The velocity (v) of a particle (under a force F) depends on its distance (x) from the origin (with x > 0)  \(v\propto \frac{1}{\sqrt{x}}\). Find how the magnitude of the force (F) on the particle depends on x.

\(\large F\propto \frac{1}{x^{\frac{3}{2}}}\)

\(\large F\propto \frac{1}{x}\)

\(\large F\propto \frac{1}{x^{2}}\)

\(\large F\propto x\)

The ratio of accelerations due to gravity \(g_{1}:g_{2}\) on the surfaces of two planets is 5 : 2 and the ratio of  their respective average densities \(\rho _{1}:\rho _{2}\) is 2 : 1. What is the ratio of respective escape velocities \(v_{1}:v_{2}\) from the surface of the plants ?

5: 2

\(\sqrt {5} :\sqrt {2}\)

5 : \(2\sqrt {2}\)

25: 4

A spherical liquid drop is placed on a horizontal plane. A small disturbance causes the volume of the drop to oscillate. The time period of oscillation (T) of the liquid drop depends on radius (r) of the drop,  density (\(\rho\)) and surface tension (s) of the liquid. Which among the following will be a possible expression for T (where k is a dimensionless constant) ?

\(\large k\sqrt {\frac {\rho r}{s}}\)

\(\large k\sqrt {\frac {\rho^{2} r}{s}}\)

\(\large k\sqrt {\frac {\rho r^{3}}{s}}\)

\(\large k\sqrt {\frac {\rho r^{3}}{s^{2}}}\)

The stress along the length of a rod (with rectangular cross section) is 1% of the Young's modulus of its  material. What is the approximate percentage of change of its volume ? (Poisson's ratio of the material of the rod is 0.3) .
 

3â„…

1â„…

0.7â„…

0.4â„…

What will be the approximate terminal velocity of a rain drop of diameter 1.8 × \(10^{-3}\)m, when density of rain water \(\approx \)\(10^{3}\)kg\(m^{-3}\) and the co-efficient of viscosity of air \(\approx\) 1.8 × \(10^{-5}\) Ns\(m^{-2}\) ? (Neglect buoyancy of air).

\(49 ms^{-1}\)

\(98ms^{-1}\)

\(392ms^{-1}\)

\(980ms^{-1}\)

 The water equivalent of a calorimeter is 10 g and it  contains 50 g of water at 15°C. Some amount of ice,  initially at –10°C is dropped in it and half of the ice melts till equilibrium is reached. What was the initial amount of ice that was dropped (when specific heat of ice = 0.5 cal \(gm^{-1}\)°\(C^{-1}\), specific heat of water = 1.0 cal \(gm^{-1}\)°\(C^{-1}\) and latent heat of melting of ice = 80 cal \(gm^{-1}\)) ?

10g

18g

20g

30g

One mole of a mono-atomic ideal gas undergoes a quasi-static process, which is depicted by a straight line joining points (\(V_{o},T_{o}\)) and (\(2V_{o},3T_{o}\)) in a V-T diagram. What is the value of the heat capacity of the gas at the point (\(V_{o},T_{o}\)) ?
 

R

\(\large \frac {3}{2}R\)

2R

0

For an ideal gas with initial pressure and volume \(P_{i}\) and \(V_{i}\) , respectively, a reversible isothermal expansion happens, when its volume becomes \(V_{o}\). Then it is compressed to its original volume \(V_{i}\) by a reversible adiabatic process. If the final pressure is \(P_{f}\) then which of the following statements is true ?

\(P_{f}=P_{i}\)

\(P_{f}>P_{i}\)

\(P_{f}<P_{i}\)

\(\large \frac {P_{f}}{V_{o}}= \frac {P_{i}}{V_{i}}\)

A point charge – q is carried from a point A to another point B on the axis of a charged ring of radius ‘r’ carrying a charge +q. If the point A is at a distance \(\frac {4}{3}\)r from the centre of the ring and the point B is \(\frac {3}{4}\)r from the centre but on the opposite side, what is the net work that need to be done for this ?
 

