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One of the refracting surfaces of a prism of angle ${30}^{\circ }$ is silvered. A ray of light incident at an angle of ${60}^{\circ }$ retraces its path. The refractive index of the material of prism is

A $10\mu F$ capacitor charged to $20V$ is connected across a $15V$ battery through a switch such that positive terminal of battery is connected to negative charged plate of capacitor. When switch is closed, work done by battery of $15V$ in long time will be

Between two stations, a train accelerates uniformly at first, then moves with constant speed and finally retards uniformly. If the ratios of time taken are $2:1:2$ and the greatest speed is $10\text{km/hr}$, find the average speed over the whole journey.

A block of mass $m$ is placed on a surface with a vertical cross section given by $y=\frac{x^3}{6}$. If the coefficient of friction is $0.5$, the maximum height above the ground at which the block can be placed without slipping is

The figure below shows the position of a ball at t = 0, t = 1s, t = 2s, t = 3 s and t = 4s.



Which of the following position-time graph is possibly correct for the situation shown above?

A bullet of mass $50\text{g}$ is fired from below into the bob of mass $450\text{g}$ of a long simple pendulum, as shown in the figure. The bullet remains inside the bob and bob rises through a height of $1.8\text{m}$. Find the speed of the bullet.

[Take $g=10{\text{ms}}^{-2}$]

Two metallic spheres of radii $2\text{cm}$ and $14\text{cm}$ are connected by a conducting wire. If the sum of charges on both the spheres is $8\text{C}$, then the charge on the smaller sphere is :

Two vectors $\overline{a}\text{and}\overline{b}$ are at an angle of 60⁰ with each other. Their resultant makes an angle of 45⁰ with $\overline{a}$ . If $|\overline{b}|$ =2 units, then $|\overline{a}|$ is

A particle executes SHM of amplitude $25\text{cm}$ and time period $3\text{s}$. What is the minimum time required for the particle to move between two points located at $12.5\text{cm}$ from the mean position on either side?

A plane wall is $25\text{cm}$ thick with an area of $1{\text{m}}^2$ and has a thermal conductivity of $0.5\text{W/mK}$. If a temperature difference of ${60}^{\circ }\text{C}$ is maintained across it, then find the rate of heat flow.

Calculate the no. of molecules present in 6oo cm³ of a gas at 99 kPa pressure at 25 degrees Celsius temperature? Given $R=8.314J{K}^{-1}mo{l}^{-1}$.
In this problem how 99 kPa is converted to 0.99 atm and why?

Five homogeneous bricks, each of length $L=9\text{m}$ are arranged as shown in the figure. Each of the bricks is displaced with respect to the next one in contact by $\frac{L}{6}$ as shown in the figure below. Find the $x-$ coordinate of the centre of mass relative to origin $\text{O}$.

An extended object is placed at point $O,10\text{cm}$ in front of a convex lens $L_1$ and a concave lens $L_2$ is placed $10\text{cm}$ behind it, as shown in the figure. The radii of curvature of all the curved surfaces in both the lenses are $20\text{cm}$. The refractive index of both the lenses is $1.5$. The total magnification of this lens system is

A particle of mass $m$ is projected in vertical plane at an angle of $\theta$ with the horizontal at a speed $V_0$. Match the physical quantities in column I with its variation in Column II
$$\begin{array}{cccc}& \text{Column I}& & \text{Column - II}\\ \text{(A)}& \text{Magnitude of momentum}& \text{(p)}& \text{Continuously increasing linearly with time}\\ \text{(B)}& \text{Magnitude of angular momentum about}& \text{(q)}& \text{First decreases and then increases}\\ & \text{point of projection}& & \\ \text{(C)}& \text{Distance of the particle from point of projection}& \text{(r)}& \text{Continuously increasing but not linearly with time}\\ \text{(D)}& \text{Magnitude of Torque about point of projection}& \text{(s)}& \text{increasing linearly with horizontal displacement}\end{array}$$

A hypothetical spring type object, exerts a force dependent on diplacement x, given by $4x^3$. What is the average force exterted by it while being displaced 2 units from its natural position?

The magnetic induction at the center $\text{O}$ in the figure is,

Define potential energy.

An object thrown at a certain angle to the ground moves in a curved path and falls back to the ground. The initial and the final points of the path of the object lie on the same horizontal line. What is the work done by the force of gravity on the object? [3 MARKS]

Spring force is a restoring force.

A force acts for $10\text{sec}$ on a body of mass $100\text{kg}$ which was at rest, after which it ceases to act. If the body travels $160\text{m}$ in the next $8\text{sec}$, find the magnitude of the force.

Real image of real object is formed by a convex lens. If the distance between real object and its real image is $80\text{cm}$, then power of lens may be

For a small temperature difference between the body and the surroundings, the relation between the rate of heat loss $R$ and the temperature of the body is depicted by

A ball is projected in a direction inclined to the horizontal and it bounces on a horizontal plane. If the range of first rebound is $R$ and coefficient of restitution is $e$, then range of the next rebound is

One mole of a monoatomic ideal gas undergoes the process $A\to B$ in the given $P-V$ diagram. The specific heat for this process is

The transfer ratio of a transistor is 50. The input resistance of the transistor when used in common-emitter configuration is 1 $k\Omega$. The peak value of collector current for an A.C input voltage of 0.01 $V$ peak is

A ring shaped tube contains two ideal gases with equal masses and molecular weight $m_1=32\text{g}$ and $m_2=28\text{g}$. The gases are separated by one fixed partition $P$ and another movable conducting partition $S$ which can move freely without friction inside the ring. The angle $\alpha$ as shown in the figure in equilibrium is:

A radioactive substance is being produced at a constant rate of $200$ nuclei. The decay constant of the substance is $1{\text{s}}^{-1}.$ Assuming that initially there are no nuclei present, the time (in second) after which the number of nuclei will become $100$ is

A particle of mass $m$ and charge $q$ moves with constant velocity $v$ along positive $x-$direction. It enters a region containing a uniform magnetic field $B$ directed along the negative $z-$direction, extending from $x=a$ to $x=b$. The minimum value of $v$ required so that the particle can just enter the region $x>b$ is:

You are given two
circuits as shown in Fig. 14.46, which consist of NAND gates.
Identify the logic operation carried out by the two circuits.

A ball is thrown up from the top of a tower at 20 m/s velocity. It took the ball 6 sec to fall to the bottom. For how long was the ball above the tower?

Obtain the first Bohr’s
radius and the ground state energy of a muonic hydrogen
atom [i.e., an atom in which a negatively charged muon (μ⁻)
of mass about 207m_e orbits around a proton].