JEE Main 2022 (Online) 29th July Morning Shift

If \(\frac{1}{(20-a)(40-a)}+\frac{1}{(40-a)(60-a)}+\ldots+\frac{1}{(180-a)(200-a)}=\frac{1}{256}\), then the maximum value of \(\mathrm{a}\) is :

Let \(a_{1}, a_{2}, a_{3}, \ldots\) be an A.P. If \(\sum\limits_{r=1}^{\infty} \frac{a_{r}}{2^{r}}=4\), then \(4 a_{2}\) is equal to _________.

If \(\frac{1}{2 \times 3 \times 4}+\frac{1}{3 \times 4 \times 5}+\frac{1}{4 \times 5 \times 6}+\ldots+\frac{1}{100 \times 101 \times 102}=\frac{\mathrm{k}}{101}\), then 34 k is equal to _________.

Let R be a relation from the set \(\{1,2,3, \ldots, 60\}\) to itself such that \(R=\{(a, b): b=p q\), where \(p, q \geqslant 3\) are prime numbers}. Then, the number of elements in R is :

Let \(S=\{4,6,9\}\) and \(T=\{9,10,11, \ldots, 1000\}\). If \(A=\left\{a_{1}+a_{2}+\ldots+a_{k}: k \in \mathbf{N}, a_{1}, a_{2}, a_{3}, \ldots, a_{k}\right.\) \(\epsilon S\}\), then the sum of all the elements in the set \(T-A\) is equal to...

Let the mean and the variance of 20 observations \(x_{1}, x_{2}, \ldots, x_{20}\) be 15 and 9 , respectively. For \(\alpha \in \mathbf{R}\), if the mean of $$\left(x_{1}+\alpha\right)^{2},\left(x_{2}+\alpha\right)^{2}, \ldots,\left(x_{20}+\...

Let the circumcentre of a triangle with vertices A(a, 3), B(b, 5) and C(a, b), ab > 0 be P(1,1). If the line AP intersects the line BC at the point Q\(\left(k_{1}, k_{2}\right)\), then \(k_{1}+k_{2}\) is equal to :

Let \(S=\left\{\theta \in(0,2 \pi): 7 \cos ^{2} \theta-3 \sin ^{2} \theta-2 \cos ^{2} 2 \theta=2\right\}\). Then, the sum of roots of all the equations $$x^{2}-2\left(\tan ^{2} \theta+\cot ^{2} \theta\right) x+6 \sin ^{2} \theta=0, \theta \...

Let \(\overrightarrow{\mathrm{a}}=3 \hat{i}+\hat{j}\) and \(\overrightarrow{\mathrm{b}}=\hat{i}+2 \hat{j}+\hat{k}\). Let \(\overrightarrow{\mathrm{c}}\) be a vector satisfying $$\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}}...

Let \(\hat{a}\) and \(\hat{b}\) be two unit vectors such that the angle between them is \(\frac{\pi}{4}\). If \(\theta\) is the angle between the vectors \((\hat{a}+\hat{b})\) and \((\hat{a}+2 \hat{b}+2(\hat{a} \times \hat{b}))\), then the ...

An alternating emf \(\mathrm{E}=440 \sin 100 \pi \mathrm{t}\) is applied to a circuit containing an inductance of \(\frac{\sqrt{2}}{\pi} \mathrm{H}\). If an a.c. ammeter is connected in the circuit, its reading will be :

A coil of inductance 1 H and resistance \(100 \,\Omega\) is connected to a battery of 6 V. Determine approximately :
(a) The time elapsed before the current acquires half of its steady - state value.
(b) The energy stored in the magnetic fi...

Find the ratio of energies of photons produced due to transition of an electron of hydrogen atom from its (i) second permitted energy level to the first level, and (ii) the highest permitted energy level to the first permitted level.

Two bodies of mass \(1 \mathrm{~kg}\) and \(3 \mathrm{~kg}\) have position vectors \(\hat{i}+2 \hat{j}+\hat{k}\) and \(-3 \hat{i}-2 \hat{j}+\hat{k}\) respectively. The magnitude of position vector of centre of mass of this system will be si...

A smooth circular groove has a smooth vertical wall as shown in figure. A block of mass m moves against the wall with a speed v. Which of the following curve represents the correct relation between the normal reaction on the block by the wa...

Find the modulation index of an AM wave having \(8 \mathrm{~V}\) variation where maximum amplitude of the AM wave is \(9 \mathrm{~V}\).

Two metallic wires of identical dimensions are connected in series. If \(\sigma_{1}\) and \(\sigma_{2}\) are the conductivities of the these wires respectively, the effective conductivity of the combination is :

The current I flowing through the given circuit will be __________A.

