JEE Main 2024 (Online) 29th January Evening Shift

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

If \(\log _e \mathrm{a}, \log _e \mathrm{~b}, \log _e \mathrm{c}\) are in an A.P. and \(\log _e \mathrm{a}-\log _e 2 \mathrm{~b}, \log _e 2 \mathrm{~b}-\log _e 3 \mathrm{c}, \log _e 3 \mathrm{c} -\log _e\) a are also in an A.P, then $$a: b:...

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

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

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

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

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

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

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}{\...

Let \(\overrightarrow{O A}=\vec{a}, \overrightarrow{O B}=12 \vec{a}+4 \vec{b} \text { and } \overrightarrow{O C}=\vec{b}\), where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then $$\mathrm{{{area\,of\,the\,quad...

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

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 clou...

Hydrogen atom is bombarded with electrons accelerated through a potential difference of \(\mathrm{V}\), which causes excitation of hydrogen atoms. If the experiment is being performed at \(\mathrm{T}=0 \mathrm{~K}\), the minimum potential d...

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

A stone of mass \(900 \mathrm{~g}\) is tied to a string and moved in a vertical circle of radius \(1 \mathrm{~m}\) making \(10 \mathrm{~rpm}\). The tension in the string, when the stone is at the lowest point is (if \(\pi^2=9.8\) and $$g=9....

A particle is moving in a circle of radius \(50 \mathrm{~cm}\) in such a way that at any instant the normal and tangential components of it's acceleration are equal. If its speed at \(\mathrm{t}=0\) is \(4 \mathrm{~m} / \mathrm{s}\), the ti...

In the given circuit, the current in resistance R\(_3\) is :

In the given circuit, the current flowing through the resistance \(20 \Omega\) is \(0.3 \mathrm{~A}\), while the ammeter reads \(0.9 \mathrm{~A}\). The value of \(\mathrm{R}_1\) is _________ \(\Omega\).

Two sources of light emit with a power of \(200 \mathrm{~W}\). The ratio of number of photons of visible light emitted by each source having wavelengths \(300 \mathrm{~nm}\) and \(500 \mathrm{~nm}\) respectively, will be :

A horizontal straight wire \(5 \mathrm{~m}\) long extending from east to west falling freely at right angle to horizontal component of earths magnetic field \(0.60 \times 10^{-4} \mathrm{~Wbm}^{-2}\). The instantaneous value of emf induced ...

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

The truth table for this given circuit is :

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

If the distance between object and its two times magnified virtual image produced by a curved mirror is \(15 \mathrm{~cm}\), the focal length of the mirror must be:

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 :

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

\(N\) 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 :

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 \(R_2\) r...

A charge of \(4.0 \mu \mathrm{C}\) is moving with a velocity of \(4.0 \times 10^6 \mathrm{~ms}^{-1}\) along the positive \(y\) axis under a magnetic field \(\vec{B}\) of strength \((2 \hat{k}) \mathrm{T}\). The force acting on the charge is...

A particle is moving in a straight line. The variation of position '\(x\)' as a function of time '\(t\)' is given as \(x=\left(t^3-6 t^2+20 t+15\right) m\). The velocity of the body when its acceleration becomes zero is :

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 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 kee...

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...

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

The bob of a pendulum was released from a horizontal position. The length of the pendulum is \(10 \mathrm{~m}\). If it dissipates \(10 \%\) of its initial energy against air resistance, the speed with which the bob arrives at the lowest poi...

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}...

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 \(Q\) i...

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 \(7 \lambda / 4\). The ratio of intensity of fringe at this p...

In a single slit diffraction pattern, a light of wavelength 6000\(\mathop A\limits^o\) is used. The distance between the first and third minima in the diffraction pattern is found to be \(3 \mathrm{~mm}\) when the screen in placed $$50 \mat...

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 en...