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A ray of light incident parallel to the base PQ of an isosceles right-angled triangular prism PQR suffers two successive total internal reflections at the faces PQ and QR before emerging reversed in direction as shown

If the refractive index of the material of the prism is $\mu$, then

The Laplace transform of $e^{i5t}$, where $i=\sqrt{-1}$ is

A stone is attached at the end of a string and rotated in a horizontal circle by a student. If the final speed of the stone is $2$ times the initial speed, then the final centripetal force is $x$ times the initial centripetal force. The value of $x$ is:

Monochromatic green light of wavelength 5×10^-7 m illuminates a pair of slits 1 mm apart. The separation of bright lines on the interference pattern formed on a screen 2 m away is

If ${}^{\prime }{f}^{\prime }$ denotes the ratio of the number of nuclei decayed $\left(N_d\right)$ to the number of nuclei at $t=0\left(N_0\right)$ then for a collection of radioactive nuclei, the rate of change of ${}^{\prime }{f}^{\prime }$ with respect to time is given as :

[$\lambda$ is the radioactive decay constant]

${}_{83}^{212}$ Bi can disintegrate either by emitting an $\alpha$ -particle or
by emitting a ${\beta }^{-}$ -particle. (a) Write the two equations showing the products of the decays. (b) The probabilities of disintegration by $\alpha$ - and $\beta$ -decays are in the ratio $\frac{7}{13}$. The overall half-life of ${}^{212}$ Bi is one hour. If 1 g of pure ${}^{212}$ Bi is taken at 12.00 noon, what will be the composition of this sample at 1 p.m. the same day?

If $x=(3y^2+4y+3)$, then $\int xdy$ will be

A uniform rod pivoted at its upped end hangs vertically. It is displaced through an angle of 60${}^{\circ }$ and then released. Find the magnitude of the force acting on a particle of mass dm at the tip of the rod when the rod makes an angle of 37${}^{\circ }$ with the vertical.

Two similar plano-convex lenses are combined together in three different ways as shown in the figure below. The ratio of the focal lengths in the three cases will be

Two balls of masses $m_1$ and $m_2$ are moving towards each other with speeds $u_1$ and $u_2$ respectively. After they collide head–on, their speeds are $v_1$ and $v_2$ respectively. (Take $m_1=8\text{kg},m_2=2\text{kg},u_2=3\text{m/s}$) All quantities are in SI units and $e$ is the coefficient of restitution.
Match the statements in List $1$ with results in List $2$
$\begin{array}{cc}\text{List 1}& \text{List 2}\\ \text{A) The speed}{u}_1\text{so that both the balls are moving in the same direction after collision is (}e=0.5)& \text{p)}1/14\\ \text{B) The speed}{u}_1\text{so that maximum fraction of energy is transferred to mass}{m}_2\text{(Assume}e=1\text{) is}& \text{q)}1.5\\ \text{C) If}{u}_1=0.5m/s\text{and}{m}_2\text{stops, then the value of e is}& \text{r)}2\\ \text{D) If}{u}_1=3m/s\text{and e=0,}\text{then fractional loss of KE after collision is}& \text{s)}0.64\end{array}$

Explain quality factor in an AC circuit

A point $P$ having the coordinate in space $(4,2,3)$. Find the position vector of $P$ ?

Find the total charge stored by the combination of capacitors shown in the figure below.

A block of mass m slides along a frictionless surface as shown in the figure. If it released from rest at A, what is its speed at B?

Two blocks $A$ and $B$ of masses $1\text{kg}$ each, are lying on a smooth horizontal surface. A spring of force constant $k=200\text{N/m}$ is fixed at one end of the block $A$. Block $B$ collides with block $A$ with an initial velocity of $v=2.0\text{m/s}$. Find the maximum compression in the spring.

Two bodies $A$ and $B$ having equal surface areas are maintained at temperatures ${10}^{\circ }\text{C}$ and ${20}^{\circ }\text{C}$. The thermal radiation emitted in a given time by $A$ and $B$ are in the ratio,

►The $x-\frac{1}{T}$ graph for an alloy of paramagnetic nature is shown in Fig. The curie constant is, then

The electric and the magnetic field, associated with an $EM$ wave, propagating along the $+z$-axis, can be represented by

If a force F is applied on a body and it moves with a velocity v, the power will be

The percentage increase in the impedance of an a.c circuit, when its power factor changes form $0.866$ to $0.5$ is (if resistance of the circuit remains constant)

The figure represents a solid uniform sphere of mass M and radius R. A spherical cavity of radius r is at a distance a from the centre of the sphere. The gravitational field inside the cavity is

In the Young’s double slit experiment the resultant intensity at a point on the screen is 75% of the maximum intensity of the bright fringe. Then the phase difference between the two interfering rays at that point is

An electron revolving in a circular path with initial angular velocity $6\text{rad/s}$ and constant angular acceleration of $4{\text{rad/s}}^2$. Find the ratio of ${\theta }_{5^{th}}:{\theta }_{7^{th}}:{\theta }_{9^{th}}$ (where ${\theta }_{n^{th}}$ is the angular displacement during $n^{th}$ second).

Equal masses of oxygen, hydrogen and methane are taken in identical conditions. What is the ratio of volumes of gases under identical conditions?

A circular copper disc, of radius $5\text{cm}$, rotates with $10\text{rad/s}$ about an axis through its center and perpendicular to the disc. A uniform magnetic field of $2\text{T}$ acts perpendicular to the plane of the disc. What is the induced current in the disc if a resistor of resistance $1\Omega$ is connected between the midpoint of a radius and the center of the disc?

Four massless springs whose force constants are 2k, 2k, k and 2k respectively are attached to mass M kept on a frictionless plane (as shown in figure). If the mass M is displaced in the horizontal direction, then the frequency of the system is

Two insulating small spheres are rubbed against each other and placed 1 cm apart. If they attract each other with a force of 0.1 N, how many electrons were transferred from one sphere to the other during rubbing?

A stone of mass $1\text{kg}$ is connected to a light inextensible string of length $2\text{m}$ and the other end of the string is fixed. If the stone is moving in a vertical circular path, having a ratio of maximum velocity to the minimum velocity as $3$. The value of kinetic energy at the bottommost point on the vertical circle is [Take $g=10{\text{m/s}}^2]$