Instructions

For the following questions answer them individually

Question 61

An AC voltage source of variable angular frequency $$\omega$$ and fixed amplitude $$V_0$$ is connected in series with a capacitance C and anelectric bulb of resistance R (inductance zero). When $$\omega$$ is increased

Question 62

A thin flexible wire of length L is connected to two adjacent fixed points and carries a current I in the clockwise direction, as shown in the figure. When the system is put in a uniform magnetic field of strength B going into the plane of the paper, the wire takes the shape of a circle. The tension in the wire is

Question 63

A block of mass m is on an inclined plane of angle $$\theta$$. The coefficient of friction between the block and the plane is $$\mu$$ and $$\tan \theta > \mu$$. The block is held stationary by applying a force P parallel to the plane. The direction of force pointing up the plane is taken to be positive. As P is varied from $$P_1 = mg(\sin \theta - \mu \cos \theta)$$ to $$P_2 = mg(\sin \theta + \mu \cos \theta)$$, the frictional force f versus P graph will look like

Question 64

A thin uniform annular disc (see figure) of mass M has outer radius 4R and inner radius 3R. The work required to take a unit mass from point P on its axis to infinity is

Question 65

A few electric field lines for a system of two charges $$Q_1$$ and $$Q_2$$ fixed at two different points on the x-axis are shown in the figure. These lines suggest that

Question 66

A student uses a simple pendulumof exactly 1m length to determine g, the acceleration due to gravity. He uses a stop watch with the least count of 1 sec for this and records 40 secondsfor 20 oscillations. For this observation, which of the following statement(s) is (are) true ?

Question 67

A point mass of 1 kg collides elastically with a stationary point mass of 5 kg. After their collision, the 1 kg mass reverses its direction and moves with a speed of 2 $$ms^{-1}$$.

Which of the following statement(s) is (are) correct for the system of these two masses ?

Question 68

A ray OP of monochromatic light is incident on the face AB of prism ABCD near vertex B at an incident angle of $$60^\circ$$ (see figure). If the refractive index of the material of the prism is $$\sqrt{3}$$, which of the following is (are) correct ?

Question 69

One mole of an ideal gas in initial state A undergoes a cyclic process ABCA, as shown in the figure. Its pressure at A is $$P_0$$. Choose the correct option(s) from the following

Instructions

When a particle of mass m moves on the x-axis in a potential of the form $$V(x) = kx^2$$, it performs simple harmonic motion. The corresponding time period is proportional to $$\sqrt{\frac{m}{k}}$$, as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even whenits potential energy increases on both sides of x = O in a way different from $$kx^2$$ and its total energy is such that the particle does not escape to infinity. Consider a particle of mass m moving on the x-axis. Its potential energy is $$V(x) = \alpha x^4 (\alpha > 0)$$ for $$\mid x \mid$$ near the origin and becomes a constant equal to $$V_0$$ for $$\mid x \mid \geq X_0$$ (see figure).