A particle executes simple harmonic motion represented by displacement function as $x ( t ) = A \sin ( \omega t + \phi )$. If the position and velocity of the particle at $t = 0 \mathrm {~s}$ are 2 cm and $2 \omega \mathrm {~cm} \mathrm {~s} ^ { - 1 }$ respectively, then its amplitude is $x \sqrt { 2 } \mathrm {~cm}$ where the value of $x$ is $\_\_\_\_$.

The width of one of the two slits in a Young's double slit experiment is three times the other slit. If the amplitude of the light coming from a slit is proportional to the slit-width, the ratio of minimum to maximum intensity in the interference pattern is $x$ : 4 where $x$ is $\_\_\_\_$.

The mass per unit length of a uniform wire is $0.135 \mathrm {~g} \mathrm {~cm} ^ { - 1 }$. A transverse wave of the form $y = - 0.21 \sin ( x + 30 t )$ is produced in it, where $x$ is in meter and $t$ is in second. Then, the expected value of tension in the wire is $x \times 10 ^ { - 2 } \mathrm {~N}$. Value of $x$ is $\_\_\_\_$ (Round-off to the nearest integer)

A particle of mass 1 mg and charge $q$ is lying at the mid-point of two stationary particles kept at a distance 2 m when each is carrying same charge $q$. If the free charged particle is displaced from its equilibrium position through distance $x$ ($x \ll 1\mathrm{~m}$), the particle executes SHM. Its angular frequency of oscillation will be \_\_\_\_ $\times 10^{5}\mathrm{~rad~s}^{-1}$ (if $q^{2} = 10\mathrm{~C}^{2}$).

If the maximum value of accelerating potential provided by a radio frequency oscillator is 12 kV. The number of revolution made by a proton in a cyclotron to achieve one sixth of the speed of light is:
$[\mathrm{m_p} = 1.67 \times 10^{-27}\text{ kg},\ e = 1.6 \times 10^{-19}\text{ C},\ \text{Speed of light} = 3 \times 10^8\text{ m s}^{-1}]$

Two cars $X$ and $Y$ are approaching each other with velocities $36\mathrm{~km~h}^{-1}$ and $72\mathrm{~km~h}^{-1}$ respectively. The frequency of a whistle sound as emitted by a passenger in car $X$, heard by the passenger in car $Y$ is 1320 Hz. If the velocity of sound in air is $340\mathrm{~ms}^{-1}$, the actual frequency of the whistle sound produced is $\_\_\_\_$ Hz.

27 similar drops of mercury are maintained at 10 V each. All these spherical drops combine into a single big drop. The potential energy of the bigger drop is $\_\_\_\_$ times that of a smaller drop.

For an ideal heat engine, the temperature of the source is $127 ^ { \circ } \mathrm { C }$. In order to have $60 \%$ efficiency the temperature of the sink should be $\_\_\_\_$ ${}^{\circ}$C. (Round off to the nearest integer)

A particle performs simple harmonic motion with a period of 2 second. The time taken by the particle to cover a displacement equal to half of its amplitude from the mean position is $\frac { 1 } { a } \mathrm {~s}$. The value of $a$ to the nearest integer is

The amplitude of wave disturbance propagating in the positive $x$-direction is given by $y = \frac { 1 } { ( 1 + x ) ^ { 2 } }$ at time $t = 0$ and $y = \frac { 1 } { 1 + ( x - 2 ) ^ { 2 } }$ at $t = 1 s$, where $x$ and $y$ are in metres. The shape of wave does not change during the propagation. The velocity of the wave will be $\mathrm { m } \mathrm { s } ^ { - 1 }$.

A uniform heating wire of resistance $36 \Omega$ is connected across a potential difference of 240 V. The wire is then cut into half and a potential difference of 240 V is applied across each half separately. The ratio of power dissipation in first case to the total power dissipation in the second case would be $1 : x$, where $x$ is $\_\_\_\_$.

In an electrical circuit, a battery is connected to pass 20 C of charge through it in a certain given time. The potential difference between two plates of the battery is maintained at 15 V. The workdone by the battery is $\_\_\_\_$ J

An electric bulb rated as 200 W at 100 V is used in a circuit having 200 V supply. The resistance $R$ that must be put in series with the bulb so that the bulb delivers the same power is \_\_\_\_ $\Omega$.

