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A bead \( P \) sliding on a frictionless semi-circular string \( (ACB) \) and it is at point \( S \) at \( t = 0 \) and at this instant the horizontal component of its velocity is \( v \). Another bead \( Q \) of the same mass as \( P \) is ejected from point \( A \) at \( t = 0 \) along the horizontal string \( AB \), with the speed \( v \), friction between the beads and the respective strings may be neglected in both cases. Let \( t_{p} \) and to be the respective times taken by beads \( P \) and \( Q \) to reach the point \( B \), then the relation between \( t_{p} \) and \( t_{Q} \) is:

The internal energy of a monoatomic gas is \( 3nRT \). One mole of helium is kept in a cylinder having internal cross section area of \( 17\,\text{cm}^{2} \) and fitted with a light movable frictionless piston. The gas is heated slowly by supplying \( 126\,\text{J} \) heat. If the temperature rises by \( 4^{\circ}\text{C} \), then the piston will move ______ cm.
\( \text{(atmospheric pressure } = 10^{5}\ \text{Pa)} \)

The current passing through a conducting loop in the form of equilateral triangle of side \( 4\sqrt{3}\ \text{cm} \) is \( 2\,\text{A} \). The magnetic field at its centroid is \( a \times 10^{-5}\ \text{T} \). The value of \( a \) is ______.
(Given: \( \mu_{0} = 4\pi \times 10^{-7} \) SI units)

Which of the following pair of nuclei are isobars of the element?

A paratrooper jumps from an aero plane and opens a parachute after \( 2 \, \text{s} \) of free fall and starts deaccelerating with \( 3 \, \text{m/s}^2 \). At \( 10 \, \text{m} \) height from ground, while descending with the help of parachute, the speed of paratrooper is \( 5 \, \text{m/s} \). The initial height of the airplane is _____ \( \text{m} \). \( (g = 10 \, \text{m/s}^2) \)

One mole of an ideal diatomic gas expands from volume \( V \) to \( 2V \) isothermally at a temperature \( 27^\circ \text{C} \) and does \( W \) joule of work. If the gas undergoes same magnitude of expansion adiabatically from \( 27^\circ \text{C} \) doing the same amount of work \( W \), then its final temperature will be (close to) _______ \( ^\circ \text{C} \). \( (\log_{e} 2 = 0.693) \)

A circular loop of radius \( 7 \, \text{cm} \) is placed in uniform magnetic field of \( 0.2 \, \text{T} \) directed perpendicular to plane of loop. The loop is converted into a square loop in \( 0.5 \, \text{s} \). The EMF induced in the loop is _______ \( \text{mV} \).

Two shorts dipoles (A, B), A having charges \( \pm 2 \, \mu \text{C} \) and length \( 1 \, \text{cm} \) and B having charges \( \pm 4 \, \mu \text{C} \) and length \( 1 \, \text{cm} \) are placed with their centres \( 80 \, \text{cm} \) apart as shown in the figure. The electric field at a point P, equi-distant from the centres of both dipoles is _______ \( \text{N/C} \).

The ratio of speeds of electromagnetic waves in vacuum and a medium, having dielectric constant \( k = 3 \) and permeability of \( \mu = 2\mu_0 \) is: \( (\mu_0 = \text{permeability of vacuum}) \)

A parallel plate capacitor with plate separation \( 5 \, \text{mm} \) is charged by a battery. On introducing a mica sheet of \( 2 \, \text{mm} \) and maintaining the connections of the plates with the terminals of the battery, it is found that it draws \( 25\% \) more charge from the battery. The dielectric constant of mica is _______.

A small metallic sphere of diameter \( 2 \, \text{mm} \) and density \( 10.5 \, \text{g/cm}^3 \) is dropped in glycerine having viscosity \( 10 \, \text{Poise} \) and density \( 1.5 \, \text{g/cm}^3 \) respectively. The terminal velocity by the sphere is ______ \( \text{cm/s} \).
\( (\pi = \frac{22}{7} \text{ and } g = 10 \, \text{m/s}^2) \)

A prism of angle \( 75^\circ \) and refractive index \( \sqrt{3} \) is coated with thin film of refractive index \( 1.5 \) only at the back exit surface. To have total internal reflection at the back exit surface the incident angle must be ______.
\( (\sin 15^\circ = 0.25 \text{ and } \sin 25^\circ = 0.43) \)

