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

In a spring gun having spring constant $100 \mathrm {~N} \mathrm {~m} ^ { - 1 }$ a small ball $B$ of mass 100 g is put in its barrel (as shown in figure) by compressing the spring through 0.05 m . There should be a box placed at a distance $d$ on the ground so that the ball falls in it. If the ball leaves the gun horizontally at a height of 2 m above the ground. The value of $d$ is $\underline{\hspace{1cm}}$ m.
$$\left( g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 } \right)$$

The average translational kinetic energy of $N _ { 2 }$ gas molecules at $\_\_\_\_$ ${ } ^ { \circ } \mathrm { C }$ becomes equal to the K.E. of an electron accelerated from rest through a potential difference of 0.1 volt. (Given $k _ { B } = 1.38 \times 10 ^ { - 23 } \mathrm {~J} \mathrm {~K} ^ { - 1 }$) (Fill the nearest integer).

A boy pushes a box of mass 2 kg with a force $\vec { F } = ( 20 \hat { \mathrm { i } } + 10 \hat { \mathrm { j } } ) \mathrm { N }$ on a frictionless surface. If the box was initially at rest, then $\_\_\_\_$ m is displacement along the $x$-axis after 10 s

Suppose two planets (spherical in shape) of radii $R$ and $2R$, but mass $M$ and $9M$ respectively have a centre to centre separation $8R$ as shown in the figure. A satellite of mass $m$ is projected from the surface of the planet of mass $M$ directly towards the centre of the second planet. The minimum speed $v$ required for the satellite to reach the surface of the second planet is $\sqrt { \frac { a } { 7 } \frac { G M } { R } }$, then the value of $a$ is. [Given: The two planets are fixed in their position]

The acceleration due to gravity is found up to an accuracy of $4\%$ on a planet. The energy supplied to a simple pendulum of known mass $m$ to undertake oscillations of time period $T$ is being estimated. If time period is measured to an accuracy of $3\%$, the accuracy to which $E$ is known as $\_\_\_\_$ \%

If the velocity of a body related to displacement $x$ is given by $v = \sqrt{5000 + 24x}\mathrm{~m~s}^{-1}$, then the acceleration of the body is $\_\_\_\_$ $\mathrm{m~s}^{-2}$.

1 mole of rigid diatomic gas performs a work of $\frac { Q } { 5 }$ when heat $Q$ is supplied to it. The molar heat capacity of the gas during this transformation is $\frac { x R } { 8 }$. The value of $x$ is [ $R$ universal gas constant]

The angular speed of truck wheel is increased from 900 rpm to 2460 rpm in 26 seconds. The number of revolutions by the truck engine during this time is $\_\_\_\_$. (Assuming the acceleration to be uniform).