Given below are two statements. One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A : Two identical balls A and B thrown with same velocity 'u' at two different angles with horizontal attained the same range R. IF A and B reached the maximum height h₁ and h₂ respectively, then $$R = 4\sqrt {{h_1}{h_2}} $$

Reason R : Product of said heights.

$${h_1}{h_2} = \left( {{{{u^2}{{\sin }^2}\theta } \over {2g}}} \right)\,.\,\left( {{{{u^2}{{\cos }^2}\theta } \over {2g}}} \right)$$

Choose the correct answer :

Two buses P and Q start from a point at the same time and move in a straight line and their positions are represented by $${X_P}(t) = \alpha t + \beta {t^2}$$ and $${X_Q}(t) = ft - {t^2}$$. At what time, both the buses have same velocity?

A disc with a flat small bottom beaker placed on it at a distance R from its center is revolving about an axis passing through the center and perpendicular to its plane with an angular velocity $$\omega$$. The coefficient of static friction between the bottom of the beaker and the surface of the disc is $$\mu$$. The beaker will revolve with the disc if :

A solid metallic cube having total surface area 24 m² is uniformly heated. If its temperature is increased by 10$$^\circ$$C, calculate the increase in volume of the cube. (Given $$\alpha$$ = 5.0 $$\times$$ 10^$$-$$4 $$^\circ$$C^$$-$$1).

A copper block of mass 5.0 kg is heated to a temperature of 500$$^\circ$$C and is placed on a large ice block. What is the maximum amount of ice that can melt? [Specific heat of copper : 0.39 J g^$$-$$1 $$^\circ$$C^$$-$$1 and latent heat of fusion of water : 335 J g^$$-$$1]

The ratio of specific heats $$\left( {{{{C_P}} \over {{C_V}}}} \right)$$ in terms of degree of freedom (f) is given by :

For a particle in uniform circular motion, the acceleration $$\overrightarrow a $$ at any point P(R, $$\theta$$) on the circular path of radius R is (when $$\theta$$ is measured from the positive x-axis and v is uniform speed) :

Two metallic plates form a parallel plate capacitor. The distance between the plates is 'd'. A metal sheet of thickness $${d \over 2}$$ and of area equal to area of each plate is introduced between the plates. What will be the ratio of the new capacitance to the original capacitance of the capacitor?

Two cells of same emf but different internal resistances r₁ and r₂ are connected in series with a resistance R. The value of resistance R, for which the potential difference across second cell is zero, is :

Given below are two statements :

Statement I : Susceptibilities of paramagnetic and ferromagnetic substances increase with decrease in temperature.

Statement II : Diamagnetism is a result of orbital motions of electrons developing magnetic moments opposite to the applied magnetic field.

Choose the correct answer from the options given below :

A long solenoid carrying a current produces a magnetic field B along its axis. If the current is doubled and the number of turns per cm is halved, the new value of magnetic field will be equal to

A sinusoidal voltage V(t) = 210 sin 3000 t volt is applied to a series LCR circuit in which L = 10 mH, C = 25 $$\mu$$F and R = 100 $$\Omega$$. The phase difference ($$\Phi $$) between the applied voltage and resultant current will be :

The electromagnetic waves travel in a medium at a speed of 2.0 $$\times$$ 10⁸ m/s. The relative permeability of the medium is 1.0. The relative permittivity of the medium will be :

The interference pattern is obtained with two coherent light sources of intensity ratio 4 : 1. And the ratio $${{{I_{\max }} + {I_{\min }}} \over {{I_{\max }} - {I_{\min }}}}$$ is $${5 \over x}$$. Then, the value of x will be equal to :

A light whose electric field vectors are completely removed by using a good polaroid, allowed to incident on the surface of the prism at Brewster's angle. Choose the most suitable option for the phenomenon related to the prism.

A proton, a neutron, an electron and an $$\alpha$$ particle have same energy. If $$\lambda$$_p, $$\lambda$$_n, $$\lambda$$_e and $$\lambda$$_a are the de Broglie's wavelengths of proton, neutron, electron and $$\alpha$$ particle respectively, then choose the correct relation from the following :

Identify the logic operation performed by the given circuit:

If n represents the actual number of deflections in a converted galvanometer of resistance G and shunt resistance S. Then the total current I when its figure of merit is K will be:

For $$z = {a^2}{x^3}{y^{{1 \over 2}}}$$, where 'a' is a constant. If percentage error in measurement of 'x' and 'y' are 4% and 12% respectively, then the percentage error for 'z' will be _______________%.

A curved in a level road has a radius 75 m. The maximum speed of a car turning this curved road can be 30 m/s without skidding. If radius of curved road is changed to 48 m and the coefficient of friction between the tyres and the road remains same, then maximum allowed speed would be ___________ m/s.

A block of mass 200 g is kept stationary on a smooth inclined plane by applying a minimum horizontal force F = $$\sqrt{x}$$N as shown in figure.

The value of x = _____________.

Moment of Inertia (M.I.) of four bodies having same mass 'M' and radius '2R' are as follows:

I₁ = M.I. of solid sphere about its diameter

I₂ = M.I. of solid cylinder about its axis

I₃ = M.I. of solid circular disc about its diameter

I₄ = M.I. of thin circular ring about its diameter

If 2(I₂ + I₃) + I₄ = x . I₁, then the value of x will be __________.

Two satellites S₁ and S₂ are revolving in circular orbits around a planet with radius R₁ = 3200 km and R₂ = 800 km respectively. The ratio of speed of satellite S₁ to be speed of satellite S₂ in their respective orbits would be $${1 \over x}$$ where x = ___________.

When a gas filled in a closed vessel is heated by raising the temperature by 1$$^\circ$$C, its pressure increases by 0.4%. The initial temperature of the gas is ___________ K.

27 identical drops are charged at 22V each. They combine to form a bigger drop. The potential of the bigger drop will be _____________ V.

The length of a given cylindrical wire is increased to double of its original length. The percentage increase in the resistance of the wire will be ____________ %.

In a series LCR circuit, the inductance, capacitance and resistance are L = 100 mH, C = 100 $$\mu$$F and R = 10 $$\Omega$$ respectively. They are connected to an AC source of voltage 220 V and frequency of 50 Hz. The approximate value of current in the circuit will be ___________ A.