The two thin coaxial rings, each of radius $a$ and having charges $+Q$ and $-Q$ respectively are separated by a distance of $s$. The potential difference between the centres of the two rings is:
(1) $\frac{Q}{2\pi\varepsilon_0}\left[\frac{1}{a} - \frac{1}{\sqrt{s^2 + a^2}}\right]$
(2) $\frac{Q}{4\pi\varepsilon_0}\left[\frac{1}{a} - \frac{1}{\sqrt{s^2 + a^2}}\right]$
(3) $\frac{Q}{4\pi\varepsilon_0}\left[\frac{1}{a} + \frac{1}{\sqrt{s^2 + a^2}}\right]$
(4) $\frac{Q}{2\pi\varepsilon_0}\left[\frac{1}{a} + \frac{1}{\sqrt{s^2 + a^2}}\right]$

A parallel-plate capacitor with plate area $A$ has separation $d$ between the plates. Two dielectric slabs of dielectric constant $K_1$ and $K_2$ of same area $\frac{A}{2}$ and thickness $\frac{d}{2}$ are inserted in the space between the plates. The capacitance of the capacitor will be given by:
(1) $\frac{\varepsilon_0 A}{d}\left(\frac{1}{2} + \frac{K_1 K_2}{K_1 + K_2}\right)$
(2) $\frac{\varepsilon_0 A}{d}\left(\frac{1}{2} + \frac{2(K_1 + K_2)}{K_1 K_2}\right)$
(3) $\frac{\varepsilon_0 A}{d}\left(\frac{1}{2} + \frac{K_1 + K_2}{K_1 K_2}\right)$
(4) $\frac{\varepsilon_0 A}{d}\left(\frac{1}{2} + \frac{K_1 K_2}{2(K_1 + K_2)}\right)$

Five identical cells each of internal resistance $1\Omega$ and emf 5 V are connected in series and in parallel with an external resistance $R$. For what value of $R$, current in series and parallel combination will remain the same?
(1) $1\Omega$
(2) $5\Omega$
(3) $25\Omega$
(4) $10\Omega$

Two resistors $R _ { 1 } = ( 4 \pm 0.8 ) \Omega$ and $R _ { 2 } = ( 4 \pm 0.4 ) \Omega$ are connected in parallel. The equivalent resistance of their parallel combination will be:
(1) $( 4 \pm 0.4 ) \Omega$
(2) $( 2 \pm 0.4 ) \Omega$
(3) $( 4 \pm 0.3 ) \Omega$
(4) $( 2 \pm 0.3 ) \Omega$

Consider the combination of two capacitors $C _ { 1 }$ and $C _ { 2 }$, with $C _ { 2 } > C _ { 1 }$, when connected in parallel, the equivalent capacitance is 10 times the equivalent capacitance of the same connected in series. Calculate the ratio of capacitors, $\frac { C _ { 2 } } { C _ { 1 } }$.
(1) $4 + \sqrt { 15 }$
(2) $2 + \sqrt { 15 }$
(3) 9
(4) $\frac { 15 } { 4 }$

An electric bulb of 500 W at 100 V is used in a circuit having a 200 V supply. Calculate the resistance $R$ to be connected in series with the bulb so that the power delivered by the bulb is 500 W.
(1) $30\,\Omega$
(2) $5\,\Omega$
(3) $20\,\Omega$
(4) $10\,\Omega$

Two ions of masses 4 amu and 16 amu have charges $+2e$ and $+3e$ respectively. These ions pass through the region of the constant perpendicular magnetic field. The kinetic energy of both ions is the same. Then:
(1) lighter ion will be deflected less than heavier ion
(2) lighter ion will be deflected more than heavier ion
(3) both ions will be deflected equally
(4) no ion will be deflected

An $AC$ current is given by $I = I _ { 1 } \sin \omega t + I _ { 2 } \cos \omega t$. A hot wire ammeter will give a reading:
(1) $\sqrt { \frac { I _ { 1 } ^ { 2 } - I _ { 2 } ^ { 2 } } { 2 } }$
(2) $\sqrt { \frac { I _ { 1 } ^ { 2 } + I _ { 2 } ^ { 2 } } { 2 } }$
(3) $\frac { I _ { 1 } + I _ { 2 } } { \sqrt { 2 } }$
(4) $\frac { I _ { 1 } + I _ { 2 } } { 2 \sqrt { 2 } }$

