A wave travelling along the $x$-axis is described by the equation $y ( x , t ) = 0.005 \cos ( \alpha x - \beta t )$. If the wavelength and the time period of the wave are 0.08 m and 2.0 s, respectively, then $\alpha$ and $\beta$ in appropriate units are
(1) $\alpha = 25.00 \pi , \beta = \pi$
(2) $\alpha = \frac { 0.08 } { \pi } , \beta = \frac { 2.0 } { \pi }$
(3) $\alpha = \frac { 0.04 } { \pi } , \beta = \frac { 1.0 } { \pi }$
(4) $\alpha = 12.50 \pi , \beta = \frac { \pi } { 2.0 }$

Paragraph: Consider a block of conducting material of resistivity ' $\rho$ ' shown in the figure. Current ' $I$ ' enters at 'A' and leaves from 'D'. We apply superposition principle to find voltage ' $\Delta V$ ' developed between 'B' and 'C'. The calculation is done in the following steps: (i) Take current ' $I$ ' entering from 'A' and assume it to spread over a hemispherical surface in the block. (ii) Calculate field $E(r)$ at distance ' $r$ ' from $A$ by using Ohm's law $E = \rho j$, where $j$ is the current per unit area at ' $r$ '. (iii) From the ' $r$ ' dependence of $E(r)$, obtain the potential $V(r)$ at $r$. (iv) Repeat (i), (ii) and (iii) for current ' $I$ ' leaving 'D' and superpose results for 'A' and 'D'.
Question: $\Delta V$ measured between B and C is
(1) $\frac { \rho I } { \pi a } - \frac { \rho I } { \pi ( a + b ) }$
(2) $\frac { \rho I } { a } - \frac { \rho I } { ( a + b ) }$
(3) $\frac { \rho I } { 2 \pi a } - \frac { \rho I } { 2 \pi ( a + b ) }$
(4) $\frac { \rho I } { 2 \pi ( a - b ) }$

Paragraph: Consider a block of conducting material of resistivity ' $\rho$ ' shown in the figure. Current ' $I$ ' enters at 'A' and leaves from 'D'. We apply superposition principle to find voltage ' $\Delta V$ ' developed between 'B' and 'C'. The calculation is done in the following steps: (i) Take current ' $I$ ' entering from 'A' and assume it to spread over a hemispherical surface in the block. (ii) Calculate field $E(r)$ at distance ' $r$ ' from $A$ by using Ohm's law $E = \rho j$, where $j$ is the current per unit area at ' $r$ '. (iii) From the ' $r$ ' dependence of $E(r)$, obtain the potential $V(r)$ at $r$. (iv) Repeat (i), (ii) and (iii) for current ' $I$ ' leaving 'D' and superpose results for 'A' and 'D'.
Question: For current entering at $A$, the electric field at a distance ' $r$ ' from $A$ is
(1) $\frac { \rho I } { 8 \pi r ^ { 2 } }$
(2) $\frac { \rho I } { r ^ { 2 } }$
(3) $\frac { \rho I } { 2 \pi r ^ { 2 } }$
(4) $\frac { \rho I } { 4 \pi r ^ { 2 } }$

Suppose an electron is attracted towards the origin by a force $k/r$ where ' $k$ ' is a constant and ' $r$ ' is the distance of the electron from the origin. By applying Bohr model to this system, the radius of the $n^{\text{th}}$ orbital of the electron is found to be ' $r_n$ ' and the kinetic energy of the electron to be $T_n$. Then which of the following is true?
(1) $T_n \propto 1/n^{2}, \quad r_n \propto n^{2}$
(2) $T_n$ independent of $n, \quad r_n \propto n$
(3) $T_n \propto 1/n, \quad r_n \propto n$
(4) $T_n \propto 1/n, \quad r_n \propto n^{2}$

Two full turns of the circular scale of a screw gauge cover a distance of 1 mm on its main scale. The total number of divisions on the circular scale is 50. Further, it is found that the screw gauge has a zero error of $-0.03$ mm. While measuring the diameter of a thin wire, a student notes the main scale reading of 3 mm and the number of circular scale divisions in line with the main scale as 35. The diameter of the wire is
(1) 3.32 mm
(2) 3.73 mm
(3) 3.67 mm
(4) 3.38 mm

A thin uniform rod of length $\ell$ and mass $m$ is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is $\omega$. Its centre of mass rises to a maximum height of
(1) $\frac { 1 } { 3 } \frac { \ell ^ { 2 } \omega ^ { 2 } } { g }$
(2) $\frac { 1 } { 6 } \frac { \ell \omega } { g }$
(3) $\frac { 1 } { 2 } \frac { \ell ^ { 2 } \omega ^ { 2 } } { g }$
(4) $\frac { 1 } { 6 } \frac { \ell ^ { 2 } \omega ^ { 2 } } { g }$

