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A block of mass $1\text{kg}$ is connected to a spring of spring constant ${\pi }^2\text{N/m}$ fixed at the other end and kept on smooth level ground. The block is pulled by a distance of $1\text{cm}$ from its natural length position and released. After what time does the block compress the spring by $\frac{1}{2}\text{cm}$?

A tunnel is dug along a diameter of the earth. If $M_e$ and $R_e$ are the mass and radius of the earth respectively , then the force on a particle of mass $m$ placed in the tunnel at a distance $r$ from the centre is :



[Assume , walls of the tunnel are frictionless]

A torch light kept at a depth h below the surface of an ocean (n = 4/3) facing upward is turned ON. Determine the illuminated surface area at the free surface of ocean.

Define the term ‘mobility’ of charge carriers. Write its S.I. unit

If the radii of $A^{+}$ and $B^{-}$ are $95pm$ and $181pm$ respectively, then the coordination number of $A^{+}$ will be?

1. 2982.6

If the potential energy of a spring is V on stretching it by 2 cm, then its potential energy when it is stretched by 10 cm will be

A rigid beam is hinged at one end and supported on linear elasitc spring (both having a stiffness of 'k') at points '1' and'2', and an inclined load acts at '2', as shown in figure









If the load P equals 100 kN, which of the follwoing options represents forces $R_{1}and{R}_2$ in springs at points 1 and 2?

Which of the following combination should be selected for better tuning of an LCR circuit used for communication?

The velocity-displacement graph for a car on a straight road is shown in figure


Determine the acceleration of the car at $s=150\text{m}$.

A solid sphere of density $\rho ={\rho }_0(1-\frac{r^2}{R^2}),0<r\le R$ (where $r$ is the distance from the centre of the sphere) just floats in a liquid. Then, density of the liquid is -

The wire shown in Fig. carries a current of 40 A. If r = 3.14 cm, the magnetic field at point P will be

A small coil C with N = 200 turns is mounted on one end of a balance beam and introduced between the poles of an electromagnet as shown in figure. The cross sectional area of coil is $A=1.0c{m}^2$, length of arm OA of the balance beam is l=30cm When there is no current in the coil the balance is in equilibrium. On passing a current l = 22 mA through the coil the equilibrium is restored by putting the additional counter weight of mass $\Delta m=60mg$ on the balance pan. Find the magnetic induction at the spot where coil is located.

Trajectories are shown in figure for three footballs. Initially vertical and horizontal velocity components are $u_y$ and $u_x$ respectively. Ignoring air resistance, choose the correct statements from List-II for the value of variable in List-I



$\begin{array}{cccc}& \text{List-I}& & \text{List-II}\\ \text{(I)}& \text{Time of flight}& \text{(P)}& \text{greater for A only}\\ \text{(II)}& \frac{u_y}{u_x}& \text{(Q)}& \text{greater for C only}\\ \text{(III)}& u_x& \text{(R)}& \text{Equal for A and B}\\ \text{(IV)}& u_x{u}_y& \text{(S)}& \text{Equal for B and C}\end{array}$

A variable air capacitor in a tuning circuit is made up of $N$ semicircular plates, each of radius $R$ and positioned at a distance $d$ from its neighbours, to which it is electrically connected as shown in figure, a second identical set of plates is enmeshed with its plates halfway between those of the first set. The second set can rotate as a unit. Determine the capacitance as a function of the angle of rotation $\theta$, where $\theta =0^{\circ }$ corresponds to the maximum capacitance.

Find the minimum and maximum value of the function $y=x^3-3x^2+6$. Find the values of y at which it occurs.

A particle is fired at an angle of ${30}^{\circ }$ to the horizontal such that the vertical component of its initial velocity is $80m/s$. Its time of flight is $T$. Its velocity at $t=\frac{T}{4}$ has a magnitude of nearly
(Take $g=10m/s^2$)

In a rigid body in plane motion, the point R is accelerating with respect to point P at $10\angle {180}^{o}m/s^2$. If the instantaneous acceleration of point Q is zero acceleration (in m\s${}^2$) of point R is

If $y=x^2+5x^{\frac{3}{2}}+\frac{2}{x},then\frac{dy}{dx}=$ ?

A mass $m$ moves in a circle on a smooth horizontal plane with velocity $v_0$ at a radius $R_0$. The mass is attached to a string which passes through a smooth hole in the plane as shown.


The tension in the string is increased gradually and finally $m$ moves in a
circle of radius $R_0/2$. Then, the final value of the kinetic energy of particle is

A particle of mass $m$ is projected at time $t=0$ from a point $P$ on the ground with a speed $v_0$, at an angle of ${45}^{\circ }$ to the horizontal. What is the magnitude of the angular momentum of the particle about $P$ at time $t=\frac{v_0}{g}$?