In the Young's double slit experiment, the distance between the slits varies in time as $d(t) = d_{0} + a_{0}\sin\omega t$; where $d_{0}$, $\omega$ and $a_{0}$ are constants. The difference between the largest fringe width and the smallest fringe width obtained over time is given as:
(1) $\frac{2\lambda D d_{0}}{d_{0}^{2} - a_{0}^{2}}$
(2) $\frac{2\lambda D a_{0}}{d_{0}^{2} - a_{0}^{2}}$
(3) $\frac{\lambda D}{d_{0}^{2}} a_{0}$
(4) $\frac{\lambda D}{d_{0} + a_{0}}$

If you are provided a set of resistances, $2\,\Omega$, $4\,\Omega$, $6\,\Omega$ and $8\,\Omega$. Connect these resistances to obtain an equivalent resistance of $\frac{46}{3}\,\Omega$.
(1) $2\,\Omega$ and $6\,\Omega$ are in parallel with $4\,\Omega$ and $8\,\Omega$ in series
(2) $4\,\Omega$ and $6\,\Omega$ are in parallel with $2\,\Omega$ and $8\,\Omega$ in series
(3) $2\,\Omega$ and $4\,\Omega$ are in parallel with $6\,\Omega$ and $8\,\Omega$ in series
(4) $6\,\Omega$ and $8\,\Omega$ are in parallel with $2\,\Omega$ and $4\,\Omega$ in series

An alternating current is given by the equation $i = i _ { 1 } \sin \omega t + i _ { 2 } \cos \omega t$. The rms current will be:
(1) $\frac { 1 } { \sqrt { 2 } } \left( i _ { 1 } + i _ { 2 } \right) ^ { 2 }$
(2) $\frac { 1 } { \sqrt { 2 } } \left( i _ { 1 } + i _ { 2 } \right)$
(3) $\frac { 1 } { 2 } \left( i _ { 1 } ^ { 2 } + i _ { 2 } ^ { 2 } \right) ^ { \frac { 1 } { 2 } }$
(4) $\frac { 1 } { \sqrt { 2 } } \left( i _ { 1 } ^ { 2 } + i _ { 2 } ^ { 2 } \right) ^ { \frac { 1 } { 2 } }$

What should be the order of arrangement of de-Broglie wavelength of electron $\lambda_{\mathrm{e}}$, an $\alpha$-particle $\lambda_{\alpha}$ and proton $\lambda_{\mathrm{p}}$ given that all have the same kinetic energy?
(1) $\lambda_{e} = \lambda_{p} = \lambda_{\alpha}$
(2) $\lambda_{\mathrm{e}} < \lambda_{\mathrm{p}} < \lambda_{\alpha}$
(3) $\lambda_{\mathrm{e}} > \lambda_{\mathrm{p}} > \lambda_{\alpha}$
(4) $\lambda_{\mathrm{e}} = \lambda_{\mathrm{p}} > \lambda_{\alpha}$

An electron of mass $m$ and a photon have same energy $E$. The ratio of wavelength of electron to that of photon is: ($c$ being the velocity of light)
(1) $\frac { 1 } { c } \left( \frac { 2 m } { E } \right) ^ { \frac { 1 } { 2 } }$
(2) $\frac { 1 } { c } \left( \frac { E } { 2 m } \right) ^ { \frac { 1 } { 2 } }$
(3) $\left( \frac { E } { 2 m } \right) ^ { \frac { 1 } { 2 } }$
(4) $c ( 2 m E ) ^ { \frac { 1 } { 2 } }$

A square loop of side 20 cm and resistance $1 \Omega$ is moved towards right with a constant speed $v _ { 0 }$. The right arm of the loop is in a uniform magnetic field of 5 T. The field is perpendicular to the plane of the loop and is going into it. The loop is connected to a network of resistors each of value $4 \Omega$. What should be the value of $v _ { 0 }$ so that a steady current of 2 mA flows in the loop?
(1) $10 ^ { - 2 } \mathrm {~cm} \mathrm {~s} ^ { - 1 }$
(2) $1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $1 \mathrm {~cm} \mathrm {~s} ^ { - 1 }$
(4) $10 ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$

