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A ball is spun with angular acceleration \( \alpha = 6t^2 - 2t \) where \( t \) is in second and \( \alpha \) is in \( \text{rad s}^{-2} \). At \( t = 0 \), the ball has angular velocity of \( 10 \, \text{rad s}^{-1} \) and angular position of \( 4 \, \text{rad} \). The most appropriate expression for the angular position of the ball is:

A block of mass \( 2 \, \text{kg} \) moving on a horizontal surface with speed of \( 4 \, \text{m s}^{-1} \) enters a rough surface ranging from \( x = 0.5 \, \text{m} \) to \( x = 1.5 \, \text{m} \). The retarding force in this range of rough surface is related to distance by \( F = -kx \) where \( k = 12 \, \text{N m}^{-1} \). The speed of the block as it just crosses the rough surface will be:

A \( \sqrt{34} \, \text{m} \) long ladder weighing \( 10 \, \text{kg} \) leans on a frictionless wall. Its feet rest on the floor \( 3 \, \text{m} \) away from the wall as shown in the figure. If \( F_f \) and \( F_w \) are the reaction forces of the floor and the wall, then ratio of \( \frac{F_w}{F_f} \) will be: (Use \( g = 10 \, \text{m} / \text{s}^2 \).)

Water falls from a \( 40 \, \text{m} \) high dam at the rate of \( 9 \times 10^4 \, \text{kg} \) per hour. Fifty percentage of gravitational potential energy can be converted into electrical energy. Using this hydro electric energy number of \( 100 \, \text{W} \) lamps, that can be lit, is: (Take \( g = 10 \, \text{m s}^{-2} \))

Two objects of equal masses placed at certain distance from each other attracts each other with a force of \( F \). If one-third mass of one object is transferred to the other object, then the new force will be:

A water drop of radius \( 1 \, \mu \text{m} \) falls in a situation where the effect of buoyant force is negligible. Co-efficient of viscosity of air is \( 1.8 \times 10^{-5} \, \text{N s m}^{-2} \) and its density is negligible as compared to that of water \( 10^{6} \, \text{g m}^{-3} \). Terminal velocity of the water drop is: (Take acceleration due to gravity \( = 10 \, \text{m s}^{-2} \))

A sample of an ideal gas is taken through the cyclic process \( ABCA \) as shown in figure. It absorbs, \( 40 \, \text{J} \) of heat during the part \( AB \), no heat during \( BC \) and rejects \( 60 \, \text{J} \) of heat during \( CA \). A work of \( 50 \, \text{J} \) is done on the gas during the part \( BC \). The internal energy of the gas at \( A \) is \( 1560 \, \text{J} \). The workdone by the gas during the part \( CA \) is:

What will be the effect on the root mean square velocity of oxygen molecules if the temperature is doubled and oxygen molecule dissociates into atomic oxygen?

Two point charges \( A \) and \( B \) of magnitude \( +8 \times 10^{-6} \, \text{C} \) and \( -8 \times 10^{-6} \, \text{C} \) respectively are placed at a distance \( d \) apart. The electric field at the middle point \( O \) between the charges is \( 6.4 \times 10^{4} \, \text{N C}^{-1} \). The distance \( 'd' \) between the point charges \( A \) and \( B \) is:

Resistance of the wire is measured as \( 2 \, \Omega \) and \( 3 \, \Omega \) at \( 10^{\circ}\text{C} \) and \( 30^{\circ}\text{C} \) respectively. Temperature co-efficient of resistance of the material of the wire is:

The space inside a straight current carrying solenoid is filled with a magnetic material having magnetic susceptibility equal to \( 1.2 \times 10^{-5} \). What is fractional increase in the magnetic field inside solenoid with respect to air as medium inside the solenoid?

Two parallel, long wires are kept \( 0.20 \, \text{m} \) apart in vacuum, each carrying current of \( x \, \text{A} \) in the same direction. If the force of attraction per meter of each wire is \( 2 \times 10^{-6} \, \text{N} \), then the value of \( x \) is approximately:

A coil is placed in a time varying magnetic field. If the number of turns in the coil were to be halved and the radius of wire doubled, the electrical power dissipated due to the current induced in the coil would be: (Assume the coil to be short circuited.)

An EM wave propagating in \( x \)–direction has a wavelength of \( 8 \, \text{mm} \). The electric field vibrating \( y \)–direction has maximum magnitude of \( 60 \, \text{V m}^{-1} \). Choose the correct equations for electric and magnetic fields if the EM wave is propagating in vacuum:

In young’s double slit experiment performed using a monochromatic light of wavelength \( \lambda \), when a glass plate \( ( \mu = 1.5 ) \) of thickness \( x \lambda \) is introduced in the path of the one of the interfering beams, the intensity at the position where the central maximum occurred previously remains unchanged. The value of \( x \) will be

Let \( K_{1} \) and \( K_{2} \) be the maximum kinetic energies of photo-electrons emitted when two monochromatic beams of wavelength \( \lambda_{1} \) and \( \lambda_{2} \), respectively are incident on a metallic surface. If \( \lambda_{1} = 3 \lambda_{2} \) then:

