Q. Two light identical spring of spring constant k are attached horizontally at the two ends of a uniform horizontally rod AB of length $\ell$ and mass m. The rod is pivoted at its centre 'O' and can rotate freely in horizontal plane. The other ends of the two spring are fixed to rigid supports as shown in figure. The rod is gently pushed through a small angle and released. The frequency of resulting oscillation is:

1. $\frac{1}{2\pi }\sqrt{\frac{2k}{m}}$
2. $\frac{1}{2\pi }\sqrt{\frac{6k}{m}}$
3. $\frac{1}{2\pi }\sqrt{\frac{k}{m}}$
4. $\frac{1}{2\pi }\sqrt{\frac{3k}{m}}$

Q. Let the moment of inertia of a hollow cylinder if length$0cm$ (inner radius $10cm$ and outer radius$20cm$), about its axis, be I. The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also I, is____?

1. 12cm
2. 18cm
3. 16cm
4. 14cm

Q. An ideal battery of $4V$ and resistance $R$ are connected in series in the primary circuit of a potentiometer of length $1m$ and resistance $5\Omega$. The value of $R$, to given a potential difference of $5mV$across $10cm$ of potentiometer wire is:

1. $490\Omega$
2. $480\Omega$
3. $395\Omega$
4. $495\Omega$

Q. The least count of the main scale of a screw gauge is $1mm$.The minimum number of division on its circular scale required to measure $5\mu m$ diameter of wire is:

1. $50$
2. $100$
3. $200$
4. $500$

Q. In a meter bridge, the wire of length $1$ m has a non-uniform cross-section such that, the variation $\frac{dR}{dl}$ of its resistance R with length $l$ is $\frac{dR}{dl}\propto \frac{1}{\sqrt{l}}$ . Two equal resistances are connected as shown in the figure. The galvanometer has zero deflection when the jockey is at point P. What is the length AP:

1. $0.2m$
2. $0.3m$
3. $0.35m$
4. $0.25m$

Q. Two electric bulbs, rated at (25 W, 220 V) and (100 W, 220 V), are connected in series across a 220 V voltage source. If the 25W and 100 W bulbs draw powers $P_1$ and $P_2$ respectively then__________?

1. $P1=9W,P2=16W$
2. $P1=16W,P2=4W$
3. $P1=16W,P2=9W$
4. $P1=4W,P2=16W$

Q. As shown in the figure, two infinitely long, identical wires are bent by$90$ and placed in such a way that the segments $LP$ and $QM$ are along the x-axis, while segments $PS$ and QN are parallel to the $y-axis$. If $OP=OQ=4cm$, and the magnitude of the magnetic field at $O$ is ${10}^{4}T$, and the two wires carry equal currents (see figure), the magnitude of the current in each wire and the direction of the magnetic field at $O$ will be ($\mu 0=4\pi {10}^{7}NA2$)

1. $40A$, perpendicular into the page
2. $40A$, perpendicular out of the page
3. $20A$, perpendicular out of the page
4. $20A$, perpendicular into the page

Q. A light wave is incident normally on a glass slab of refractive index $1.5$. If $4\%$ of light gets reflected and the amplitude of the electric field of the incident light is $30V/m$, then the amplitude of the electric field for the wave propogating in the glass medium will be:

1. $10V/m$
2. $24V/m$
3. $6V/m$
4. $30V/m$

Q. There is a uniform spherically symmetric surface charge density at a distance $R_0$ from the origin. The charge distribution is initially at rest and starts expanding because of mutual repulsion. The figure that represents best the speed $V\left(R\right(t\left)\right)$ of the distribution as a function of its instantaneous radius $R\left(t\right)$ is

1. 
   
2. 
   
3. 
   
4. 
   

Q. A straight rod of length L extends from x=a to x=L +a. The gravitational force is exerted on a point mass 'm' at x=0, if the mass per unit length of the rod is $A+B{x}^2$ is given by.

