : $$\la...

A particle of mass \(m\) moves on a straight line with its velocity increasing with distance according to the equation \(v=\alpha \sqrt{x}\), where \(\alpha\) is a constant. The total work done by all the forces applied on the particle duri...