JEE Main 2021 (Online) 25th July Evening Shift

If a rectangle is inscribed in an equilateral triangle of side length \(2\sqrt 2\) as shown in the figure, then the square of the largest area of such a rectangle is _____________.

If [x] be the greatest integer less than or equal to x, then \(\sum\limits_{n = 8}^{100} {\left[ {{{{{( - 1)}^n}n} \over 2}} \right]}\) is equal to :

The number of real solutions of the equation, x2 \(-\) |x| \(-\) 12 = 0 is :

If a + b + c = 1, ab + bc + ca = 2 and abc = 3, then the value of a4 + b4 + c4 is equal to ______________.

The first of the two samples in a group has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15.6 and standard deviation \(\sqrt {13.44}\), then the standard deviation of the second sample is :

Let the equation of the pair of lines, y = px and y = qx, can be written as (y \(-\) px) (y \(-\) qx) = 0. Then the equation of the pair of the angle bisectors of the lines x2 \(-\) 4xy \(-\) 5y2 = 0 is :

The value of \(\cot {\pi \over {24}}\) is :

Let a, b and c be distinct positive numbers. If the vectors \(a\widehat i + a\widehat j + c\widehat k,\widehat i+\widehat k\) and \(c\widehat i + c\widehat j + b\widehat k\) are co-planar, then c is equal to :

If \(\left| {\overrightarrow a } \right| = 2,\left| {\overrightarrow b } \right| = 5\) and \(\left| {\overrightarrow a \times \overrightarrow b } \right|\) = 8, then \(\left| {\overrightarrow a .\,\overrightarrow b } \right|\) is equal to ...

If \(\left( {\overrightarrow a + 3\overrightarrow b } \right)\) is perpendicular to \(\left( {7\overrightarrow a - 5\overrightarrow b } \right)\) and \(\left( {\overrightarrow a - 4\overrightarrow b } \right)\) is perpendicular to $$\lef...

A 10 \(\Omega\) resistance is connected across 220V \(-\) 50 Hz AC supply. The time taken by the current to change from its maximum value to the rms value is :

Two circuits are shown in the figure (a) & (b). At a frequency of ____________ rad/s the average power dissipated in one cycle will be same in both the circuits.

From the given data, the amount of energy required to break the nucleus of aluminium \(_{13}^{27}\)Al is __________ x \(\times\) 10\(-\)3 J.Mass of neutron = 1.00866 uMass of proton = 1.00726 uMass of Aluminium nucleus = 27.18846 u(Assume 1...

The nuclear activity of a radioactive element becomes \({\left( {{1 \over 8}} \right)^{th}}\) of its initial value in 30 years. The half-life of radioactive element is _____________ years.

If qf is the free charge on the capacitor plates and qb is the bound charge on the dielectric slab of dielectric constant k placed between the capacitor plates, then bound charge qb an be expressed as :

A message signal of frequency 20 kHz and peak voltage of 20 volt is used to modulate a carrier wave of frequency 1 MHz and peak voltage of 20 volt. The modulation index will be :

In the given potentiometer circuit arrangement, the balancing length AC is measured to be 250 cm. When the galvanometer connection is shifted from point (1) to point (2) in the given diagram, the balancing length becomes 400 cm. The ratio o...

The given potentiometer has its wire of resistance 10\(\Omega\). When the sliding contact is in the middle of the potentiometer wire, the potential drop across 2\(\Omega\) resistor is :

A 16 \(\Omega\) wire is bend to form a square loop. A 9V supply having internal resistance of 1\(\Omega\) is connected across one of its sides. The potential drop across the diagonals of the square loop is _______________ \(\times\) 10\(-\)...

An electron moving with speed v and a photon moving with speed c, have same D-Broglie wavelength. The ratio of kinetic energy of electron to that of photon is :

When radiation of wavelength \(\lambda\) is incident on a metallic surface, the stopping potential of ejected photoelectrons is 4.8 V. If the same surface is illuminated by radiation of double the previous wavelength, then the stopping pote...

A light beam of wavelength 500 nm is incident on a metal having work function of 1.25 eV, placed in a magnetic field of intensity B. The electrons emitted perpendicular to the magnetic field B, with maximum kinetic energy are bent into circ...

In a semiconductor, the number density of intrinsic charge carries at 27\(^\circ\)C is 1.5 \(\times\) 1016/m3. If the semiconductor is doped with impurity atom, the hole density increases to 4.5 \(\times\) 1022/m3. The electron density in t...

Two ideal electric dipoles A and B, having their dipole moment p1 and p2 respectively are placed on a plane with their centres at O as shown in the figure. At point C on the axis of dipole A, the resultant electric field is making an angle ...

A prism of refractive index \(\mu\) and angle of prism A is placed in the position of minimum angle of deviation. If minimum angle of deviation is also A, then in terms of refractive index

A ray of light entering from air into a denser medium of refractive index \({4 \over 3}\), as shown in figure. The light ray suffers total internal reflection at the adjacent surface as shown. The maximum value of angle \(\theta\) should be...

Consider a planet in some solar system which has a mass double the mass of earth and density equal to the average density of earth. If the weight of an object on earth is W, the weight of the same object on that planet will be :

A heat engine has an efficiency of \({1 \over 6}\). When the temperature of sink is reduced by 62\(^\circ\)C, its efficiency get doubled. The temperature of the source is :

A system consists of two types of gas molecules A and B having same number density 2 \(\times\) 1025/m3. The diameter of A and B are 10 \(\mathop A\limits^o\) and 5 \(\mathop A\limits^o\) respectively. They suffer collision at room temper...

A force \(\overrightarrow F = (40\widehat i + 10\widehat j)N\) acts on a body of mass 5 kg. If the body starts from rest, its position vector \(\overrightarrow r\) at time t = 10 s, will be :

Two ions having same mass have charges in the ratio 1 : 2. They are projected normally in a uniform magnetic field with their speeds in the ratio 2 : 3. The ratio of the radii of their circular trajectories is :

The relation between time t and distance x for a moving body is given as t = mx2 + nx, where m and n are constants. The retardation of the motion is : (When v stands for velocity)

A balloon was moving upwards with a uniform velocity of 10 m/s. An object of finite mass is dropped from the balloon when it was at a height of 75 m from the ground level. The height of the balloon from the ground when object strikes the gr...

The instantaneous velocity of a particle moving in a straight line is given as \(V = \alpha t + \beta {t^2}\), where \(\alpha\) and \(\beta\) are constants. The distance travelled by the particle between 1s and 2s is :

Two spherical soap bubbles of radii r1 and r2 in vacuum combine under isothermal conditions. The resulting bubble has a radius equal to :

A solid disc of radius 20 cm and mass 10 kg is rotating with an angular velocity of 600 rpm, about an axis normal to its circular plane and passing through its centre of mass. The retarding torque required to bring the disc at rest in 10 s ...

In a simple harmonic oscillation, what fraction of total mechanical energy is in the form of kinetic energy, when the particle is midway between mean and extreme position.

The force is given in terms of time t and displacement x by the equation F = A cos Bx + C sin DtThe dimensional formula of \({{AD} \over B}\) is :

Two vectors \(\overrightarrow X\) and \(\overrightarrow Y\) have equal magnitude. The magnitude of (\(\overrightarrow X\) \(-\) \(\overrightarrow Y\)) is n times the magnitude of (\(\overrightarrow X\) + \(\overrightarrow Y\)). The an...

A force of F = (5y + 20)\(\widehat j\) N acts on a particle. The work done by this force when the particle is moved from y = 0 m to y = 10 m is ___________ J.