The SI unit of a physical quantity is pascal-second. The dimensional formula of this quantity will be :

The distance of the Sun from earth is 1.5 $$\times$$ 10¹¹ m and its angular diameter is (2000) s when observed from the earth. The diameter of the Sun will be :

When a ball is dropped into a lake from a height 4.9 m above the water level, it hits the water with a velocity v and then sinks to the bottom with the constant velocity v. It reaches the bottom of the lake 4.0 s after it is dropped. The approximate depth of the lake is :

One end of a massless spring of spring constant k and natural length l₀ is fixed while the other end is connected to a small object of mass m lying on a frictionless table. The spring remains horizontal on the table. If the object is made to rotate at an angular velocity $$\omega$$ about an axis passing through fixed end, then the elongation of the spring will be :

A stone tide to a spring of length L is whirled in a vertical circle with the other end of the spring at the centre. At a certain instant of time, the stone is at its lowest position and has a speed u. The magnitude of change in its velocity, as it reaches a position where the string is horizontal, is $$\sqrt {x({u^2} - gL)} $$. The value of x is -

Four spheres each of mass m from a square of side d (as shown in figure). A fifth sphere of mass M is situated at the centre of square. The total gravitational potential energy of the system is :

For a perfect gas, two pressures P₁ and P₂ are shown in figure. The graph shows :

According to kinetic theory of gases,

A. The motion of the gas molecules freezes at 0$$^\circ$$C.

B. The mean free path of gas molecules decreases if the density of molecules is increased.

C. The mean free path of gas molecules increases if temperature is increased keeping pressure constant.

D. Average kinetic energy per molecule per degree of freedom is $${3 \over 2}{k_B}T$$ (for monoatomic gases).

Choose the most appropriate answer from the options given below :

A lead bullet penetrates into a solid object and melts. Assuming that 40% of its kinetic energy is used to heat it, the initial speed of bullet is :

(Given : initial temperature of the bullet = 127$$^\circ$$C, Melting point of the bullet = 327$$^\circ$$C, Latent heat of fusion of lead = 2.5 $$\times$$ 10⁴ J kg^$$-$$1, Specific heat capacity of lead = 125 J/kg K)

The equation of a particle executing simple harmonic motion is given by $$x = \sin \pi \left( {t + {1 \over 3}} \right)m$$. At t = 1s, the speed of particle will be

(Given : $$\pi$$ = 3.14)

If a charge q is placed at the centre of a closed hemispherical non-conducting surface, the total flux passing through the flat surface would be :

Three identical charged balls each of charge 2 C are suspended from a common point P by silk threads of 2 m each (as shown in figure). They form an equilateral triangle of side 1m.

The ratio of net force on a charged ball to the force between any two charged balls will be :

Two long parallel conductors S₁ and S₂ are separated by a distance 10 cm and carrying currents of 4A and 2A respectively. The conductors are placed along x-axis in X-Y plane. There is a point P located between the conductors (as shown in figure).

A charge particle of 3$$\pi$$ coulomb is passing through the point P with velocity $$\overrightarrow v = (2\widehat i + 3\widehat j)$$ m/s; where $$\widehat i$$ and $$\widehat j$$ represents unit vector along x & y axis respectively.

The force acting on the charge particle is $$4\pi \times {10^{ - 5}}( - x\widehat i + 2\widehat j)$$ N. The value of x is :

If L, C and R are the self inductance, capacitance and resistance respectively, which of the following does not have the dimension of time?

Given below are two statements :

Statement I : A time varying electric field is a source of changing magnetic field and vice-versa. Thus a disturbance in electric or magnetic field creates EM waves.

Statement II : In a material medium, the EM wave travels with speed $$v = {1 \over {\sqrt {{\mu _0}{ \in _0}} }}$$. In the light of the above statements, choose the correct answer from the options given below.

A convex lens has power P. It is cut into two halves along its principal axis. Further one piece (out of the two halves) is cut into two halves perpendicular to the principal axis (as shown in figures). Choose the incorrect option for the reported pieces.

If a wave gets refracted into a denser medium, then which of the following is true?

Given below are two statements :

Statement I : In hydrogen atom, the frequency of radiation emitted when an electron jumps from lower energy orbit (E₁) to higher energy orbit (E₂), is given as hf = E₁ $$-$$ E₂

Statement II : The jumping of electron from higher energy orbit (E₂) to lower energy orbit (E₁) is associated with frequency of radiation given as f = (E₂ $$-$$ E₁)/h

This condition is Bohr's frequency condition.

In the light of the above statements, choose the correct answer from the options given below :

A mass of 10 kg is suspended vertically by a rope of length 5 m from the roof. A force of 30 N is applied at the middle point of rope in horizontal direction. The angle made by upper half of the rope with vertical is $$\theta$$ = tan^$$-$$1 (x $$\times$$ 10^$$-$$1). The value of x is ____________.

(Given, g = 10 m/s²)

A rolling wheel of 12 kg is on an inclined plane at position P and connected to a mass of 3 kg through a string of fixed length and pulley as shown in figure. Consider PR as friction free surface. The velocity of centre of mass of the wheel when it reaches at the bottom Q of the inclined plane PQ will be $${1 \over 2}\sqrt {xgh} $$ m/s. The value of x is ___________.

A diatomic gas ($$\gamma$$ = 1.4) does 400J of work when it is expanded isobarically. The heat given to the gas in the process is __________ J.

A particle executes simple harmonic motion. Its amplitude is 8 cm and time period is 6 s. The time it will take to travel from its position of maximum displacement to the point corresponding to half of its amplitude, is ___________ s.

A parallel plate capacitor is made up of stair like structure with a plate area A of each stair and that is connected with a wire of length b, as shown in the figure. The capacitance of the arrangement is $${x \over {15}}{{{ \in _0}A} \over b}$$. The value of x is ____________.

The current density in a cylindrical wire of radius r = 4.0 mm is 1.0 $$\times$$ 10⁶ A/m². The current through the outer portion of the wire between radial distances $${r \over 2}$$ and r is x$$\pi$$ A; where x is __________.

In the given circuit 'a' is an arbitrary constant. The value of m for which the equivalent circuit resistance is minimum, will be $$\sqrt {{x \over 2}} $$. The value of x is __________.

A deuteron and a proton moving with equal kinetic energy enter into a uniform magnetic field at right angle to the field. If r_d and r_p are the radii of their circular paths respectively, then the ratio $${{{r_d}} \over {{r_p}}}$$ will be $$\sqrt{x}$$ : 1 where x is __________.

A metallic rod of length 20 cm is placed in North-South direction and is moved at a constant speed of 20 m/s towards East. The horizontal component of the Earth's magnetic field at that place is 4 $$\times$$ 10^$$-$$3 T and the angle of dip is 45$$^\circ$$. The emf induced in the rod is ___________ mV.

The cut-off voltage of the diodes (shown in figure) in forward bias is 0.6 V. The current through the resister of 40 $$\Omega$$ is __________ mA.