The height at which the acceleration due to gravity becomes $\frac { \mathrm { g } } { 9 }$ (where $\mathrm { g } =$ the acceleration due to gravity on the surface of the earth) in terms of $R$, the radius of the earth is
(1) $2 R$
(2) $\frac { R } { \sqrt { 2 } }$
(3) $\frac { R } { 2 }$
(4) $\sqrt { 2 } \mathrm { R }$

Two wires are made of the same material and have the same volume. However wire 1 has cross-sectional area $A$ and wire-2 has cross-sectional area $3A$. If the length of wire 1 increases by $\Delta x$ on applying force $F$, how much force is needed to stretch wire 2 by the same amount?
(1) F
(2) 4 F
(3) 6 F
(4) 9 F

If $x , v$ and a denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period T, then, which of the following does not change with time?
(1) $a ^ { 2 } T ^ { 2 } + 4 \pi ^ { 2 } v ^ { 2 }$
(2) $\frac { a T } { x }$
(3) $a T + 2 \pi v$
(4) $\frac { a T } { v }$

A motor cycle starts from rest and accelerates along a straight path at $2 \mathrm {~m} / \mathrm { s } ^ { 2 }$. At the starting point of the motor cycle there is a stationary electric siren. How far has the motor cycle gone when the driver hears the frequency of the siren at $94\%$ of its value when the motor cycle was at rest? (speed of sound $= 330 \mathrm {~ms} ^ { - 1 }$)
(1) 49 m
(2) 98 m
(3) 147 m
(4) 196 m

Three sound waves of equal amplitudes have frequencies $( v - 1 ) , v , ( v + 1 )$. They superpose to give beats. The number of beats produced per second will be
(1) 4
(2) 3
(3) 2
(4) 1

Let $P ( r ) = \frac { Q } { \pi R ^ { 4 } } r$ be the charge density distribution for a solid sphere of radius $R$ and total charge $Q$. For a point '$p$' inside the sphere at distance $r _ { 1 }$ from the centre of the sphere, the magnitude of electric field is
(1) 0
(2) $\frac { Q } { 4 \pi \varepsilon _ { 0 } r _ { 1 } ^ { 2 } }$
(3) $\frac { Q r _ { 1 } ^ { 2 } } { 4 \pi \varepsilon _ { 0 } R ^ { 4 } }$
(4) $\frac { Q r _ { 1 } ^ { 2 } } { 3 \pi \varepsilon _ { 0 } R ^ { 4 } }$

Two points $P$ and $Q$ are maintained at the potentials of 10 V and $-4$ V respectively. The work done in moving 100 electrons from $P$ to $Q$ is
(1) $- 19 \times 10 ^ { - 17 } \mathrm {~J}$
(2) $9.60 \times 10 ^ { - 17 } \mathrm {~J}$
(3) $- 2.24 \times 10 ^ { - 16 } \mathrm {~J}$
(4) $2.24 \times 10 ^ { - 16 } \mathrm {~J}$

A charge $Q$ is placed at each of the opposite corners of a square. A charge $q$ is placed at each of the other two corners. If the net electrical force on $Q$ is zero, then the $Q/q$ equals
(1) $- 2 \sqrt { 2 }$
(2) $- 1$
(3) $1$
(4) $- \frac { 1 } { \sqrt { 2 } }$

In an optics experiment, with the position of the object fixed, a student varies the position of a convex lens and for each position, the screen is adjusted to get a clear image of the object. A graph between the object distance $u$ and the image distance $v$, from the lens, is plotted using the same scale for the two axes. A straight line passing through the origin and making an angle of $45 ^ { \circ }$ with the x-axis meets the experimental curve at $P$. The coordinates of $P$ will be
(1) $( 2f , 2f )$
(2) $\left( \frac { f } { 2 } , \frac { f } { 2 } \right)$
(3) $( f , f )$
(4) $( 4f , 4f )$

A transparent solid cylindrical rod has a refractive index of $\frac { 2 } { \sqrt { 3 } }$. It is surrounded by air. A light ray is incident at the mid point of one end of the rod as shown in the figure. The incident angle $\theta$ for which the light ray grazes along the wall of the rod is
(1) $\sin ^ { - 1 } \left( \frac { 1 } { 2 } \right)$
(2) $\sin ^ { - 1 } \left( \frac { \sqrt { 3 } } { 2 } \right)$
(3) $\sin ^ { - 1 } \left( \frac { 2 } { \sqrt { 3 } } \right)$
(4) $\sin ^ { - 1 } \left( \frac { 1 } { \sqrt { 3 } } \right)$

