JEE Main 2021 (Online) 20th July Evening Shift

Let P be a variable point on the parabola \(y = 4{x^2} + 1\). Then, the locus of the mid-point of the point P and the foot of the perpendicular drawn from the point P to the line y = x is :

If the point on the curve y2 = 6x, nearest to the point \(\left( {3,{3 \over 2}} \right)\) is (\(\alpha\), \(\beta\)), then 2(\(\alpha\) + \(\beta\)) is equal to _____________.

Let A, B and C be three events such that the probability that exactly one of A and B occurs is (1 \(-\) k), the probability that exactly one of B and C occurs is (1 \(-\) 2k), the probability that exactly one of C and A occurs is (1 \(-\) k...

If sum of the first 21 terms of the series \({\log _{{9^{1/2}}}}x + {\log _{{9^{1/3}}}}x + {\log _{{9^{1/4}}}}x + .......\), where x > 0 is 504, then x is equal to

For k \(\in\) N, let \({1 \over {\alpha (\alpha + 1)(\alpha + 2).........(\alpha + 20)}} = \sum\limits_{K = 0}^{20} {{{{A_k}} \over {\alpha + k}}}\), where \(\alpha > 0\). Then the value of $$100{\left( {{{{A_{14}} + {A_{15}}} \over {...

Let \(\left\{ {{a_n}} \right\}_{n = 1}^\infty\) be a sequence such that a1 = 1, a2 = 1 and \({a_{n + 2}} = 2{a_{n + 1}} + {a_n}\) for all n \(\ge\) 1. Then the value of \(47\sum\limits_{n = 1}^\infty {{{{a_n}} \over {{2^{3n}}}}}\) is equ...

If the mean and variance of six observations 7, 10, 11, 15, a, b are 10 and \({{20} \over 3}\), respectively, then the value of | a \(-\) b | is equal to :

Consider a triangle having vertices A(\(-\)2, 3), B(1, 9) and C(3, 8). If a line L passing through the circum-centre of triangle ABC, bisects line BC, and intersects y-axis at point \(\left( {0,{\alpha \over 2}} \right)\), then the value o...

In a triangle ABC, if \(\left| {\overrightarrow {BC} } \right| = 3\), \(\left| {\overrightarrow {CA} } \right| = 5\) and \(\left| {\overrightarrow {BA} } \right| = 7\), then the projection of the vector \(\overrightarrow {BA}\) on $$\overr...

For p > 0, a vector \({\overrightarrow v _2} = 2\widehat i + (p + 1)\widehat j\) is obtained by rotating the vector \({\overrightarrow v _1} = \sqrt 3 p\widehat i + \widehat j\) by an angle \(\theta\) about origin in counter clockwise direc...

For a series LCR circuit with R = 100 \(\Omega\), L = 0.5 mH and C = 0.1 pF connected across 220V\(-\)50 Hz AC supply, the phase angle between current and supplied voltage and the nature of the circuit is :

A series LCR circuit of R = 5\(\Omega\), L = 20 mH and C = 0.5 \(\mu\)F is connected across an AC supply of 250 V, having variable frequency. The power dissipated at resonance condition is ______________ \(\times\) 102 W.

For a certain radioactive process the graph between In R and t(sec) is obtained as shown in the figure. Then the value of half life for the unknown radioactive material is approximately :

A radioactive substance decays to \({\left( {{1 \over {16}}} \right)^{th}}\) of its initial activity in 80 days. The half life of the radioactive substance expressed in days is ____________.

If the Kinetic energy of a moving body becomes four times its initial Kinetic energy, then the percentage change in its momentum will be :

In the given figure switches S1 and S2 are in open condition. The resistance across ab when the switches S1 and S2 are closed is _____________ \(\Omega\).

