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A bullet of \( 4 \, \text{g} \) mass is fired from a gun of mass \( 4 \, \text{kg} \). If the bullet moves with the muzzle speed of \( 50 \, \text{m s}^{-1} \), the impulse imparted to the gun and velocity of recoil of gun are:

A nucleus with mass number \( 184 \) initially at rest emits an \( \alpha \)-particle. If the Q value of the reaction is \( 5.5 \, \text{MeV} \), calculate the kinetic energy of the \( \alpha \)-particle.

\( T_0 \) is the time period of a simple pendulum at a place. If the length of the pendulum is reduced to \( \dfrac{1}{16} \) times of its initial value, the modified time period is:

A porter lifts a heavy suitcase of mass \( 80 \, \text{kg} \) and at the destination lowers it down by a distance of \( 80 \, \text{cm} \) with a constant velocity. Calculate the workdone by the porter in lowering the suitcase. (Take \( g = 9.8 \, \text{m s}^{-2} \) )

An electron of mass \( m_e \) and a proton of mass \( m_p \) are accelerated through the same potential difference. The ratio of the de-Broglie wavelength associated with the electron to that with the proton is:

Match List-I with List-II:

Choose the correct answer from the options given below:

An electric dipole is placed on \( x \)-axis in proximity to a line charge of linear charge density \( 3.0 \times 10^{-6} \, \text{C/m} \). Line charge is placed on \( z \)-axis and positive and negative charge of dipole is at a distance of \( 10 \, \text{mm} \) and \( 12 \, \text{mm} \) from the origin respectively. If total force of \( 4 \, \text{N} \) is exerted on the dipole, find out the amount of positive or negative charge of the dipole.

In a circuit consisting of a capacitance and a generator with alternating emf \( E_g = E_g \sin \omega t \), \( V_c \) and \( I_c \) are the voltage and current. Correct phasor diagram for such circuit is:

Consider a situation in which reverse biased current of a particular \( P-\!N \) junction increases when it is exposed to a light of wavelength \( \leq 621 \, \text{nm} \). During this process, enhancement in carrier concentration takes place due to generation of hole-electron pairs. The value of band gap is nearly.

A body is projected vertically upwards from the surface of earth with a velocity sufficient enough to carry it to infinity. The time taken by it to reach height \( h \) is _______ \( \text{s} \).

Statement I: The ferromagnetic property depends on temperature. At high temperature ferromagnet becomes paramagnet.
Statement II: At high temperature, the domain wall area of a ferromagnetic substance increases.
In the light of the above statements, choose the most appropriate answer from the options given below

What will be the projection of vector \( \vec{A} = \hat{i} + \hat{j} + \hat{k} \) on vector \( \vec{B} = \hat{i} + \hat{j} \)?

The motion of a mass on a spring, with spring constant \( K \) is as shown in figure.

The equation of motion is given by \( x(t) = A \sin \omega t + B \cos \omega t \) with \( \omega = \sqrt{\dfrac{K}{m}} \). Suppose that at time \( t = 0 \), the position of mass is \( x(0) \) and velocity \( v(0) \), then its displacement can also be represented as \( x(t) = C \cos( \omega t - \phi) \), where \( C \) and \( \phi \) are: 

What will be the average value of energy for a monoatomic gas in thermal equilibrium at temperature \( T \)?

What should be the height of transmitting antenna and the population covered if the television telecase is to cover a radius of \( 150 \, \text{km} \)? The average population density around the tower is \( {2000}/{\text{km}^2} \) and the value of \( R_e = 6.5 \times 10^{6} \, \text{m} \).

Consider a situation in which a ring, a solid cylinder and a solid sphere roll down on the same inclined plane without slipping. Assume that they start rolling from rest and having identical diameter.
The correct statement for this situation is:

A copper (Cu) rod of length \( 25 \, \mathrm{cm} \) and cross-sectional area \( 3 \, \mathrm{mm^2} \) is joined with a similar Aluminium (Al) rod as shown in figure. Find the resistance of the combination between the ends A and B.
(Take Resistivity of Copper \( = 1.7 \times 10^{-8} \, \Omega \,\mathrm{m} \) , Resistivity of Aluminium \( = 2.6 \times 10^{-8} \, \Omega \,\mathrm{m} \) )

Intensity of sunlight is observed as \( 0.092 \, \mathrm{W \, m^{-2}} \) at a point in free space. What will be the peak value of magnetic field at the point? \(( \varepsilon_{0} = 8.85 \times 10^{-12} \, \mathrm{C^{2} \, N^{-1} \, m^{-2}}) \)

A ray of light passing through a prism \( (\mu = \sqrt{3} )\) suffers minimum deviation. It is found that the angle of incidence is double the angle of refraction within the prism. Then, the angle of prism is ______ (in degrees).