A coil in the shape of an equilateral triangle of side 10 cm lies in a vertical plane between the pole pieces of permanent magnet producing a horizontal magnetic field 20 mT. The torque acting on the coil when a current of 0.2 A is passed through it and its plane becomes parallel to the magnetic field will be $\sqrt{x} \times 10^{-5}\text{ Nm}$. The value of $x$ is

First, a set of $n$ equal resistors of $10\Omega$ each are connected in series to a battery of E.M.F. 20 V and internal resistance $10\Omega$. A current $I$ is observed to flow. Then, the $n$ resistors are connected in parallel to the same battery. It is observed that the current is increased 20 times, then the value of $n$ is $\_\_\_\_$.

A point source of light $S$, placed at a distance 60 cm infront of the centre of a plane mirror of width 50 cm , hangs vertically on a wall. A man walks infront of the mirror along a line parallel to the mirror at a distance 1.2 m from it (see in the figure). The distance between the extreme points where he can see the image of the light source in the mirror is $\_\_\_\_$ cm

A closed organ pipe of length $L$ and an open organ pipe contain gases of densities $\rho _ { 1 }$ and $\rho _ { 2 }$ respectively. The compressibility of gases are equal in both the pipes. Both the pipes are vibrating in their first overtone with same frequency. The length of the open pipe is $\frac { x } { 3 } L \sqrt { \frac { \rho _ { 1 } } { \rho _ { 2 } } }$, where $x$ is $\_\_\_\_$. (Round off to the Nearest Integer)

A parallel plate capacitor has plate area $100 \mathrm {~m} ^ { 2 }$ and plate separation of 10 m. The space between the plates is filled up to a thickness 5 m with a material of dielectric constant of 10. The resultant capacitance of the system is $x \mathrm { pF }$. The value of $\varepsilon _ { 0 } = 8.85 \times 10 ^ { - 12 } \mathrm {~F} \mathrm {~m} ^ { - 1 }$. The value of $x$ to the nearest integer is $\_\_\_\_$.

The frequency of a car horn encountered a change from 400 Hz to 500 Hz . When the car approaches a vertical wall. If the speed of sound is $330 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Then the speed of car is $\_\_\_\_$ $\mathrm { km } \mathrm { h } ^ { - 1 }$.

In the given figure the magnetic flux through the loop increases according to the relation $\phi _ { B } ( t ) = 10 t ^ { 2 } + 20 t$, where $\phi _ { B }$ is in milliwebers and $t$ is in seconds. The magnitude of current through $R = 2 \Omega$ resistor at $t = 5 \mathrm {~s}$ is $\_\_\_\_$ mA.

A steel rod with $y = 2.0 \times 10 ^ { 11 } \mathrm {~N} \mathrm {~m} ^ { - 2 }$ and $\alpha = 10 ^ { - 5 } { } ^ { \circ } \mathrm { C } ^ { - 1 }$ of length 4 m and area of cross-section $10 \mathrm {~cm} ^ { 2 }$ is heated from $0 ^ { \circ } \mathrm { C }$ to $400 ^ { \circ } \mathrm { C }$ without being allowed to extend. The tension produced in the rod is $x \times 10 ^ { 5 } \mathrm {~N}$ where the value of $x$ is $\_\_\_\_$.

In a series LCR resonant circuit, the quality factor is measured as 100. If the inductance is increased by two fold and resistance is decreased by two fold, then the quality factor after this change will be $\_\_\_\_$

The value of aluminium susceptibility is $2.2 \times 10^{-5}$. The percentage increase in the magnetic field if space within a current carrying toroid is filled with aluminium is $\frac{x}{10^{4}}$. Then the value of $x$ is \_\_\_\_.

A circular coil of radius 8.0 cm and 20 turns is rotated about its vertical diameter with an angular speed of $50\text{ rad s}^{-1}$ in a uniform horizontal magnetic field of $3.0 \times 10^{-2}\text{ T}$. The maximum emf induced in the coil will be $\_\_\_\_$ $\times 10^{-2}$ volt (rounded off to the nearest integer).

A uniform conducting wire of length is $24a$, and resistance $R$ is wound up as current carrying coil in the shape of an equilateral triangle of side $a$ and then in the form of a square of side $a$. The coil is connected to a voltage source $V_{0}$. The ratio of magnetic moment of the coils in case of equilateral triangle to that for square is $1:\sqrt{y}$ where $y$ is $\_\_\_\_$.