Two projectiles are thrown with same initial velocity making an angle of $$45^{\circ}$$ and $$30^{\circ}$$ with the horizontal respectively. The ratio of their respective ranges will be :

In a Vernier Calipers, 10 divisions of Vernier scale is equal to the 9 divisions of main scale. When both jaws of Vernier calipers touch each other, the zero of the Vernier scale is shifted to the left of zero of the main scale and $$4^{\text {th }}$$ Vernier scale division exactly coincides with the main scale reading. One main scale division is equal to $$1 \mathrm{~mm}$$. While measuring diameter of a spherical body, the body is held between two jaws. It is now observed that zero of the Vernier scale lies between 30 and 31 divisions of main scale reading and $$6^{\text {th }}$$ Vernier scale division exactly coincides with the main scale reading. The diameter of the spherical body will be :

A ball of mass $$0.15 \mathrm{~kg}$$ hits the wall with its initial speed of $$12 \mathrm{~ms}^{-1}$$ and bounces back without changing its initial speed. If the force applied by the wall on the ball during the contact is $$100 \mathrm{~N}$$, calculate the time duration of the contact of ball with the wall.

A body of mass $$8 \mathrm{~kg}$$ and another of mass $$2 \mathrm{~kg}$$ are moving with equal kinetic energy. The ratio of their respective momentum will be :

Two uniformly charged spherical conductors $$A$$ and $$B$$ of radii $$5 \mathrm{~mm}$$ and $$10 \mathrm{~mm}$$ are separated by a distance of $$2 \mathrm{~cm}$$. If the spheres are connected by a conducting wire, then in equilibrium condition, the ratio of the magnitudes of the electric fields at the surface of the sphere $$A$$ and $$B$$ will be :

The oscillating magnetic field in a plane electromagnetic wave is given by

$$B_{y}=5 \times 10^{-6} \sin 1000 \pi\left(5 x-4 \times 10^{8} t\right) T$$. The amplitude of electric field will be :

Light travels in two media $$M_{1}$$ and $$M_{2}$$ with speeds $$1.5 \times 10^{8} \mathrm{~ms}^{-1}$$ and $$2.0 \times 10^{8} \mathrm{~ms}^{-1}$$ respectively. The critical angle between them is :

A body is projected vertically upwards from the surface of earth with a velocity equal to one third of escape velocity. The maximum height attained by the body will be :

(Take radius of earth $$=6400 \mathrm{~km}$$ and $$\mathrm{g}=10 \mathrm{~ms}^{-2}$$ )

A nucleus of mass $$M$$ at rest splits into two parts having masses $$\frac{M^{\prime}}{3}$$ and $${{2M'} \over 3}(M' < M)$$. The ratio of de Broglie wavelength of two parts will be :

An ice cube of dimensions $$60 \mathrm{~cm} \times 50 \mathrm{~cm} \times 20 \mathrm{~cm}$$ is placed in an insulation box of wall thickness $$1 \mathrm{~cm}$$. The box keeping the ice cube at $$0^{\circ} \mathrm{C}$$ of temperature is brought to a room of temperature $$40^{\circ} \mathrm{C}$$. The rate of melting of ice is approximately :

(Latent heat of fusion of ice is $$3.4 \times 10^{5} \mathrm{~J} \mathrm{~kg}^{-1}$$ and thermal conducting of insulation wall is $$0.05 \,\mathrm{Wm}^{-1 \circ} \mathrm{C}^{-1}$$ )

A gas has $$n$$ degrees of freedom. The ratio of specific heat of gas at constant volume to the specific heat of gas at constant pressure will be :

A transverse wave is represented by $$y=2 \sin (\omega t-k x)\, \mathrm{cm}$$. The value of wavelength (in $$\mathrm{cm}$$) for which the wave velocity becomes equal to the maximum particle velocity, will be :

A battery of $$6 \mathrm{~V}$$ is connected to the circuit as shown below. The current I drawn from the battery is :

A source of potential difference $$V$$ is connected to the combination of two identical capacitors as shown in the figure. When key '$$K$$' is closed, the total energy stored across the combination is $$E_{1}$$. Now key '$$K$$' is opened and dielectric of dielectric constant 5 is introduced between the plates of the capacitors. The total energy stored across the combination is now $$E_{2}$$. The ratio $$E_{1} / E_{2}$$ will be :

Two concentric circular loops of radii $$r_{1}=30 \mathrm{~cm}$$ and $$r_{2}=50 \mathrm{~cm}$$ are placed in $$\mathrm{X}-\mathrm{Y}$$ plane as shown in the figure. A current $$I=7 \mathrm{~A}$$ is flowing through them in the direction as shown in figure. The net magnetic moment of this system of two circular loops is approximately :