A mixture of light, consisting of wavelength 590 nm and an unknown wavelength, illuminates Young's double slit and gives rise to two overlapping interference patterns on the screen. The central maximum of both lights coincide. Further, it is observed that the third bright fringe of known light coincides with the $4 ^ { \text {th} }$ bright fringe of the unknown light. From this data, the wavelength of the unknown light is
(1) 393.4 nm
(2) 885.0 nm
(3) 442.5 nm
(4) 776.8 nm

The surface of a metal is illuminated with the light of 400 nm. The kinetic energy of the ejected photoelectrons was found to be 1.68 eV. The work function of the metal is $( \mathrm { hc } = 1240 \mathrm { eVnm } )$
(1) 3.09 eV
(2) 1.41 eV
(3) 1.51 eV
(4) 1.68 eV

The transition from the state $n = 4$ to $n = 3$ in a hydrogen like atom results in ultraviolet radiation. Infrared radiation will be obtained in the transition from
(1) $2 \rightarrow 1$
(2) $3 \rightarrow 2$
(3) $4 \rightarrow 2$
(4) $5 \rightarrow 4$

In an experiment the angles are required to be measured using an instrument. 29 divisions of the main scale exactly coincide with the 30 divisions of the vernier scale. If the smallest division of the main scale is half-a-degree $\left( = 0.5 ^ { \circ } \right)$, then the least count of the instrument is
(1) one minute
(2) half minute
(3) one degree
(4) half degree

In an atom, an electron is moving with a speed of $600 \mathrm {~m} / \mathrm { s }$ with an accuracy of $0.005\%$. Certainty with which the position of the electron can be located is $\left( h = 6.6 \times 10 ^ { - 34 } \mathrm {~kg} \mathrm {~m} ^ { 2 } \mathrm {~s} ^ { - 1 } \right.$, mass of electron, $e _ { m } = 9.1 \times 10 ^ { - 31 } \mathrm {~kg} )$
(1) $1.52 \times 10 ^ { - 4 } \mathrm {~m}$
(2) $5.10 \times 10 ^ { - 3 } \mathrm {~m}$
(3) $1.92 \times 10 ^ { - 3 } \mathrm {~m}$
(4) $3.84 \times 10 ^ { - 3 } \mathrm {~m}$

Calculate the wavelength (in nanometer) associated with a proton moving at $1.0 \times 10 ^ { 3 } \mathrm {~ms} ^ { - 1 }$ (Mass of proton $= 1.67 \times 10 ^ { - 27 } \mathrm {~kg}$ and $\mathrm { h } = 6.63 \times 10 ^ { - 34 } \mathrm { Js }$):
(1) 0.032 nm
(2) 0.40 nm
(3) 2.5 nm
(4) 14.0 nm

Copper crystallizes in fcc with a unit cell length of 361 pm. What is the radius of copper atom?
(1) 108 pm
(2) 127 pm
(3) 157 pm
(4) 181 pm

Two liquids $X$ and $Y$ form an ideal solution. At 300 K, vapour pressure of the solution containing 1 mol of $X$ and 3 mol of $Y$ is 550 mmHg. At the same temperature, if 1 mol of $Y$ is further added to this solution, vapour pressure of the solution increases by 10 mmHg. Vapour pressure (in mmHg) of $X$ and $Y$ in their pure states will be, respectively:
(1) 200 and 300
(2) 300 and 400
(3) 400 and 600
(4) 500 and 600

In a fuel cell methanol is used as fuel and oxygen gas is used as an oxidizer. The reaction is $\mathrm { CH } _ { 3 } \mathrm { OH } ( \ell ) + \frac { 3 } { 2 } \mathrm { O } _ { 2 } ( \mathrm {~g} ) \rightarrow \mathrm { CO } _ { 2 } ( \mathrm {~g} ) + 2 \mathrm { H } _ { 2 } \mathrm { O } ( \ell )$. At 298 K standard Gibb's energies of formation for $\mathrm { CH } _ { 3 } \mathrm { OH } ( \ell ) , \mathrm { H } _ { 2 } \mathrm { O } ( \ell )$ and $\mathrm { CO } _ { 2 } ( \mathrm {~g} )$ are $-166.2, -237.2$ and $-394.4 \mathrm {~kJ} \mathrm {~mol} ^ { - 1 }$ respectively. If standard enthalpy of combustion of methanol is $-726 \mathrm {~kJ} \mathrm {~mol} ^ { - 1 }$, efficiency of the fuel cell will be
(1) $80\%$
(2) $87\%$
(3) $90\%$
(4) $97\%$