\(\large -\frac{7}{5}\frac{q^{2}}{4\pi \epsilon_{o}r}\)

\(\large -\frac{1}{5}\frac{q^{2}}{4\pi \epsilon_{o}r}\)

\(\large \frac{7}{5}\frac{q^{2}}{4\pi \epsilon_{o}r}\)

\(\large \frac{1}{5}\frac{q^{2}}{4\pi \epsilon_{o}r}\)

Consider a region in free space bounded by the surfaces of an imaginary cube having sides of length ‘a’ as shown in the diagram. A charge +Q is placed at the centre ‘O’ of the cube. P is such a point outside the cube that the line OP perpendicularly intersects the surface ABCD at R and also OR = RP = a/2. A charge +Q is placed at point P also. What is the total electric flux through the five faces of the cube other than ABCD ?

\(\large \frac {Q}{\epsilon _{o}}\)

\(\large \frac {5Q}{6\epsilon _{o}}\)

\(\large \frac {10Q}{6\epsilon _{o}}\)

Zero

Four equal charges of value +Q are placed at any four vertices of a regular hexagon of side ‘a’. By suitable choosing the vertices, what can be the maximum possible magnitude of electric field at the centre of the hexagon ?
 

\(\frac {Q} {4\pi \epsilon _{o} a^{2}}\)

\(\large \frac {\sqrt {2}Q} {4\pi \epsilon _{o} a^{2}}\)

\(\large \frac {\sqrt {3}Q} {4\pi \epsilon _{o} a^{2}}\)

\(\large \frac {2Q} {4\pi \epsilon _{o} a^{2}}\)

A proton of mass 'm' moving with a speed v (<< c, velocity of light in vacuum) completes a circular orbit in time 'T' in a uniform magnetic field. If the speed of the proton is increased to \(\sqrt {2}\) v, what will be time needed to complete the circular orbit ?
 

\(\sqrt {2} \) T

T

\(\frac {T}{\sqrt{2}}\)

\(\large \frac {T}{2}\)

A uniform current is flowing along the length of an infinite, straight, thin, hollow cylinder of radius 'R'.The magnetic field 'B' produced at a perpendicular distance 'd' from the axis of the cylinder is plotted in a graph. Which of the following figures looks like the plot ?

A circular loop of radius 'r' of conducting wire connected with a voltage source of zero internal resistance produces a magnetic filed 'B' at its centre. If instead, a circular loop of radius '2r' made of same material, having the same cross section is connected to the same voltage source, what will be the magnetic field at its centre ?
 

\(\large \frac {B}{2}\)

\(\large \frac {B}{4}\)

2B

B

An alternating current is flowing through a series LCR circuit. It is found that the current reaches a value of 1 mA at both 200 Hz and 800 Hz frequency. What is the Resonance frequency of the circuit?

600 Hz

300 Hz

500 Hz

400 Hz

An electric bulb, a capacitor, battery and a switch are all in series in a circuit. How does the intensity of light very when the switch is turned on?

Continues to increase gradually

Gradually increases for some time and then becomes steady.

Sharply rises initially and then gradually decreases.
 

Gradually increases for some then time and then gradually decreases.

Four resistors, 100\(\Omega\), 200\(\Omega\), 300\(\Omega\), and 400\(\Omega\), are connected to form four sides of a square. The resistors can be connected in any order. What is the maximum possible equivalent resistance across the diagonal of the square ?
 

\(210 \Omega\)

\(240 \Omega\)

\(300 \Omega\)

\(250 \Omega\)

What will be current through the 200 \(\Omega\) resistor in the given circuit a long time after the switch 'K' is made on ?

Zero

100 mA

10 mA

1 mA

A point source is placed at co-ordinates (0, 1) in X-Y plane. A ray of light from the source is reflected on a plane along the X-axis and perpendicular to the X-Y plane. The reflected ray passes through the point (3, 3). What is the path length of the ray from (0, 1) to (3, 3) ?
 

5

\(\sqrt {13}\)

\(2\sqrt{13}\)

\(1+2\sqrt{3}\)

Two identical equi-convex lenses, each of focal length 'f' are placed side by side in contact with each other with a layer of water in between them as shown in the figure. If refractive index of the material of the lenses is greater then that of water, how the combined focal length 'F' is related to 'f' ?