1. $Gm\left[A\right(\frac{1}{a+L}-\frac{1}{a})-BL$]
2. $Gm\left[A\right(\frac{1}{a}-\frac{1}{a+L})+BL$]
3. $Gm\left[A\right(\frac{1}{a}-\frac{1}{a+L})-BL$]
4. $Gm\left[A\right(\frac{1}{a+L}-\frac{1}{a})+BL$]

Q. The position vector of the centre of mass $\overrightarrow{r}cm$ of an symmetric uniform bar of negligible area of cross section as shown in figure is:

1. $\overrightarrow{r}cm=\frac{13}{8}L\hat{x}+\frac{5}{8}L\hat{y}$
2. $\overrightarrow{r}cm=\frac{11}{8}L\hat{x}+\frac{3}{8}L\hat{y}$
3. $\overrightarrow{r}cm=\frac{13}{8}L\hat{x}+\frac{11}{8}L\hat{y}$
4. $\overrightarrow{r}cm=\frac{5}{8}L\hat{x}+\frac{13}{8}L\hat{y}$

Q. A particle A of mass 'm' and charge 'q' is accelerated by a potential difference of $50V$. Another particle B of mass '4 m' and charge 'q' is accelerated by a potential difference of $2500V$ The ratio of de-Broglie wavelengths $\frac{{\lambda }_A}{{\lambda }_B}$ is close to :

1. $10.00$
2. $14.14$
3. $4.47$
4. $0.07$

Q. The output of the given logic circuit is:

1. $\overline{A}B$
2. $A\overline{B}$
3. $AB+\overline{AB}$
4. $A\overline{B}+\overline{A}B$

Q. A satellite of mass M is in a circular orbit of radius R about the centre of the earth. A meteorite of the same mass, falling towards the earth, collides with the satellite completely inelastically. The speed of the satellite and the meteorite are the same, just before the collision. The subsequent motion of the combined body will be:

1. in a circular orbit of a different radius
2. in the same circular orbit of radius R
3. in an elliptical orbit
4. such that it escapes to infinity

Q. In the figure shown, after the switch 'S' is turned from position 'A' to position 'B', the energy dissipated in the circuit in terms of capacitance 'C' and total charge 'Q' is :

1. $\frac{3}{8}\frac{Q^2}{C}$
2. $\frac{3}{4}\frac{Q^2}{C}$
3. $\frac{1}{8}\frac{Q^2}{C}$
4. $\frac{5}{8}\frac{Q^2}{C}$

Q. A cylinder of radius R is surrounded by a cylindrical shell of inner radius R and outer radius 2R. The thermal conductivity of the material of the inner cylinder is $K_1$ and that of the outer cylinder is $K_2$. Assuming no loss of heat, the effective thermal conductivity if the system for heat flowing along the length of the cylinder is:

1. $\frac{2K_1+3K_2}{5}$
2. $\frac{K_1+K_2}{2}$
3. $\frac{K_1+3K_2}{4}$
4. $K_1+K_2$

Q. What is the position and nature of image formed by lens combination shown in figure?
$(f_1,f_2$ are focal length $)$

1. $40$cm from point $B$ at right ;real
2. $\frac{20}{3}$ cm from point $B$ at right , real
3. $70$cm from point $B$ at right , real
4. $70$ cm from point $B$ at left; virtual

Q. An ideal gas occupies a volume of $2m^3$ at a pressure of $3\times {10}^{6}Pa$. The energy of the gas is:

1. ${10}^{8}J$
2. $9\times {10}^{6}J$
3. $3\times {10}^{2}J$
4. $6\times {10}^{4}J$

Q. A person standing on an open ground hears the sound jet aeroplane, coming from north at an angle ${60}^o$ with ground level.But he finds the aeroplane right vertically above his position. If $\upsilon$ is the speed of sound, speed of the plane is:

1. $\frac{2\upsilon }{\sqrt{3}}$
2. $\upsilon$
3. $\frac{\upsilon }{2}$
4. $\frac{\sqrt{3}}{2}\upsilon$

Q. Determine the electric dipole moment of the system of three charges, placed on the vertices of an equilateral triangle , as shown in the figure:

1. $\left(ql\right)\frac{\hat{i}+\hat{j}}{\sqrt{2}}$
2. $\sqrt{3}ql\frac{\hat{j}-\hat{i}}{\sqrt{2}}$
3. $-\sqrt{3}ql\hat{j}$
4. $2ql\hat{j}$

Q. A passenger train of length 60 m travels at a speed of $80km/hr$. Another freight train of length $120m$ travels at a speed of $30km/hr$. The ratio of times taken by the passenger train to completely cross the freight train when: (1)they are moving in the same direction, and (ii) in the opposite directions is____?