A velocity selector consists of electric field $$\vec{E}=E \,\hat{k}$$ and magnetic field $$\vec{B}=B \,\hat{j}$$ with $$B=12 \,m T$$. The value of $$E$$ required for an electron of energy $$728 \,\mathrm{e} V$$ moving along the positive $$x$$-axis to pass undeflected is :

(Given, mass of electron $$=9.1 \times 10^{-31} \mathrm{~kg}$$ )

Two masses $$M_{1}$$ and $$M_{2}$$ are tied together at the two ends of a light inextensible string that passes over a frictionless pulley. When the mass $$M_{2}$$ is twice that of $$M_{1}$$, the acceleration of the system is $$a_{1}$$. When the mass $$M_{2}$$ is thrice that of $$M_{1}$$, the acceleration of the system is $$a_{2}$$. The ratio $$\frac{a_{1}}{a_{2}}$$ will be :

The area of cross section of the rope used to lift a load by a crane is $$2.5 \times 10^{-4} \mathrm{~m}^{2}$$. The maximum lifting capacity of the crane is 10 metric tons. To increase the lifting capacity of the crane to 25 metric tons, the required area of cross section of the rope should be :

(take $$g=10 \,m s^{-2}$$ )

If $$\vec{A}=(2 \hat{i}+3 \hat{j}-\hat{k})\, \mathrm{m}$$ and $$\vec{B}=(\hat{i}+2 \hat{j}+2 \hat{k}) \,\mathrm{m}$$. The magnitude of component of vector $$\vec{A}$$ along vector $$\vec{B}$$ will be ____________ $$\mathrm{m}$$.

The radius of gyration of a cylindrical rod about an axis of rotation perpendicular to its length and passing through the center will be ___________ $$\mathrm{m}$$.

Given, the length of the rod is $$10 \sqrt{3} \mathrm{~m}$$.

In the given figure, the face $$A C$$ of the equilateral prism is immersed in a liquid of refractive index '$$n$$'. For incident angle $$60^{\circ}$$ at the side $$A C$$, the refractive light beam just grazes along face $$A C$$. The refractive index of the liquid $$n=\frac{\sqrt{x}}{4}$$. The value of $$x$$ is ____________.

(Given refractive index of glass $$=1.5$$ )

Two lighter nuclei combine to form a comparatively heavier nucleus by the relation given below :

$${ }_{1}^{2} X+{ }_{1}^{2} X={ }_{2}^{4} Y$$

The binding energies per nucleon for $$\frac{2}{1} X$$ and $${ }_{2}^{4} Y$$ are $$1.1 \,\mathrm{MeV}$$ and $$7.6 \,\mathrm{MeV}$$ respectively. The energy released in this process is _______________ $$\mathrm{MeV}$$.

A uniform heavy rod of mass $$20 \mathrm{~kg}$$, cross sectional area $$0.4 \mathrm{~m}^{2}$$ and length $$20 \mathrm{~m}$$ is hanging from a fixed support. Neglecting the lateral contraction, the elongation in the rod due to its own weight is $$x \times 10^{-9} \mathrm{~m}$$. The value of $$x$$ is _______________.

(Given, young modulus Y = 2 $$\times$$ 10¹¹ Nm^$$-$$2 and g = 10 ms^$$-$$2)

Three point charges of magnitude $$5 \mu \mathrm{C}, 0.16 \mu \mathrm{C}$$ and $$0.3 \mu \mathrm{C}$$ are located at the vertices $$A, B, C$$ of a right angled triangle whose sides are $$A B=3 \mathrm{~cm}, B C=3 \sqrt{2} \mathrm{~cm}$$ and $$C A=3 \mathrm{~cm}$$ and point $$A$$ is the right angle corner. Charge at point $$\mathrm{A}$$ experiences ____________ $$\mathrm{N}$$ of electrostatic force due to the other two charges.

In a coil of resistance $$8 \,\Omega$$, the magnetic flux due to an external magnetic field varies with time as $$\phi=\frac{2}{3}\left(9-t^{2}\right)$$. The value of total heat produced in the coil, till the flux becomes zero, will be _____________ $$J$$.

As per given figures, two springs of spring constants $$k$$ and $$2 k$$ are connected to mass $$m$$. If the period of oscillation in figure (a) is $$3 \mathrm{s}$$, then the period of oscillation in figure (b) will be $$\sqrt{x}~ s$$. The value of $$x$$ is ___________.