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A NCC parade is going to a uniform speed of \( 9 \, \text{km}/\text{h} \) under a mango tree on which a monkey is sitting at a height of \( 19.6 \, \text{m} \). At any particular instant, the monkey drops a mango. A cadet will receive the mango whose distance form the tree at time of drop is:
(Given \( g = 9.8 \, \text{m}/\text{s}^{2} \) )

In two different experiments, an object of mass \( 5 \, \text{kg} \) moving with a speed of \( 25 \, \text{m s}^{-1} \) hits two different walls and comes to rest within (i) \( 3 \) second, (ii) \( 5 \) seconds, respectively.
Choose the correct option out of the following:

A balloon has mass of \( 10 \, \text{g} \) in air. The air escapes from the balloon at a uniform rate with velocity \( 4.5 \, \text{cm}/\text{s} \). if the balloon shrinks in \( 5 \, \text{s} \) completely. Then, the average force acting on that balloon will be (in dyne).

If the radius of earth shrinks by \( 2\% \) while its mass remains same. The acceleration due to gravity on the earth’s surface will approximately.

The force required to stretch a wire of cross-section \( 1 \, \text{cm}^{2} \) to double its length will be:
(Given Young’s modulus of the wire \( = 2 \times 10^{11} \, \text{N}/\text{m}^{2} \) )

A Carnot engine has efficiency of \( 50\% \). If the temperature of sink is reduced by \( 40^\circ \mathrm{C} \), its efficiency increases by \( 30\% \). The temperature of the source will be:

Given below are two statements:
Statement I: The average momentum of a molecule in a sample of an ideal depends on temperature.
Statement II: The rms speed of oxygen molecules in a gas is \( v \). If the temperature is doubled and the oxygen molecules dissociate into oxygen atoms, the rms speed will become \( 2v \).
In the light of the above statements, choose the correct answer from the options given below:

In the wave equation
\( y = 0.5 \sin  \dfrac{2\pi}{\lambda}(400t-x)  \, \mathrm{m} \)
The velocity of the wave will be:

Two capacitors, each having capacitance \( 40\,\mu \mathrm{F} \) are connected in series. The space between one of the capacitors is filled with dielectric material of dielectric constant \( K \) such that the equivalence capacitance of the system became \( 24\,\mu \mathrm{F} \). The value of \( K \) will be:

A wire of resistance \( R_1 \) is drawn out so that its length is increased by twice of its original length. The ratio of new resistance to original resistance is:

The current sensitivity of a galvanometer can be increased by:
(a) decreasing the number of turns
(b) increasing the magnetic field
(c) decreasing the area of the coil
(d) decreasing the torsional constant of the spring
Choose the most appropriate answer form the options given below:

As shown in the figure, a metallic rod of linear density \( 0.45 \, \text{kg} \, \text{m}^{-1} \) is lying horizontally on a smooth inclined plane which makes an angle of \( 45^\circ \) with the horizontal. The minimum current flowing in the rod required to keep it stationary, when \( 0.15 \, \text{T} \) magnetic field is acting on it in the vertical upward direction, will be:
{Use \( g = 10 \, \text{m/s}^2 \)}

The equation of current in a purely inductive circuit is \( 5 \sin(49\pi t - 30^\circ) \). If the inductance is \( 30 \, \text{mH} \) then the equation for the voltage across the inductor, will be:
{Let \( \pi = \dfrac{22}{7} \)}

As shown in the figure, after passing through the medium \( 1 \). The speed of light \( v_2 \) in medium \( 2 \) will be:
(Given \( c = 3 \times 10^8 \, \text{m s}^{-1} \))

In normal adjustment, for a refracting telescope, the distance between objective and eye piece is \( 30 \, \text{cm} \). The focal length of the objective, when the angular magnification of the telescope is \( 2 \), will be:

The equation \( \lambda = \dfrac{1.227}{x} \, \text{nm} \) can be used to find the de-Brogli wavelength of an electron. In this equation \( x \) stands for
Where \( m = \) mass of electron
\( P = \) momentum of electron
\( K = \) Kinetic energy of electron
\( V = \) Accelerating potential in volts for electron

The half life period of a radioactive substance is \( 60 \, \text{days} \). The time taken for \( \dfrac{7}{8} \) th of its original mass to disintegrate will be:

Identify the solar cell characteristics from the following options:

In the case of amplitude modulation to avoid distortion the modulation index \( (\mu) \) should be:

If the projection of \( 2\hat{i}+4\hat{j}-2\hat{k} \) on \( \hat{i}+2\hat{j}+\alpha \hat{k} \) is zero. Then, the value of \( \alpha \) will be _______.

A freshly prepared radioactive source of half life \( 2 \, \text{hours} \, 30 \, \text{minutes} \) emits radiation which is \( 64 \) times the permissible safe level. The minimum time, after which it would be possible to work safely with source, will be ________ \( \text{hours} \).

In a Young’s double slit experiment, a laser light of \( 560 \, \text{nm} \) produces an interference pattern with consecutive bright fringes’ separation of \( 7.2 \, \text{mm} \). Now another light is used to produce an interference pattern with consecutive bright fringes’ separation of \( 8.1 \, \text{mm} \). The wavelength of second light is _______ \( \text{nm} \).

The frequencies at which the current amplitude in an \( LCR \) series circuit becomes \( \dfrac{1}{\sqrt{2}} \) times its maximum value, are \( 212 \, \text{rad s}^{-1} \) and \( 232 \, \text{rad s}^{-1} \). The value of resistance in the circuit is \( R = 5 \, \Omega \). The self inductance in the circuit is __________ \( \text{mH} \).

As shown in the figure, a potentiometer wire of resistance \( 20 \, \Omega \) and length \( 300 \, \text{cm} \) is connected with resistance box \( (\text{R.B.}) \) and a standard cell of emf \( 4 \, \text{V} \). For a resistance '\( R \)' of resistance box introduced into the circuit, the null point for a cell of \( 20 \, \text{mV} \) is found to be \( 60 \, \text{cm} \). The value of 'R' is _______ \( \Omega \).

Two electric dipoles of dipole moments \( 1.2 \times 10^{-30} \, \text{Cm} \) and \( 2.4 \times 10^{-30} \, \text{Cm} \) are placed in two different uniform electric fields of strengths \( 5 \times 10^{4} \, \text{NC}^{-1} \) and \( 15 \times 10^{4} \, \text{NC}^{-1} \) respectively. The ratio of maximum torque experienced by the electric dipoles will be \( \dfrac{1}{x} \). The value of \( x \) is ________.

The frequency of echo will be _______ \( \text{Hz} \) if the train blowing a whistle of frequency \( 320 \, \text{Hz} \) is moving with a velocity of \( 36 \, \text{km/h} \) towards a hill from which an echo is heard by the train driver. Velocity of sound in air is \( 330 \, \text{m/s} \).

The diameter of an air bubble which was initially \( 2 \, \text{mm} \), rises steadily through a solution of density \( 1750 \, \text{kg m}^{-3} \) at the rate of \( 0.35 \, \text{cms}^{-1} \). The coefficient of viscosity of the solution is ______ poise (in nearest integer). (the density of air is negligible).

A block of mass 'm' (as shown in figure) moving with kinetic energy \( E \) compresses a spring through a distance \( 25 \, \text{cm} \) when, its speed is halved. The value of spring constant of used spring will be \( nE \, \text{N m}^{-1} \) for \( n = \) __________.

Four identical discs each of mass 'M' and diameter 'a' are arranged in a small plane as shown in figure. If the moment of inertia of the system about \( OO' \) is \( \dfrac{x}{4} \, Ma^{2} \). Then, the value of \( x \) will be __________.