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Find the time period of the oscillation of mass m in figures.

(i) $2\pi \sqrt{\frac{m}{k_1+k_2}}$

(ii) $2\pi \sqrt{\frac{m(k_1+k_2)}{k_1{k}_2}}$

(iii) $2\pi \sqrt{\frac{m(k_1-k_2)}{k_1{k}_2}}$

An equilateral triangular loop $\text{ABC}$ having some resistance, is pulled with a constant velocity $v$ out of a uniform magnetic field $B$, directed into the paper. At time $t=0$, side $BC$ of the loop is at the edge of the magnetic field. The induced current $\left(i\right)$ versus time $\left(t\right)$ graph will be -

Consider a uniform square plate fo side a and mass M. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is

The resistance in the left and right gaps of a balanced meter bridge are $R_1$, $R_2$. The balance point is at 50 cm. If a resistance of $24\Omega$ is connected in parallel to $R_2$the balance point is at 70 cm. The value of $R_1$ is

A block of mass $2kg$ is hanging with two identical massless springs as shown in figure. The acceleration of the block at the moment the right spring breaks is $(g=10m/s^2)$

1. 200

A large conducting sphere of radius $r$, having a charge $Q$ is placed in contact with a small neutral conducting sphere of radius $r^{\prime }$ and is then separated. The charge on smaller sphere will now be:

In the previous problem find the maximum intensity $E_{max}$ and the corresponding distance $r_m$

The electric potential between a proton and an electron is given by $V=V_0\ln \frac{r}{r_0},$ where $r_0$ is a constant. Assuming Bohr's model to be applicable, Find the relation between $r_n$ and $n.$ where, $n$ being the principal quantum number.

In a given scale $X$, the ice point is $40{}^{\circ }\text{X}$ and the steam point is $120{}^{\circ }\text{X}$. For another scale $Y$, the ice point and steam point are $-30{}^{\circ }\text{Y}$ and $130{}^{\circ }\text{Y}$ respectively. If $X$ reads $50{}^{\circ }\text{X}$ temperature of a body, then $Y$ would read

Two slits in Young's experiment have widths in the ratio 1 : 25. The ratio of intensity at the maxima and minima in the interference pattern, $\frac{I_{max}}{I_{min}}$ is

For a simple pendulum, a graph is plotted between its kinetinc energy $\left(KE\right)$ and potential energy $\left(PE\right)$ against its displacement $x$. Which one of the following represents these correctly?

(graphs are schematic and not drawn to scale)

A current of $2\text{A}$ is flowing through a circular coil of radius $1\text{m}$.



Find the magnitude of magnetic field at a point on its axial line which is at $\sqrt{3}\text{m}$ distance away from its center.



​​​​​​​$[$1 Mark$]$

When an ideal diatomic gas is heated at constant pressure, the fraction of the heat energy supplied which increases the internal energy of the gas, is

In the given figure, the block displaces down to the equilibrium position by an amount (K, 2K and 3K represents stiffness of the respective springs)

One end of a uniform wire of length $L$ and of weight $W$ is attached rigidly to a point in the roof and a weight $W_1$ is suspended from its lower end. If $S$ is the area of cross section of the wire, the stress in the wire at a height $3L/4$ from its lower end is

If $\alpha \text{and}\beta$ are the roots of $x^2-5x+6=0$ then $\frac{1}{\alpha }+\frac{1}{\beta }$ is

A transverse wave is described by the equation $Y=y_{0}sin2\pi (ft-\frac{x}{\lambda })$. The maximum particle velocity is four times the wave velocity if

A toy gun uses a spring of force constant K. Before being triggered in the upward direction, the spring is compressed by a distance x. If the mass of the shot is m, on being triggered, it will go up to a maximum height of

The period of oscillation of a simple pendulum of length $L$ suspended from the roof of a vehicle, which moves without friction down an inclined plane of inclination $\alpha$, is given by

A string of length $L$ is fixed at one end and carries a mass $M$ at the other end. The string makes $\frac{2}{\pi }$ revolutions per second around the vertical axis through the fixed end as shown in the figure, then tension in the string is

A particle moves according to the equation $\frac{dv}{dt}$ = $\alpha$ - $\beta$ v , where $\alpha$ and $\beta$ are constants. Find the velocity as a funtion of time. Assume body starts from rest.

In a triangle $ABC.I_1,I_2,I_3$ are excentres of triangle then the value of $I{I}_1\cdot I{I}_2\cdot I{I}_3$ is equal to:

A man can row a boat $4km/hr$ in still water. He is crossing a river where the current is $2km/hr$. If the width of the river is $4km$ , the time taken by him to reach a point directly opposite to him is

Two bodies of masses $2\text{kg}$ & $5\text{kg}$ have position vectors $\hat{i}-2\hat{j}+\hat{k}$ and $3\hat{i}+2\hat{j}-\hat{k}$ respectively. The centre of mass of the system has a position vector

The infinite sheets of uniform charge density $10\frac{\mu C}{m^2},-10\frac{\mu C}{m^2}and5\frac{\mu C}{m^2}$ are placed in vacuum as shown. The direction of field intensity at points A and B are respectively?

A hollow conducting sphere has inner radius $5\text{cm}$ and outer radius $8\text{cm}$, then find maximum possible potential of conductor, if breakdown electric field of air is $3\times {10}^6\text{V/m}$

Force acting on a body of mass 1 kg is related to its position as, F= $x^3$-3x newton. It is at rest at x=1m. Its velocity at x=3 m can be,

In space two point masses of mass $10kg$ each are fixed. The masses are separated by a distance $10m$. Another point mass of mass $1kg$ is to be projected from the point at midpoint of line joining the fixed masses such that it escapes to infinity. The minimum speed of the projection is