An electron having de-Broglie wavelength \(\lambda\) is incident on a target in a X-ray tube. Cut-off wavelength of emitted X-ray is :

A certain metallic surface is illuminated by monochromatic radiation of wavelength \(\lambda\). The stopping potential for photoelectric current for this radiation is 3V0. If the same surface is illuminated with a radiation of wavelength 2$...

In an electromagnetic wave the electric field vector and magnetic field vector are given as \(\overrightarrow E = {E_0}\widehat i\) and \(\overrightarrow B = {B_0}\widehat k\) respectively. The direction of propagation of electromagnetic ...

A zener diode having zener voltage 8 V and power dissipation rating of 0.5 W is connected across a potential divider arranged with maximum potential drop across zener diode is as shown in the diagram. The value of protective resistance Rp i...

For the forward biased diode characteristics shown in the figure, the dynamic resistance at ID = 3 mA will be __________ \(\Omega\).

A satellite is launched into a circular orbit of radius R around earth, while a second satellite is launched into a circular orbit of radius 1.02 R. The percentage difference in the time periods of the two satellites is :

Consider a binary star system of star A and star B with masses mA and mB revolving in a circular orbit of radii rA an rB, respectively. If TA and TB are the time period of star A and star B, respectively,Then :

Which of the following graphs represent the behavior of an ideal gas? Symbols have their usual meaning.

The correct relation between the degrees of freedom f and the ratio of specific heat \(\gamma\) is :

One mole of an ideal gas at 27\(^\circ\) is taken from A to B as shown in the given PV indicator diagram. The work done by the system will be _________ \(\times\) 10\(-\)1 J. [Given : R = 8.3 J/mole K, ln2 = 0.6931] (Round off to the neare...

A body of mass 'm' is launched up on a rough inclined plane making an angle of 30\(^\circ\) with the horizontal. The coefficient of friction between the body and plane is \({{\sqrt x } \over 5}\) if the time of ascent is half of the time of...

At an angle of 30\(^\circ\) to the magnetic meridian, the apparent dip is 45\(^\circ\). Find the true dip :

The magnetic susceptibility of a material of a rod is 499. Permeability in vacuum is 4\(\pi\) \(\times\) 10\(-\)7 H/m. Absolute permeability of the material of the rod is :

A boy reaches the airport and finds that the escalator is not working. He walks up the stationary escalator in time t1. If he remains stationary on a moving escalator then the escalator takes him up in time t2. The time taken by him to walk...

The length of a metal wire is l1, when the tension in it is T1 and is l2 when the tension is T2. The natural length of the wire is :

Two small drops of mercury each of radius R coalesce to form a single large drop. The ratio of total surface energy before and after the change is :

A body rolls down an inclined plane without slipping. The kinetic energy of rotation is 50% of its translational kinetic energy. The body is :

Two bodies, a ring and a solid cylinder of same material are rolling down without slipping an inclined plane. The radii of the bodies are same. The ratio of velocity of the centre of mass at the bottom of the inclined plane of the ring to t...

A body rotating with an angular speed of 600 rpm is uniformly accelerated to 1800 rpm in 10 sec. The number of rotations made in the process is ___________.

A particle is making simple harmonic motion along the X-axis. If at a distances x1 and x2 from the mean position the velocities of the particle are v1 and v2 respectively. The time period of its oscillation is given as :

If time (t), velocity (v), and angular momentum (l) are taken as the fundamental units. Then the dimension of mass (m) in terms of t, v and l is :

Two vectors \({\overrightarrow P }\) and \({\overrightarrow Q }\) have equal magnitudes. If the magnitude of \({\overrightarrow P + \overrightarrow Q }\) is n times the magnitude of \({\overrightarrow P - \overrightarrow Q }\), then angle...

With what speed should a galaxy move outward with respect to earth so that the sodium-D line at wavelength 5890 \(\mathop A\limits^o\) is observed at 5896 \(\mathop A\limits^o\) ?

A body at rest is moved along a horizontal straight line by a machine delivering a constant power. The distance moved by the body in time 't' is proportional to :