The area of cross-section of a railway track is \( 0.01 \, \mathrm{m^{2}} \). The temperature variation is \( 10^{\circ}\mathrm{C} \). Coefficient of linear expansion of material of track is \( 10^{-5} \, / ^{\circ}\mathrm{C} \). The energy stored per meter in the track is ______ \( \mathrm{J/m} \). (Young’s modulus of material of track is \( 10^{11} \, \mathrm{N \, m^{-2}} \))

The total charge enclosed in an incremental volume of \( 2 \times 10^{-9} \, \mathrm{m^{3}} \) located at the origin is ______ \( \mathrm{nC} \), if electric flux density of its field is found as:
\( \mathbf{D} = e^{-x} \sin y \, \hat{\imath} - e^{-x} \cos y \, \hat{\jmath} + 2 z \, \hat{k} \; \mathrm{C/m^{2}} \).

The position of the centre of mass of a uniform semi-circular wire of radius ‘\( R \)’ placed in x–y plane with its centre at the origin and the line joining its ends as x-axis is given by \( \left( 0, \dfrac{xR}{\pi} \right) \). Then, the value of \( |x| \) is ______.

Three students \( S_1 , S_2 \) and \( S_3 \) perform an experiment for determining the acceleration due to gravity \( (g) \) using a simple pendulum. They use different lengths of pendulum and record time for different number of oscillations. The observations are as shown in the table.

(Least count of length \( = 0.1 \, \text{cm} \), least count of time \( = 0.1 \, \text{s} \))
If \( E_1 , E_2 \) and \( E_3 \) are the percentage errors in ‘\( g \)’ for students \( 1 \), \( 2 \), and \( 3 \) respectively, then the minimum percentage error is obtained by student no. _______.

In an electric circuit, a cell of certain emf provides a potential difference of \( 1.25 \, \text{V} \) across a load resistance of \( 5 \, \Omega \). However, it provides a potential difference of \( 1 \, \text{V} \) across a load resistance of \( 2 \, \Omega \). The emf of the cell is given by \( \dfrac{x}{10} \, \text{V} \). Then the value of \( x \) is _______.

The centre of a wheel rolling on a plane surface moves with a speed \( v_0 \). A particle on the rim of the wheel at the same level as the centre will be moving at a speed \( \sqrt{x}\, v_0 \). Then the value of \( x \) is _______.

In a given circuit diagram, a \( 5 \, \text{V} \) Zener diode along with a series resistance is connected across a \( 50 \, \text{V} \) power supply. The minimum value of the resistance required, if the maximum Zener current is \( 90 \, \text{mA} \) will be _______ \( \Omega \).

Three particle \( P, Q \) and \( R \) are moving along the vectors \( \vec{A} = \hat{i} + \hat{j} \), \( \vec{B} = \hat{j} + \hat{k} \) and \( \vec{C} = -\hat{i} + \hat{j} \) respectively. They strike on a point and start to move in different directions. Now particle \( P \) is moving normal to the plane which contains vector \( \vec{A} \) and \( \vec{B} \). Similarly particle \( Q \) is moving normal to the plane which contains vector \( \vec{A} \) and \( \vec{C} \). The angle between the direction of motion of \( P \) and \( Q \) is \( \cos^{-1}\!\left( \dfrac{1}{\sqrt{x}} \right) \). Then the value of \( x \) is _______.

In \( 5 \) minutes, a body cools from \( 75^\circ \text{C} \) to \( 65^\circ \text{C} \) at room temperature of \( 25^\circ \text{C} \). The temperature of body at the end of next \( 5 \) minutes is _______ \( ^\circ \text{C} \).