A particle of mass $4M$ at rest disintegrates into two particles of mass $M$ and $3M$, respectively, having non zero velocities. The ratio of de-Broglie wavelength of particle of mass $M$ to that of mass $3M$ will be:
(1) $1 : 3$
(2) $3 : 1$
(3) $1 : \sqrt{3}$
(4) $1 : 1$

An object is placed beyond the centre of curvature $C$ of the given concave mirror. If the distance of the object is $d_{1}$ from $C$ and the distance of the image formed is $d_{2}$ from $C$, the radius of curvature of this mirror is:
(1) $\frac{2d_{1}d_{2}}{d_{1}+d_{2}}$
(2) $\frac{2d_{1}d_{2}}{d_{1}-d_{2}}$
(3) $\frac{d_{1}d_{2}}{d_{1}+d_{2}}$
(4) $\frac{d_{1}d_{2}}{d_{1}-d_{2}}$

Some nuclei of a radioactive material are undergoing radioactive decay. The time gap between the instances when a quarter of the nuclei have decayed and when half of the nuclei have decayed is given as: (where $\lambda$ is the decay constant)
(1) $\frac{1}{2}\frac{\ln 2}{\lambda}$
(2) $\frac{\ln 2}{\lambda}$
(3) $\frac{2\ln 2}{\lambda}$
(4) $\frac{\ln\frac{3}{2}}{\lambda}$

The de-Broglie wavelength of a particle having kinetic energy $E$ is $\lambda$. How much extra energy must be given to this particle so that the de-Broglie wavelength reduces to $75\%$ of the initial value?
(1) $E$
(2) $\frac{7E}{9}$
(3) $\frac{16E}{9}$
(4) $\frac{E}{9}$

Find the distance of the image from object $O$, formed by the combination of lenses in the figure:
(1) 75 cm
(2) 10 cm
(3) infinity
(4) 20 cm

The half-life of ${}^{198}\mathrm{Au}$ is 3 days. If atomic weight of ${}^{198}\mathrm{Au}$ is $198\mathrm{~g~mol}^{-1}$, then the activity of 2 mg of ${}^{198}\mathrm{Au}$ is [in disintegration second${}^{-1}$]:
(1) $2.67 \times 10^{12}$
(2) $6.06 \times 10^{18}$
(3) $32.36 \times 10^{12}$
(4) $16.18 \times 10^{12}$

There are $10^{10}$ radioactive nuclei in a given radioactive element. Its half-life time is 1 min. How many nuclei will remain after $30\mathrm{~s}$? $(\sqrt{2} = 1.414)$
(1) $7 \times 10^{9}$
(2) $10^{5}$
(3) $2 \times 10^{10}$
(4) $4 \times 10^{10}$

The vernier scale used for measurement has a positive zero error of 0.2 mm. If while taking a measurement it was noted that 0 on the vernier scale lies between 8.5 cm and 8.6 cm. Vernier coincidence is 6, then the correct value of measurement is cm. (least count=0.01 cm)
(1) 8.36 cm
(2) 8.54 cm
(3) 8.58 cm
(4) 8.56 cm

A transmitting antenna at top of a tower has a height of 50 m, and the height of receiving antenna is 80 m. What is the range of communication for the line of sight (LOS) mode?
[use radius of the earth $= 6400\text{ km}$]
(1) 80.2 km
(2) 144.1 km
(3) 57.28 km
(4) 45.5 km

In the reported figure of earth, the value of acceleration due to gravity is same at point $A$ and $C$ but it is smaller than that of its value at point $B$ (surface of the earth). The value of $O A : A B$ will be $x : 5$. The value of $x$ is

A body of mass 2 kg moves under a force of $( 2 \hat { \mathrm { i } } + 3 \hat { \mathrm { j } } + 5 \widehat { \mathrm { k } } ) \mathrm { N }$. It starts from rest and was at the origin initially. After 4 s, its new coordinates are $( 8 , b , 20 )$. The value of $b$ is $\_\_\_\_$. (Round off to the Nearest Integer)