A torque meter is calibrated to reference standards of mass, length and time each with $$5 \%$$ accuracy. After calibration, the measured torque with this torque meter will have net accuracy of :

A bullet is shot vertically downwards with an initial velocity of $$100 \mathrm{~m} / \mathrm{s}$$ from a certain height. Within 10 s, the bullet reaches the ground and instantaneously comes to rest due to the perfectly inelastic collision. The velocity-time curve for total time $$\mathrm{t}=20 \mathrm{~s}$$ will be:

(Take g = 10 m/s²).

Sand is being dropped from a stationary dropper at a rate of $$0.5 \,\mathrm{kgs}^{-1}$$ on a conveyor belt moving with a velocity of $$5 \mathrm{~ms}^{-1}$$. The power needed to keep the belt moving with the same velocity will be :

A bag is gently dropped on a conveyor belt moving at a speed of $$2 \mathrm{~m} / \mathrm{s}$$. The coefficient of friction between the conveyor belt and bag is $$0.4$$. Initially, the bag slips on the belt before it stops due to friction. The distance travelled by the bag on the belt during slipping motion, is : [Take $$\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{-2}$$ ]

Two cylindrical vessels of equal cross-sectional area $$16 \mathrm{~cm}^{2}$$ contain water upto heights $$100 \mathrm{~cm}$$ and $$150 \mathrm{~cm}$$ respectively. The vessels are interconnected so that the water levels in them become equal. The work done by the force of gravity during the process, is [Take, density of water $$=10^{3} \mathrm{~kg} / \mathrm{m}^{3}$$ and $$\mathrm{g}=10 \mathrm{~ms}^{-2}$$ ] :

Two satellites $$\mathrm{A}$$ and $$\mathrm{B}$$, having masses in the ratio $$4: 3$$, are revolving in circular orbits of radii $$3 \mathrm{r}$$ and $$4 \mathrm{r}$$ respectively around the earth. The ratio of total mechanical energy of $$\mathrm{A}$$ to $$\mathrm{B}$$ is :

If $$K_{1}$$ and $$K_{2}$$ are the thermal conductivities, $$L_{1}$$ and $$L_{2}$$ are the lengths and $$A_{1}$$ and $$A_{2}$$ are the cross sectional areas of steel and copper rods respectively such that $$\frac{K_{2}}{K_{1}}=9, \frac{A_{1}}{A_{2}}=2, \frac{L_{1}}{L_{2}}=2$$. Then, for the arrangement as shown in the figure, the value of temperature $$\mathrm{T}$$ of the steel - copper junction in the steady state will be:

Read the following statements :

A. When small temperature difference between a liquid and its surrounding is doubled, the rate of loss of heat of the liquid becomes twice.

B. Two bodies $$P$$ and $$Q$$ having equal surface areas are maintained at temperature $$10^{\circ} \mathrm{C}$$ and $$20^{\circ} \mathrm{C}$$. The thermal radiation emitted in a given time by $$\mathrm{P}$$ and $$\mathrm{Q}$$ are in the ratio $$1: 1.15$$.

C. A Carnot Engine working between $$100 \mathrm{~K}$$ and $$400 \mathrm{~K}$$ has an efficiency of $$75 \%$$.

D. When small temperature difference between a liquid and its surrounding is quadrupled, the rate of loss of heat of the liquid becomes twice.

Choose the correct answer from the options given below :

Same gas is filled in two vessels of the same volume at the same temperature. If the ratio of the number of molecules is $$1: 4$$, then

A. The r.m.s. velocity of gas molecules in two vessels will be the same.

B. The ratio of pressure in these vessels will be $$1: 4$$.

C. The ratio of pressure will be $$1: 1$$.

D. The r.m.s. velocity of gas molecules in two vessels will be in the ratio of $$1: 4$$.

Choose the correct answer from the options given below :

Two identical positive charges $$Q$$ each are fixed at a distance of '2a' apart from each other. Another point charge $$q_{0}$$ with mass 'm' is placed at midpoint between two fixed charges. For a small displacement along the line joining the fixed charges, the charge $$\mathrm{q}_{0}$$ executes $$\mathrm{SHM}$$. The time period of oscillation of charge $$\mathrm{q}_{0}$$ will be :

Two sources of equal emfs are connected in series. This combination is connected to an external resistance R. The internal resistances of the two sources are $$r_{1}$$ and $$r_{2}$$ $$\left(r_{1}>r_{2}\right)$$. If the potential difference across the source of internal resistance $$r_{1}$$ is zero, then the value of R will be :

A direct current of $$4 \mathrm{~A}$$ and an alternating current of peak value $$4 \mathrm{~A}$$ flow through resistance of $$3\, \Omega$$ and $$2\,\Omega$$ respectively. The ratio of heat produced in the two resistances in same interval of time will be :

A beam of light travelling along $$X$$-axis is described by the electric field $$E_{y}=900 \sin \omega(\mathrm{t}-x / c)$$. The ratio of electric force to magnetic force on a charge $$\mathrm{q}$$ moving along $$Y$$-axis with a speed of $$3 \times 10^{7} \mathrm{~ms}^{-1}$$ will be :

(Given speed of light $$=3 \times 10^{8} \mathrm{~ms}^{-1}$$)

A microscope was initially placed in air (refractive index 1). It is then immersed in oil (refractive index 2). For a light whose wavelength in air is $$\lambda$$, calculate the change of microscope's resolving power due to oil and choose the correct option.

