JEE Main 2024 (Online) 6th April Morning Shift

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

A company has two plants \(A\) and \(B\) to manufacture motorcycles. \(60 \%\) motorcycles are manufactured at plant \(A\) and the remaining are manufactured at plant \(B .80 \%\) of the motorcycles manufactured at plant \(A\) are rated of ...

Let \(\alpha, \beta\) be the distinct roots of the equation \(x^2-\left(t^2-5 t+6\right) x+1=0, t \in \mathbb{R}\) and \(a_n=\alpha^n+\beta^n\). Then the minimum value of \(\frac{a_{2023}+a_{2025}}{a_{2024}}\) is

Let \(x_1, x_2, x_3, x_4\) be the solution of the equation \(4 x^4+8 x^3-17 x^2-12 x+9=0\) and \(\left(4+x_1^2\right)\left(4+x_2^2\right)\left(4+x_3^2\right)\left(4+x_4^2\right)=\frac{125}{16} m\). Then the value of \(m\) is _________.

Let the first term of a series be \(T_1=6\) and its \(r^{\text {th }}\) term \(T_r=3 T_{r-1}+6^r, r=2,3\),
............ \(n\). If the sum of the first \(n\) terms of this series is $$\frac{1}{5}\left(n^2-12 n+39\right)\left(4 \cdot 6^n-5 \c...

Let the relations \(R_1\) and \(R_2\) on the set \(X=\{1,2,3, \ldots, 20\}\) be given by \(R_1=\{(x, y): 2 x-3 y=2\}\) and \(R_2=\{(x, y):-5 x+4 y=0\}\). If \(M\) and \(N\) be the minimum number of elements required to be added in \(R_1\) a...

Let \(A=\{n \in[100,700] \cap \mathrm{N}: n\) is neither a multiple of 3 nor a multiple of 4\(\}\). Then the number of elements in \(A\) is

The mean and standard deviation of 20 observations are found to be 10 and 2 , respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12. The correct standard deviation is

Let a variable line of slope \(m>0\) passing through the point \((4,-9)\) intersect the coordinate axes at the points \(A\) and \(B\). The minimum value of the sum of the distances of \(A\) and \(B\) from the origin is

Let \(\vec{a}=2 \hat{i}-3 \hat{j}+4 \hat{k}, \vec{b}=3 \hat{i}+4 \hat{j}-5 \hat{k}\) and a vector \(\vec{c}\) be such that \(\vec{a} \times(\vec{b}+\vec{c})+\vec{b} \times \vec{c}=\hat{i}+8 \hat{j}+13 \hat{k}\). If $$\vec{a} \cdot \vec{c}=1...

Given below are two statements :
Statement I : In an LCR series circuit, current is maximum at resonance.
Statement II : Current in a purely resistive circuit can never be less than that in a series LCR circuit when connected to same voltag...

When a \(d c\) voltage of \(100 \mathrm{~V}\) is applied to an inductor, a \(d c\) current of \(5 \mathrm{~A}\) flows through it. When an ac voltage of \(200 \mathrm{~V}\) peak value is connected to inductor, its inductive reactance is foun...

Radius of a certain orbit of hydrogen atom is 8.48 \(\mathop A\limits^o\). If energy of electron in this orbit is \(E / x\). then \(x=\) ________ (Given \(\mathrm{a}_0=0.529\) \(\mathop A\limits^o\), \(E=\) energy of electron in ground stat...

The ratio of the shortest wavelength of Balmer series to the shortest wavelength of Lyman series for hydrogen atom is :

A wire of resistance \(R\) and radius \(r\) is stretched till its radius became \(r / 2\). If new resistance of the stretched wire is \(x ~R\), then value of \(x\) is ________.