The kinetic energy of emitted electron is E when the light incident on the metal has wavelength \(\lambda\). To double the kinetic energy, the incident light must have wavelength:

Match List - I with List - II :

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;padding:10px 5px;word-break:nor...

If the potential barrier across a p-n junction is \(0.6 \mathrm{~V}\). Then the electric field intensity, in the depletion region having the width of \(6 \times 10^{-6} \mathrm{~m}\), will be __________ $$\times 10^{5} \mathrm{~N} / \mathrm...

A spherically symmetric charge distribution is considered with charge density varying as
$$\rho(r)= \begin{cases}\rho_{0}\left(\frac{3}{4}-\frac{r}{R}\right) & \text { for } r \leq R \\ \text { zero } & \text { for } r>R\end{cases}$$
Where,...

Given below are two statements.
Statement I : Electric potential is constant within and at the surface of each conductor.
Statement II : Electric field just outside a charged conductor is perpendicular to the surface of the conductor at eve...

The X-Y plane be taken as the boundary between two transparent media \(\mathrm{M}_{1}\) and \(\mathrm{M}_{2}\). \(\mathrm{M}_{1}\) in \(Z \geqslant 0\) has a refractive index of \(\sqrt{2}\) and \(M_{2}\) with \(Z<0\) has a refractive index...

If the acceleration due to gravity experienced by a point mass at a height h above the surface of earth is same as that of the acceleration due to gravity at a depth \(\alpha \mathrm{h}\left(\mathrm{h}<<\mathrm{R}_{\mathrm{e}}\right)\) from...

The pressure \(\mathrm{P}_{1}\) and density \(\mathrm{d}_{1}\) of diatomic gas \(\left(\gamma=\frac{7}{5}\right)\) changes suddenly to \(\mathrm{P}_{2}\left(>\mathrm{P}_{1}\right)\) and \(\mathrm{d}_{2}\) respectively during an adiabatic pr...

One mole of a monoatomic gas is mixed with three moles of a diatomic gas. The molecular specific heat of mixture at constant volume is \(\frac{\alpha^{2}}{4} \mathrm{R} \,\mathrm{J} / \mathrm{mol} \,\mathrm{K}\); then the value of $$\alpha$...

A closely wounded circular coil of radius 5 cm produces a magnetic field of \(37.68 \times 10^{-4} \mathrm{~T}\) at its center. The current through the coil is _________A.
[Given, number of turns in the coil is 100 and \(\pi=3.14\)]

A wire of length \(314 \mathrm{~cm}\) carrying current of \(14 \mathrm{~A}\) is bent to form a circle. The magnetic moment of the coil is ________ A \(-\mathrm{m}^{2}\). [Given \(\pi=3.14\)]

An object is projected in the air with initial velocity u at an angle \(\theta\). The projectile motion is such that the horizontal range R, is maximum. Another object is projected in the air with a horizontal range half of the range of fir...

A ball is thrown up vertically with a certain velocity so that, it reaches a maximum height h. Find the ratio of the times in which it is at height \(\frac{h}{3}\) while going up and coming down respectively.

If \(\mathrm{t}=\sqrt{x}+4\), then \(\left(\frac{\mathrm{d} x}{\mathrm{~d} t}\right)_{\mathrm{t}=4}\) is :

Given below are two statements : One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Clothes containing oil or grease stains cannot be cleaned by water wash.
Reason (R): Because the angle of contact betw...

If the length of a wire is made double and radius is halved of its respective values. Then, the Young's modulus of the material of the wire will :

The time period of oscillation of a simple pendulum of length L suspended from the roof of a vehicle, which moves without friction down an inclined plane of inclination \(\alpha\), is given by :

Given below are two statements : One is labelled as Assertion (A) and other is labelled as Reason (R).
Assertion (A) : Time period of oscillation of a liquid drop depends on surface tension (S), if density of the liquid is \(\rho\) and radi...

A travelling microscope has 20 divisions per \(\mathrm{cm}\) on the main scale while its vernier scale has total 50 divisions and 25 vernier scale divisions are equal to 24 main scale divisions, what is the least count of the travelling mic...

In an experiment to find out the diameter of wire using screw gauge, the following observations were noted :

(A) Screw moves \(0.5 \mathrm{~mm}\) on main scale in one complete rotation
(B) Total divisions on circular scale \(=50\)
(C) Main...

Two light beams of intensities 4I and 9I interfere on a screen. The phase difference between these beams on the screen at point A is zero and at point B is \(\pi\). The difference of resultant intensities, at the point A and B, will be ____...

A ball is projected with kinetic energy E, at an angle of \(60^{\circ}\) to the horizontal. The kinetic energy of this ball at the highest point of its flight will become :