An air bubble of volume \( 2.9 \, \text{cm}^3 \) rises from the bottom of a swimming pool of \( 5 \, \text{m} \) deep. At the bottom of the pool water temperature is \( 17^\circ\text{C} \). The volume of the bubble when it reaches the surface, where the water temperature is \( 27^\circ\text{C} \), is ______ \( \text{cm}^3 \).
\( (g = 10 \, \text{m/s}^2 , \ \text{density of water} = 10^3 \, \text{kg/m}^3 , \text{ and } 1 \, \text{atm pressure is } 10^5 \, \text{Pa}) \)

A block is sliding down on an inclined plane of slope \( \theta \) and at an instant \( t = 0 \) this block is given an upward momentum so that it starts moving up on the inclined surface with velocity \( u \). The distance \( (S) \) travelled by the block before its velocity become zero, is ________.
\( (g = \text{gravitational acceleration}) \)

To compare EMF of two cells using potentiometer the balancing lengths obtained are \( 200 \, \text{cm} \) and \( 150 \, \text{cm} \). The least count of scale is \( 1 \, \text{cm} \). The percentage error in the ratio of EMFs is ______.

Two charges \( 7 \, \mu \text{C} \) and \( -2 \, \mu \text{C} \) are placed at \( (-9, 0, 0) \, \text{cm} \) and \( (9, 0, 0) \, \text{cm} \) respectively in an external field \( {E} = \dfrac{A}{r^2} \, \hat{r} \), where \( A = 9 \times 10^{5} \, \text{N}/\text{C} \cdot \text{m}^{2} \). Considering the potential at infinity is \( 0 \), the electrostatic energy of the configuration is ______ \( \text{J} \).

For the given logic gate circuit, which of the following is the correct truth table?

A body of mass \( 14 \, \text{kg} \) initially at rest explodes and breaks into three fragments of masses in the ratio \( 2:2:3 \). The two pieces of equal masses fly off perpendicular to each other with a speed of \( 18 \, \text{m/s} \) each. The velocity of the heavier fragment is ______ \( \text{m/s} \).

When an unpolarized light falls at a particular angle on a glass plate (placed in air), it is observed that the reflected beam is linearly polarized. The angle of refracted beam with respect to the normal is ______.
\(( \tan^{-1}(1.52) = 57.7^\circ \), refractive indices of air and glass are \( 1.00 \) and \( 1.52 \), respectively.

Suppose a long solenoid of \( 100 \, \text{cm} \) length, radius \( 2 \, \text{cm} \) having \( 500 \) turns per unit length, carries a current \( I = 10 \sin (\omega t) \, \text{A} \), where \( \omega = 1000 \, \text{rad/s} \). A circular conducting loop (B) of radius \( 1 \, \text{cm} \) coaxially slided through the solenoid at a speed \( v = 1 \, \text{cm/s} \). The r.m.s. current through the loop when the coil B is inserted \( 10 \, \text{cm} \) inside the solenoid is \( \alpha / \sqrt{2} \, \mu \text{A} \). The value of \( \alpha \) is _______. [Resistance of the loop \( 10 \, \Omega \)]

The average energy released per fission for the nucleus of \( {}^{235}_{92}\text{U} \) is \( 190 \, \text{MeV} \). When all the atoms of \( 47 \, \text{g} \) pure \( {}^{235}_{92}\text{U} \) undergo fission process, the energy released is \( a \times 10^{23} \, \text{MeV} \). The value of \( a \) is ______.
(Avogadro Number \( = 6 \times 10^{23} \) per mole)

A ball of radius \( r \) and density \( \rho \) dropped through a viscous liquid of density \( \sigma \) and viscosity \( \eta \) attains its terminal velocity at time \( t \), given by \( t = A \rho^{a} r^{b} \eta^{c} \sigma^{d} \), where \( A \) is a constant and \( a, b, c \) and \( d \) are integers. The value of \( \frac{b+c}{a+d} \) is _______.

Suppose there is a uniform circular disc of mass \( M \; \mathrm{kg} \) and radius \( r \; \mathrm{m} \) shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis \( A \) of the disc is given by \( \dfrac{x}{256} M r^{2} \). The value of \( x \) is ________.

The size of the image of an object, formed by a thin lens are equal when the object is placed at two different positions \( 8 \; \mathrm{cm} \) and \( 24 \; \mathrm{cm} \) from the lens. The focal length of the lens is ______ \( \mathrm{cm} \).

The velocity of sound in air is doubled when the temperature is raised from \( 0^{\circ}\mathrm{C} \) to \( \alpha^{\circ}\mathrm{C} \). The value of \( \alpha \) is ______.

JEE Main 2026 23rd Jan Evening Shift

Mathematics Physics Chemistry

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For the given logic gate circuit, which of the following is the correct truth table?