The value of unknown resistance \((x)\) for which the potential difference between \(B\) and \(D\) will be zero in the arrangement shown, is :

In photoelectric experiment energy of \(2.48 \mathrm{~eV}\) irradiates a photo sensitive material. The stopping potential was measured to be \(0.5 \mathrm{~V}\). Work function of the photo sensitive material is :

Which of the following phenomena does not explain by wave nature of light.
A. reflection
B. diffraction
C. photoelectric effect
D. interference
E. polarization
Choose the most appropriate answer from the options given below:

Electromagnetic waves travel in a medium with speed of \(1.5 \times 10^8 \mathrm{~m} \mathrm{~s}^{-1}\). The relative permeability of the medium is 2.0. The relative permittivity will be:

The correct truth table for the following logic circuit is :

\(\sigma\) is the uniform surface charge density of a thin spherical shell of radius R. The electric field at any point on the surface of the spherical shell is :

Three infinitely long charged thin sheets are placed as shown in figure. The magnitude of electric field at the point \(P\) is \(\frac{x \sigma}{\epsilon_0}\). The value of \(x\) is _________ (all quantities are measured in SI units).

The refractive index of prism is \(\mu=\sqrt{3}\) and the ratio of the angle of minimum deviation to the angle of prism is one. The value of angle of prism is _________\(^\circ\).

If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be ________ hours 30 minutes.

To project a body of mass \(m\) from earth's surface to infinity, the required kinetic energy is (assume, the radius of earth is \(R_E, g=\) acceleration due to gravity on the surface of earth):

The specific heat at constant pressure of a real gas obeying \(P V^2=R T\) equation is:

A sample contains mixture of helium and oxygen gas. The ratio of root mean square speed of helium and oxygen in the sample, is :

A light string passing over a smooth light pulley connects two blocks of masses \(m_1\) and \(m_2\left(\right.\) where \(\left.m_2>m_1\right)\). If the acceleration of the system is \(\frac{g}{\sqrt{2}}\), then the ratio of the masses $$\fr...

A circular coil having 200 turns, \(2.5 \times 10^{-4} \mathrm{~m}^2\) area and carrying \(100 \mu \mathrm{A}\) current is placed in a uniform magnetic field of \(1 \mathrm{~T}\). Initially the magnetic dipole moment \((\vec{M})\) was direc...

An element \(\Delta l=\Delta x\hat{i}\) is placed at the origin and carries a large current \(I=10 \mathrm{~A}\). The magnetic field on the \(y\)-axis at a distance of \(0.5 \mathrm{~m}\) from the elements \(\Delta x\) of \(1 \mathrm{~cm}\)...

A train starting from rest first accelerates uniformly up to a speed of \(80 \mathrm{~km} / \mathrm{h}\) for time \(t\), then it moves with a constant speed for time \(3 t\). The average speed of the train for this duration of journey will ...

A small ball of mass \(m\) and density \(\rho\) is dropped in a viscous liquid of density \(\rho_0\). After sometime, the ball falls with constant velocity. The viscous force on the ball is :

A big drop is formed by coalescing 1000 small droplets of water. The ratio of surface energy of 1000 droplets to that of energy of big drop is \(\frac{10}{x}\). The value of \(x\) is ________.

A particle is doing simple harmonic motion of amplitude \(0.06 \mathrm{~m}\) and time period \(3.14 \mathrm{~s}\). The maximum velocity of the particle is _________ \(\mathrm{cm} / \mathrm{s}\).

Match List I with List II

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While measuring diameter of wire using screw gauge the following readings were noted. Main scale reading is \(1 \mathrm{~mm}\) and circular scale reading is equal to 42 divisions. Pitch of screw gauge is \(1 \mathrm{~mm}\) and it has 100 di...

To find the spring constant \((k)\) of a spring experimentally, a student commits \(2 \%\) positive error in the measurement of time and \(1 \%\) negative error in measurement of mass. The percentage error in determining value of \(k\) is :

For three vectors \(\vec{A}=(-x \hat{i}-6 \hat{j}-2 \hat{k}), \vec{B}=(-\hat{i}+4 \hat{j}+3 \hat{k})\) and \(\vec{C}=(-8 \hat{i}-\hat{j}+3 \hat{k})\), if \(\vec{A} \cdot(\vec{B} \times \vec{C})=0\), then value of \(x\) is ________.

Four particles \(A, B, C, D\) of mass \(\frac{m}{2}, m, 2 m, 4 m\), have same momentum, respectively. The particle with maximum kinetic energy is :

A bullet of mass \(50 \mathrm{~g}\) is fired with a speed \(100 \mathrm{~m} / \mathrm{s}\) on a plywood and emerges with \(40 \mathrm{~m} / \mathrm{s}\). The percentage loss of kinetic energy is :