An electron (mass $$\mathrm{m}$$) with an initial velocity $$\vec{v}=v_{0} \hat{i}\left(v_{0}>0\right)$$ is moving in an electric field $$\vec{E}=-E_{0} \hat{i}\left(E_{0}>0\right)$$ where $$E_{0}$$ is constant. If at $$\mathrm{t}=0$$ de Broglie wavelength is $$\lambda_{0}=\frac{h}{m v_{0}}$$, then its de Broglie wavelength after time t is given by

A logic gate circuit has two inputs A and B and output Y. The voltage waveforms of A, B and Y are shown below.

The logic gate circuit is :

In a meter bridge experiment, for measuring unknown resistance 'S', the null point is obtained at a distance $$30 \mathrm{~cm}$$ from the left side as shown at point D. If R is $$5.6$$ $$\mathrm{k} \Omega$$, then the value of unknown resistance 'S' will be __________ $$\Omega$$.

The one division of main scale of Vernier callipers reads $$1 \mathrm{~mm}$$ and 10 divisions of Vernier scale is equal to the 9 divisions on main scale. When the two jaws of the instrument touch each other, the zero of the Vernier lies to the right of zero of the main scale and its fourth division coincides with a main scale division. When a spherical bob is tightly placed between the two jaws, the zero of the Vernier scale lies in between $$4.1 \mathrm{~cm}$$ and $$4.2 \mathrm{~cm}$$ and $$6^{\text {th }}$$ Vernier division coincides scale division. The diameter of the bob will be ____________ $$\times$$ 10^$$-$$2 cm.

Two beams of light having intensities I and 4I interfere to produce a fringe pattern on a screen. The phase difference between the two beams are $$\pi / 2$$ and $$\pi / 3$$ at points $$\mathrm{A}$$ and $$\mathrm{B}$$ respectively. The difference between the resultant intensities at the two points is $$x I$$. The value of $$x$$ will be ________.

To light, a $$50 \mathrm{~W}, 100 \mathrm{~V}$$ lamp is connected, in series with a capacitor of capacitance $$\frac{50}{\pi \sqrt{x}} \mu F$$, with $$200 \mathrm{~V}, 50 \mathrm{~Hz} \,\mathrm{AC}$$ source. The value of $$x$$ will be ___________.

A $$1 \mathrm{~m}$$ long copper wire carries a current of $$1 \mathrm{~A}$$. If the cross section of the wire is $$2.0 \mathrm{~mm}^{2}$$ and the resistivity of copper is $$1.7 \times 10^{-8}\, \Omega \mathrm{m}$$, the force experienced by moving electron in the wire is ____________ $$\times 10^{-23} \mathrm{~N}$$.

(charge on electorn $$=1.6 \times 10^{-19} \,\mathrm{C}$$)

A long cylindrical volume contains a uniformly distributed charge of density $$\rho \,\mathrm{Cm}^{-3}$$. The electric field inside the cylindrical volume at a distance $$x=\frac{2 \varepsilon_{0}}{\rho} \mathrm{m}$$ from its axis is ________ $$\mathrm{Vm}^{-1}$$.

A mass $$0.9 \mathrm{~kg}$$, attached to a horizontal spring, executes SHM with an amplitude $$\mathrm{A}_{1}$$. When this mass passes through its mean position, then a smaller mass of $$124 \mathrm{~g}$$ is placed over it and both masses move together with amplitude $$A_{2}$$. If the ratio $$\frac{A_{1}}{A_{2}}$$ is $$\frac{\alpha}{\alpha-1}$$, then the value of $$\alpha$$ will be ___________.

A square aluminum (shear modulus is $$25 \times 10^{9}\, \mathrm{Nm}^{-2}$$) slab of side $$60 \mathrm{~cm}$$ and thickness $$15 \mathrm{~cm}$$ is subjected to a shearing force (on its narrow face) of $$18.0 \times 10^{4}$$ $$\mathrm{N}$$. The lower edge is riveted to the floor. The displacement of the upper edge is ____________ $$\mu$$m.

A pulley of radius $$1.5 \mathrm{~m}$$ is rotated about its axis by a force $$F=\left(12 \mathrm{t}-3 \mathrm{t}^{2}\right) N$$ applied tangentially (while t is measured in seconds). If moment of inertia of the pulley about its axis of rotation is $$4.5 \mathrm{~kg} \mathrm{~m}^{2}$$, the number of rotations made by the pulley before its direction of motion is reversed, will be $$\frac{K}{\pi}$$. The value of K is ___________.

A ball of mass m is thrown vertically upward. Another ball of mass $$2 \mathrm{~m}$$ is thrown at an angle $$\theta$$ with the vertical. Both the balls stay in air for the same period of time. The ratio of the heights attained by the two balls respectively is $$\frac{1}{x}$$. The value of x is _____________.