1. $\frac{5}{2}$
2. $\frac{25}{11}$
3. $\frac{3}{2}$
4. $\frac{11}{5}$

Q. A particle of mass $m$ moves in a circular orbit in a central potential field $U\left(r\right)=\frac{1}{2}k{r}^2$. If Bohr's quantization condition is applied, radii of possible orbitals and energy levels vary with quantum number $n$ as:

1. $r_n\propto n^2,E_n\propto \frac{1}{n^2}$
2. $r_n\propto n,E_n\propto n$
3. $r_n\propto \sqrt{n},E_n\propto \frac{1}{n}$
4. $r_n\propto \sqrt{n},E_n\propto n$

Q. For the given cyclic process CAB as shown for a gas, the work done is :

1. $1J$
2. $30J$
3. $5J$
4. $10J$

Q. A point source of light, S is placed at a distance $L$ in front of the centre of plane mirror of width d which is hanging vertically on a wall. A man walks in front of the mirror along a line parallel to the mirror, at a distance $2L$ as shown below. The distance over which the man can see the image of the light source in the mirror is :

1. $3d$
2. $\frac{d}{2}$
3. $d$
4. $2d$

Q. A simple pendulum, made of a string of length $l$ and a bob of mass $m$, is released from a small angle ${\theta }_0$. It strikes a block of mass $M$, kept on a horizontal surface at its lowest point of oscillations, elastically. It bounces back and goes up to an angle ${\theta }_1$. Then $M$ is given by :

1. $\frac{m}{2}\left(\frac{{\theta }_o-{\theta }_1}{{\theta }_o+{\theta }_1}\right)$
2. $m\left(\frac{{\theta }_o+{\theta }_1}{{\theta }_o-{\theta }_1}\right)$
3. $\frac{m}{2}\left(\frac{{\theta }_o+{\theta }_1}{{\theta }_o-{\theta }_1}\right)$
4. $m\left(\frac{{\theta }_o-{\theta }_1}{{\theta }_o+{\theta }_1}\right)$

Q. A $100V$ carrier wave is made to vary berween $160V$ and $40V$ by a modulating signal. What is the modulation index?

1. 0.6
2. 0.5
3. 0.3
4. 0.4

Q. A travelling harmonic wave is represented by the equation $y(x,t)={10}^{-3}$ $\sin (50t+2x),$, where x and y are in meter and t is in seconds. Which of the following is a correct statement about the wave? The wave is propagating along the__?

1. negative x-axis with speed $25m{s}^{-1}$
2. The wave is propagating along the positive x-axis with speed $25m{s}^{-1}$
3. The wave is propagating along the positive x-axis with speed $100m{s}^{-1}$
4. The wave is propagating along the negative x-axis with speed $25m{s}^{-1}$

Q. In the figure shown, a circuit contains two identical resistors with resistance $R=5\Omega$ and an inductance with $L=2mH$. An ideal battery of $15V$ is connected in the circuit. What will be the current through the battery long after the switch is closed?

1. $3A$
2. $6A$
3. $5.5A$
4. $7.5A$

Q. A proton and an $\alpha -$ particle (with their masses in the ratio of $1:4$ and charges in the ratio of $1:2$) are accelerated from rest through a potential difference V. If a uniform magnetic field (B) is set up perpendicular to their velocities, the ratio of the radii $r_p:r_{\alpha }$ of the circular path described by them will be :

1. $1:\sqrt{2}$
2. $1:2$
3. $1:3$
4. $1:\sqrt{3}$

Q. The galvanometer deflection, when key $K_1$ is closed but $K_2$ is open, equals ${\theta }_0$ (see figure). On closing $K_2$ also and adjusting $R_{2}to5\Omega$ , the deflection in galvanometer becomes $\frac{{\theta }_0}{5}$ .The resistance of the galvanometer is, then, given by [Neglect the internal resistance of battery]:

1. $12\Omega$
2. $5\Omega$
3. $22\Omega$
4. $25\Omega$