F>f

\(\frac {f}{2}<F<f\)

\(F<\frac{f}{2}\)

F=f

 There is a small air bubble at the centre of a solid glass sphere of radius 'r' and refractive index '\(\mu\)'. What will be the apparent distance of the bubble from the centre of the sphere, when viewed from outside ?

r

\(\large \frac{r}{\mu}\)

\(r\left (1-\frac{1}{\mu} \right )\)

Zero

 If Young's double slit experiment is done with white light,  which of the following statements will be true?
 

All the bright fringes will be coloured.

 All the bright fringes will be white.

The central fringe will be white.

No stable interference pattern will be white.

How the linear velocity 'v' of an electron in the Bohr orbit is related to its quantum number 'n' ?
 

\(v\propto \frac{1}{n}\)

\(v\propto \frac{1}{n^{2}}\)

\(v\propto \frac{1}{\sqrt{n}}\)

\(v\propto n\)

If the half life of a radioactive nucleus is 3 days, nearly what fraction of the initial number of nuclei will decay on the \(3^{rd}\) day ? (Given that \(\sqrt[3]{0.25}\)= 0.63)
 

0.63

0.5

0.37

0.13

An electron accelerated through a potential of 10,000 V from rest has a de-Broglie wave length '\(\lambda\)'.What should be the accelerating potential so that the wave length is doubled ?
 

20,000 V

40,000 V

5,000 V

2,500 V

 ln the circuit shown, inputs A and B are in states '1' and'0' respectively. What is the only possible stable state of the outputs 'X' and 'Y' ?

X='1' ,Y='1'

X='1', Y='0'

X='0', Y='1'

X='0' ,Y='0'

What will be the current flowing through the 6K\(\Omega\) resistor in the circuit shown, where the breakdown voltage of the zener is 6 V ?

\(\frac {2}{3} mA\)

1 mA

10 mA

\(\frac {3}{2} mA\)

In case of a simple harmonic motion, if the velocity is plotted along the X-axis and the displacement (from the equilibrium position) is plotted along the Y-axis, the resultant curve happens to be an ellipse with the ratio :

\(\large \frac {major \,axis \,\left (along X\right )}{minor \, axis \, \left (along Y\right )}=20\pi\)

What is the frequency of the simple harmonic motion ?
 

100 Hz

20 Hz

10 Hz

\(\large \frac{1}{10} Hz\)

A block of mass \(m_{2}\) is placed on a horizontal table and another block of mass \(m_{1}\) is placed on top of it. An increasing horizontal force F = \(\alpha\)t is exerted on the upper block but the lower block never moves as a result. If the co-efficient of friction between the blocks is \(\large \mu_{1}\) and that between the lower block and the table is \(\large \mu_{2}\), then what is the maximum possible value of \(\large \frac {\mu_{1}}{\mu_{2}}\) ?

\(\large \frac {m_{2}}{m_{1}}\)

\(\large 1+\frac {m_{2}}{m_{1}}\)

\(\large \frac {m_{1}}{m_{2}}\)

\(\large 1+\frac {m_{1}}{m_{2}}\)

In a triangle ABC, the sides AB and AC are represented by the vectors \(3\hat{i}+\hat{j}+\hat{k}\, and\,\hat{i}+2\hat{j}+\hat{k} \) respectively. Calculate the angle \(\angle\)ABC.

\(\cos^{-1} \sqrt{\frac{5}{11}}\)

\(\cos^{-1} \sqrt{\frac{6}{11}}\)

\(\left (90-\cos^{-1} \sqrt{\frac{5}{11}}\right )\)

\(\left (180-\cos^{-1} \sqrt{\frac{5}{11}}\right )\)

The insulated plates of a charged parallel plate capacitor (with small separation between the plates) are approaching each other due to electrostatic attraction. Assuming no other force to be operative and no radiation taking place, which of the following graphs approximately shows the variation with time (t) of the potential difference (V) between the plates ?