Following statements related to radioactivity are given below:
(\( a \)) Radioactivity is a random and spontaneous process and is dependent on physical and chemical conditions.
(\( b \)) The number of un-decayed nuclei in the radioactive sample decays exponentially with time.
(\( c \)) Slope of the graph of \( \log_{e} \) (no. of undecayed nuclei) Vs. time represents the reciprocal of mean life time \( ( \tau ) \).
(\( d \)) Product of decay constant \( ( \lambda ) \) and half-life time \( ( T_{1/2} ) \) is not constant.
Choose the most appropriate answer from the options given below:

In the given circuit the input voltage \( V_{in} \) is shown in figure. The cut-in voltage of p–n junction diode (\( D_1 \) or \( D_2 \)) is \( 0.6 \, \text{V} \). Which of the following output voltage (\( V_0 \)) waveform across the diode is correct?

Amplitude modulated wave is represented by \( V_{AM} = 10 \left[ 1 + 0.4 \cos \left( 2\pi \times 10^{4} \, t \right) \right] \cos \left( 2\pi \times 10^{7} \, t \right) \). The total bandwidth of the amplitude modulated wave is:

A student in the laboratory measures thickness of a wire using screw gauge. The readings are \( 1.22 \, \text{mm} \), \( 1.23 \, \text{mm} \), \( 1.19 \, \text{mm} \) and \( 1.20 \, \text{mm} \). The percentage error is \( \frac{x}{121} \% \). The value of \( x \) is ________.

A zener of breakdown voltage \( V_Z = 8 \, \text{V} \) and maximum zener current, \( I_{ZM} = 10 \, \text{mA} \) is subjected to an input voltage \( V_i = 10 \, \text{V} \) with series resistance \( R = 100 \, \Omega \). In the given circuit \( R_L \) represents the variable load resistance. The ratio of maximum and minimum value of \( R_L \) is __________.

In a Young’s double slit experiment, an angular width of the fringe is \( 0.35^\circ \) on a screen placed at \( 2 \, \text{m} \) away for particular wavelength of \( 450 \, \text{nm} \). The angular width of the fringe, when whole system is immersed in a medium of refractive index \( 7/5 \), is \( \dfrac{1}{\alpha} \). The value of \( \alpha \) is __________.

In the given circuit, the magnitude of \( V_L \) and \( V_C \) are twice that of \( V_R \). Given that \( f = 50 \, \text{Hz} \), the inductance of the coil is \( \dfrac{1}{K \pi} \, \text{mH} \). The value of \( K \) is __________.

All resistances in figure are \( 1\,\Omega \) each. The value of current 'I' is \( \frac{a}{5} \) \( \text{A} \). The value of \( a \) is ____________.

A capacitor \( C_1 \) of capacitance \( 5\,\mu\text{F} \) is charged to a potential of \( 30\,\text{V} \) using a battery. The battery is then removed and the charged capacitor is connected to an uncharged capacitor \( C_2 \) of capacitance \( 10\,\mu\text{F} \) as shown in figure. When the switch is closed charge flows between the capacitors. At equilibrium, the charge on the capacitor \( C_2 \) is ____________ \( \mu\text{C} \).

A tunning fork of frequency \( 340\,\text{Hz} \) resonates in the fundamental mode with an air column of length \( 125\,\text{cm} \) in a cylindrical tube closed at one end. When water is slowly poured in it, the minimum height of water required for observing resonance once again is ____________ \( \text{cm} \).
(Velocity of sound in air is \( 340\,\text{ms}^{-1} \))

A liquid of density \( 750\,\text{kg m}^{-3} \) flows smoothly through a horizontal pipe that tapers in cross-sectional area from \( A_1 = 1.2 \times 10^{-2}\,\text{m}^2 \) to \( A_2 = \frac{A_1}{2} \). The pressure difference between the wide and narrow sections of the pipe is \( 4500\,\text{Pa} \). The rate of flow of liquid is ____________ \( \times 10^{-3}\,\text{m}^3\text{s}^{-1} \).

A uniform disc with mass \( M = 4\,\text{kg} \) and radius \( R = 10\,\text{cm} \) is mounted on a fixed horizontal axle as shown in figure. A block with mass \( m = 2\,\text{kg} \) hangs from a massless cord that is wrapped around the rim of the disc. During the fall of the block, the cord does not slip and there is no friction at the axle. The tension in the cord is _______ \( \text{N} \). (Take \( g = 10\,\text{ms}^{-2} \)).

A car covers AB distance with first one-third at velocity \( v_1 \, \text{ms}^{-1} \), second one-third at \( v_2 \, \text{ms}^{-1} \) and last one-third at \( v_3 \, \text{ms}^{-1} \). If \( v_3 = 3v_1 \), \( v_2 = 2v_1 \) and \( v_1 = 11 \, \text{ms}^{-1} \) then the average velocity of the car is ________ \( \text{ms}^{-1} \).