1 A particle is moving with a uniform speed in a circular orbit of radius R in a central force inversely proportional to the n th power of R. If the period of rotation of the particle is T, then A) $T\propto R^{\frac{3}{2}}$ For any n B) $T\propto R^{\frac{n}{2}}$ C) $T\propto R^{\frac{n}{2}+1}$ D) $T\propto R^{\frac{(n+1)}{2}}$
2 From a unifrom circular disc of radius R and mass 9 M, a small disc of radius $\frac{R}{3}$ is removed as shown in the figure. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through centre of disc is A) $4MR^{2}$ B) $10MR^{2}$ C) $\frac{40}{9}MR^{2}$ D) $\frac{37}{9}MR^{2}$
3 A solid sphere of radius r made of a soft material of bulk modulus K is surrounded by a liquid in a cylindrical container. A massless piston of area a floats on the surface of the liquid, covering entire cross- section of cylindrical container. When a mass m is placed on the surface of the piston to compress the liquid, the fractional decrement in the radius of the sphere, ($\frac{\text{d}r}{\text{}r}$) is A) $\frac{Ka}{mg}$ B) $\frac{Ka}{3mg}$ C) $\frac{mg}{3Ka}$ D) $\frac{mg}{Ka}$
4 If the series limit frequency of the Lyman series is vL, the the series limit frequency of the Pfund series is A) $25v_{L}$ B) $16v_{L}$ C) $\frac{v_{L}}{16}$ D) $\frac{v_{L}}{25}$
5 A uniform capillary tube of inner radius r is dipped vertically in to a beaker filled with water. The water rises to a height h in the capillary tube above the water surface in the beaker. The surface tension of the water is $\sigma$ . The angle of contact between water and the wall of the capillary tube is θ . Ignore the mass of water in the meniscus. Which of the following statements is (are) true ? A) For a given material of the capillary tube, h decreases with increase in r B) For a given material of the capillary tube , h is independent of $\sigma$ C) If this experiment is performed in a lift going up with a constant acceleration, then h decreases D) h is proportional to contact angle$\theta$
6 In the figure below , the switches S1 and S2 are closed simultaneously at t=0 and a current starts to flow in the circuit. Both the batteries have the same magnitude of the electromotive force (emf) and the polarities are as indicated in the figure. Ignore mutual inductance between the inductors. The current I in the middle wires reaches its maximum magnitude Imax at time $t=\tau$ . which of the following statements is (are) true ? A) $I_{max}=\frac{V}{2R}$ B) $I_{max}=\frac{V}{4R}$ C) $\tau =\frac{L}{R} In 2$ D) $\tau =\frac{2L}{R} In 2$
7 One mole of a monoatomic ideal gas undergoes a cyclic process as shown in the figure (where ,V is the volume and T is the temperature ). Which of the statements below is (are) true? A) Process I is an isochoric process B) In process II, gas absorbs heat C) In process IV, gas releases heat D) Processes I and III are not isobaric
8 Two infinitely long straight wires lie in the xy-plane along the lines x=± R. The wire located at x=± R carries a constant current I1 and the wire located at x= -R carries a constant cureent I2. A circular loop of radius R is suspended with its centre at (0,0,√3R) and in a plane parallel to the xy-plane . This loop carries a constant current I in the clockwise direction as seen from above the loop.The current in the wire taken to be postive , if it is in the +$\hat{j}$- direction . Which of the following statements regarding the magnetic field B is (are) true ? A) If $l_{1}=l_{2}$, then B cannot be equal to zero at the origin (0,0,0) B) If $l_{1}\gt0$ and $l_{2}\lt0$ , then B can be equal to zero at the origin (0,0,0) C) If $l_{1}\lt0$ and $l_{2}\lt0$, then B can be equal to zero at the origin (0,0,0) D) If $l_{1}=l_{2}$ , then the z-component of the magnetic field at the centre of the loop is $(-\frac{\mu_{0}l}{2R})$
9 Two men are walking along a horizontal straight line in the same direction.The man in front walks at speed 1.0 ms-1 and the man behind walks at a speed 2.0 ms-1 . A third man is standing at a height 12 m above the same horizontal line such that all three men are in a vertical plane. The two walking men are blowing identical whistles which emit a sound of frequency 1430Hz. The speed of sound in air 330 ms-1. At the instant , when the moving men are 10 m apart, the stationary man is equidistant from them. The frequency of beats in Hz, heard by the stationary man at this instant, is.... A) 3 Hz B) 5 Hz C) 7 Hz D) 6 Hz
10 A ring and a disc are initially at rest , side by side , at the top of an inclined plane which makes an angle 60° with the horizontal . They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground is $\frac{(2-\sqrt{3})}{\sqrt{10}}$ s, then the height of the top of the inclined plane, in meters is ... [Take, g=10 m s-2 ] A) 0.50 m B) 0.95m C) 0.75m D) 1.00m
11 Three identical capacitors C1 ,C2 and C3 have a capacitance of 1.0 μF each and they are uncharged initially. They are connected in a circuit as shown in the figure and C1 is then filled completely with a dielectric material of relative permittivity εr. The cell electromotive force (emf) V0=8V. First the switch S1 is closed while the switch S2 is kept open. When the capacitor C3 is fully charged, S1 is opened and S2 is closed simultaneously. When all the capacitors reach equilibrium, the acitors reach equilibrium, the charge on C3 is found to be 5μC . The value of εr= ....... A) 2 B) 1 C) 1.25 D) 1.50
12 In the XY-plane , the region y >0 has a uniform magnetic field $B_{1}\hat{K}$ and the region y<0 has another uniform magnetic field $B_{2}\hat{K}$ . A positively charged particle is projected from origin along the positive Y-axis with speed $v_{0}=\pi$ms-1 at t=0, as shown in figure. Neglect gravity in this problem. Let t=T be the time when the particle crosses the X-axis from below for the first time . If B2 =4B1, the average speed of the particle , in ms-1 , along the X-axis in the time interval T is........ A) 2 m/s B) 5 m/s C) 6 m/s D) 1 m/s
13 Sunlight of intensity 1.3kWm-2 is incident normally on a thin convex lens of focal length 20 cm. Ignore the energy loss of light due to the lens and assume that the lens aperture size is much smaller than its focal length. The average intensity of light , in kW m-2, at a distance 22 cm from the lens on the other side is ..... A) $130kWm^{-2}$ B) $140kWm^{-2}$ C) $120kWm^{-2}$ D) $150kWm^{-2}$
14 A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass m= 0.4 kg is at rest on this surface . An impulse of 1.0 N s is applied to the block at time t=0, so that it starts moving along the x- axis with a velocity $v_{t}=v_{0}e^{\frac{-t}{\tau}}$ where v0 is a constant and $\tau=4$ s. The displacement of the block, in metres at $t=\tau$ is............... ( Take , e-1 =0.37) A) 6.30 B) 6.75 C) 5.60 D) 5.36
15 One mole of monoatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is 100K and the universal gas constant R= 8.0 j mol-1 K-1 , the decrease in its internal energy in joule, is .............. A) 450 J B) 900 J C) 850 J D) 750 J
