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If an incident ray directed towards a corner reflector as shown, at what angle$\left(\theta \right)$ does the reflected ray emerge?

Prove the result that the velocity v of translation of a rolling body (like a ring, disc, cylinder or sphere) at the bottom of an inclined plane of a height h is given by .

Using dynamical consideration (i.e. by consideration of forces and torques). Note k is the radius of gyration of the body about its symmetry axis, and R is the radius of the body. The body starts from rest at the top of the plane.

Indium antimonide has a band gap of $0.23\text{eV}$ between the valence and the conduction band. Find the temperature at which the thermal energy $kT$ equals the indium antimonide’s band gap.

$(k=\text{Boltzmann constant},T=\text{temperature})$

$(\text{Take}k=1.38\times {10}^{-23}{\text{JK}}^{-1})$

The time period of a magnet, oscillating in an external magnetic field, is T. If it is divided in four equal parts, along its axis and perpendicular to its axis (as shown in the diagram), then the time period of each part will be,

A coin is placed on a turntable which is making $40$ revolutions per minute. The distance of the coin from the center of the turntable is $\frac{3}{{\pi }^2}\text{cm}$. The minimum friction coefficient between the turntable and the coin is [Take $g=10{\text{m/s}}^2$]

The displacement of a particle of a string carrying a travelling wave is given by y= (3.0 cm) sin 6.28(0.50x - 50 t),

where x is in centimeter and t in second. Find

(i) the amplitude (p) 50

(ii) the wavelength (q) 3

(iii) the frequency(r) 2

(iv) the speed (s) 100

A Kundt's tube apparatus has a copper rod of length 1.0 m clamped at 25 cm from one of the ends. The tube contains air in which the speed of sound is 340 in/s. The powder collects in heaps separated by a distance of 5.0 cm. Find the speed of sound waves in copper.

A particle of mass $10\text{g}$ is kept on the surface of a uniform sphere of mass $100\text{kg}$ and radius $10\text{cm}$. Find the work to be done against the gravitational force between them, to take the particle far away from the sphere (you may take $G=6.67\times {10}^{-11}{\text{Nm}}^2{\text{kg}}^{-2}$

A refrigerator is thermally equivalent to a box of corkboard 90mm thick and $6m^2$ in inner surface area, the thermal conductivity of cork being 0.05 W/mK, the motor of the refrigerator runs $15\%$ of the time while the door is closed. The inside wall of the door, when it is closed, is kept, on an average, ${22}^{\circ }C$ below the temperature of the outside wall. The rate at which heat is taken from the interior wall while the motor is running is

Two soap bubbles, one of radius 50 mm and the other of radius 80 mm, are brought in contact so that they have a common interface. The radius of the curvature of the common interface is

Two coils of self inductance $L_1$ and $L_2$ are placed closer to each other so that total flux in one coil is completely linked with other. If $M$ is mutual inductance between them, then

The smallest division on the main scale of a Vernier calipers is $0.1\text{cm}$. Ten divisions of the Vernier scale correspond to nine divisions of the main scale. The figure below on the left shows the reading of this calipers with no gap between its two jaws. The figure on the right shows the reading with a solid sphere held between the jaws. The correct diameter of the sphere is

The maximum height reached by projectile is $4\text{m}$. The horizontal range is $12\text{m}$. Velocity of projection in $\left({\text{ms}}^{-1}\right)$ is
($g$ is acceleration due to gravity)

Two wires $A$ and $B$ of same length, same area of cross-section having the same Young’s modulus are heated to the same range of temperature. If the coefficient of linear expansion of $A$ is $3/2$ times of that of wire $B$. The ratio of the forces produced in two wires will be

Two particles of masses m₁ and m₂ are connected to a rigid massless rod of length r to constitute a dumb bell which is free to move in the plane. The moment of inertia of the dumb bell about an axis perpendicular to the length of rod and passing through the centre of mass is:

A fish looking up through water sees the outside world contained in a circular horizon. If the fish is $\sqrt{7}\text{cm}$ below the surface of water $(\mu =\frac{4}{3})$, the radius of the circle will be

Pure Si at $300K$ has equal electron and hole concentration of $1.5\times {10}^{16}/m^3$. Doping by indium increases $n_h$ to $4.5\times {10}^{22}/m^3$. What is $n_e$ in doped silicon?

For matrices of same dimension M, N and scalar c, which one of these properties DOES NOT ALWAYS hold?

If AB is the object and CD is the image formed. If AP is the object distance and CP is the image distance from the mirror. Then choose the correct sign convention?

$\begin{array}{cc}1.heightofimage=+ve& 2.heightofimage=-ve\\ 3.heightofobject=+ve& 4.heightofobject=-ve\\ 5.AP=+ve& 6.AP=-ve\\ 7.PC=+ve& 8.PC=-ve\end{array}$

Two blocks of $4\text{kg}$ and $6\text{kg}$ are attached by springs and they are hanging in vertical position. Lower spring breaks due to excessive force. Acceleration of $4\text{kg}$ block just after breaking will be (Take $g=10m/s^2$)

A uniform thermometer scale is at steady state with its $0\text{cm}$ mark at ${20}^{\circ }\text{C}$ and $100\text{cm}$ mark at ${100}^{\circ }\text{C}$. Temperature of the $60\text{cm}$ mark is

A rectangular coil of $20$ turns and area of cross-section $25c{m}^2$ has a resistance of $100\Omega$. If a magnetic field which is perpendicular to the plane of coil changes at a rate of $1000T/s$ , the current in the coil is

A closed organ pipe can vibrate at a minimum frequency of 500 Hz. Find the length of the tube. Speed of sound in air = $340m{s}^{-1}$.

Lift A is going up with velocity 20m/s and acceleration of $8m/s^2$ in the downward direction. While lift B is going down with 10m/s and acceleration of $+2m/s^2$ in upward direction. At this moment a bolt in the ceiling of lift A gets loose and falls. Height of the lift is 16 m.Find the displacement of bolt relative to B when it hits the floor of A.

Find the time period of oscillation of the spring - mass system assuming strings, pulleys and springs are light.