The bob of a pendulum of mass 'm', suspended by an inextensible string of length 'L' as shown in the figure carries a small charge 'a'. An infinite horizontal plane conductor with uniform surface charge density '\(\sigma\)'. is placed below it. What will be the time period of the pendulum for small amplitude oscillations 

\(\large 2\pi \sqrt{\frac{L}{\left ( g-\frac{mq}{\epsilon_{o}\sigma} \right )}}\)

\( \large \sqrt{\frac{L}{\left ( g-\frac{mq\sigma}{\epsilon_{o}} \right )}}\)

\( \large \frac{1}{2\pi}\sqrt{\frac{L}{\left ( g-\frac{q\sigma}{\epsilon_{o}m} \right )}}\)

\(\Large 2\pi \sqrt{\frac{L}{\left ( g-\frac{q\sigma}{\epsilon_{o}m} \right )}}\)

A light charged particle is revolving in a circle of radius 'r' in electrostatic attraction of a static heavy particle with opposite charge. How does the magnetic field 'B' at the centre of the circle due to the moving charge depend on 'r' ?
 

\(\large B\propto \frac{1}{r}\)

\(\large B\propto \frac{1}{r^{2}}\)

\(\large B\propto \frac{1}{r^{\frac{3}{2}}}\)

\(\large B\propto \frac{1}{r^{\frac {5}{2}}}\)

As shown in the figure, a rectangular loop of a conducting wire is moving away with a constant velocity 'v' in a perpendicular direction from a very long straight conductor carrying a steady current 'l'. When the breadth of the rectangular loop is very small compared to its distance from the straight conductor, how does the e.m.f. 'E' induced in the loop vary with time 't' ?

\(E\propto \frac {1}{t^{2}}\)

\(E\propto \frac {1}{t}\)

\(E\propto ln(t)\)

\(E\propto \frac {1}{t^{3}}\)

A solid spherical ball and a hollow spherical ball of two different materials of densities \(\rho_{1}\, and\,\rho_{2}\) respectively have same outer radii and same mass. What will be the ratio the moment of inertia (about an axis passing through the centre) of the hollow sphere to that of the solid sphere ?
 

\(\frac{\rho_{2}}{\rho_{1}}\left ( 1-\frac{\rho_{2}}{\rho_{1}} \right )^{\frac{5}{3}}\)

\(\frac{\rho_{2}}{\rho_{1}}\left [ 1- \left ( 1-\frac{\rho_{2}}{\rho_{1}} \right )^{\frac{5}{3}}\right ]\)

\(\frac{\rho_{2}}{\rho_{1}} \left ( 1-\frac{\rho_{1}}{\rho_{2}} \right )^{\frac{5}{3}}\)

\(\frac{\rho_{2}}{\rho_{1}}\left [1- \left ( 1-\frac{\rho_{1}}{\rho_{2}} \right )^{\frac{5}{3}}\right ]\)

 Which of the following statement(s) is/are true ?
"Internal energy of an ideal gas ………………"
 

decreases in an isobaric process.

 remains constant in an isothermal process.
 

 increases in an isobaric process.

decreases in an isobaric expansion.
 

Two positive charges Q and 4Q are placed at points A and B respectively, where B is at a distance 'd' units to the right of A. The total electric potential due to these charges is minimum at P on the line through A and B, What is (are) the distance(s) of P from A ?

\(\large \frac {d}{3} \) units to the right of A

\(\large \frac {d}{3} \) units to the left of A

\(\large \frac {d}{5} \) units to the right of A

d units to the left of A

A non–zero current passes through the galvanometer G shown in the circuit when the key 'K' is closed and its value does not change when the key is opened. Then which of the following statement(s) is/are true ?

The galvanometer resistance is infinite.
 

The current through the galvanometer is 40 mA.

After the key is closed, the current through the 200\(\Omega\)resistor is higher as the current through the 300\(\Omega\) resistor

The galvanometer resistance is 100\(\Omega\).

A ray of light is incident on a right angled isosceles prism parallel to its base as shown in the figure. Refractive index of the material of the prism is \(\sqrt{2}\) . Then which of the following statement(s) is/are true?

The reflection at P is total internal.
 

The reflection at Q is total internal.

The ray emerging at R is not parallel to the ray incident at S.
 

Total deviation of the ray is 150°

The intensity of a sound appears to an observer to be periodic. Which of the following can be the cause of it ?

The intensity of the source is periodic.

The source is moving towards the observer.
 

The observer is moving away from the source.