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A car is negotiating a curve of radius R = 20 m on a banked road with banking angle $\theta$ and coefficient of friction $\mu$. Take speed of car as v.

$\begin{array}{cccccc}& \text{Column I}& & \text{Column II}& & \text{Column III}\\ \left(I\right)& v=10\sqrt{2}m/s& \left(i\right)& \mu =0.2& \left(P\right)& \theta ={30}^{\circ }\\ \left(II\right)& v=20m/s& \left(ii\right)& \mu =0.4& \left(Q\right)& \theta ={45}^{\circ }\\ \left(III\right)& v=50m/s& \left(iii\right)& \mu =0.6& \left(R\right)& \theta ={37}^{\circ }\\ \left(IV\right)& v=5m/s& \left(iv\right)& \mu =0.8& \left(S\right)& \theta ={60}^{\circ }\end{array}$

In which of the following cases, sliding occurs

In the figure shown, assume the current in the wire to be 5 A and flowing from A to E and if B = 0.15T. Find the force on the section CD.
BC=16cm and CD =20cm.

Assuming that the junction diode is ideal in the circuit shown in fig, the current through the diode is

A steel scale measures the length of a copper rod as $L\text{cm}$ when both are at ${20}^{\circ }\text{C}$, the calibration temperature for the scale. If the coefficients of linear expansion for steel and copper are ${\alpha }_s$ and ${\alpha }_c$ respectively, what would be the scale reading (in cm) when both are at ${21}^{\circ }\text{C}$?

A diwali rocket is ejecting 0.05 kg of gases per second at a velocity of 400 m/sec. The accelerating force on the rocket is

A linear harmonic oscillator of force constant $2\times {10}^6\text{N/m}$ and amplitude $0.01\text{m}$ has a total mechanical energy of $160$ Joules. Which of the following is true?

A bullet is fired from a cannon with velocity 500 m/s. If the angle of projection is ${15}^{\circ }$ and $g=10m/s^2$. Then the range is

A particle moves under the effect of a force F = Cx from x = 0 to $x=x_1$ . The work done in the process is

A simple pendulum is made of a string of length $l$ and a bob of mass $m$, is released from a small angle ${\theta }_0$. It strikes a block of mass $M$, kept on a horizontal surface at its lowest point of oscillations, elastically. It bounces back and goes up to an angle ${\theta }_1$. Then, $M$ is given by

A particle is projected with a speed $u$ at an angle $\theta$ with horizontal from point $A$. It strikes elastically with a vertical wall at height $h/2$. It rebounds and reaches maximum height $h$ and falls back on the ground at point $B$ as shown in Figure. Distances from $A$ to wall and from wall to $B$ are $x_1$ and $x_2$, and time to cover is $t_1$ and $t_2$ respectively. Match the values in column I with the expressions in column II.


$$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{i.}\sqrt{2}& \text{a.}\frac{x_2-x_1}{x_2+x_1}\text{or}\frac{x_2+x_1}{x_2-x_1}\\ \text{ii.}\frac{1}{\sqrt{2}}& \text{b.}\frac{t_2-t_1}{t_2+t_1}\text{or}\frac{t_2+t_1}{t_2-t_1}\\ \text{iii.}1& \text{c.}\frac{u\sin \theta }{g(t_2+t_1)}\\ \text{iv.}\frac{1}{2}& \text{d.}\frac{u\cos \theta (t_1+t_2)}{x_1+x_2}\end{array}$$

A body $A$ is thrown vertically upward with the initial velocity $v_1$. Another body $B$ is dropped from a height $h$. Find how the distance $x$ between the bodies depends on the time $t$ if the body begin to move simultaneously.

The option(s) with the value of $a$ and $L$ that satisfy the following equation is(are)

$\frac{\underset{0}{\overset{4\pi }{\int }}{e}^t({\sin }^{6}at+{\cos }^{4}at)dt}{\underset{0}{\overset{\pi }{\int }}{e}^t({\sin }^{6}at+{\cos }^{4}at)dt}=L?$

A sphere of mass m rolls on a plane surface. Find its kinetic energy at an instant when its centre moves with speed v.

The separation between a node and the next antinode in a vibrating air column is 25 cm. If the speed of sound 1 in air is $340m{s}^{-1}$, find the frequency of vibration of the air column.

Two long strings A and B, each having linear mass density $1.2\times {10}^{-2}kg{m}^{-1},$ are stretched by different tensions 4.8 N and 7.5 N respectively and are kept parallel to each other with their left ends at x = 0. Wave pulses are produced on the strings at the left ends at t = 0 on string A and at t = 20 ms on string B. When and where will the pulse on B overtake that on A ?

For given half cell; $A{l}^{3+}+3e^{-}\to Al$: on increasing $\left[A{l}^{3+}\right]$, the electrode potential

Two identical blocks A and B, each of mass 'm' resting on smooth floor are connected by a light spring of natural length L and spring constant K, with the spring at its natural length. A third identical block 'C' (mass m) moving with a speed v along the line joining A and B collides with A. the maximum compression in the spring is

A 1.0 kg block collides with a horizontal light spring of force constant 2 $\frac{N}{m}$. The block compresses the spring 4m from the rest position. Assuming that the coefficient of kinetic friction between the block and the horizontal surface is 0.25, what was the speed of the block at the instant of collision?

Can you give some evidence to prove that health infrastructure has improved in India ?

A black body radiates heat energy at the rate of $2\times {10}^5{\text{J-s}}^{-1}{\text{m}}^{-2}$ at a temperature of ${127}^{\circ }\text{C}$. The temperature of black body, at which the rate of heat radiation is $32\times {10}^5{\text{J-s}}^{-1}{\text{m}}^{-2}$ is

Find the ratio of moment of inertia of a ring and a disc about an axis passing through their center and perpendicular to their plane. Both ring and disc have the same mass and same radius.

Suppose a tunnel is dug along a diameter of the earth. A particle is dropped from a point, a distance h directly above the tunnel. The motion of the particle as seen from the earth is

A mosquito net over a 7 ft \times 4 ft bed is 3 ft high. The net has a hole at one corner of the bed through which a mosquito enters the net. It flies and sits at the diagonally opposite upper corner of the net.

(a) Find the magnitude of the displacement of the mosquito.

(b) Taking the hole as the origin, the length. of the bed as the X-axis, its width as the Y-axis, and vertically up as the Z-axis, write the components of the displacement vector.

The force of buoyancy exerted by the atmosphere on a balloon is B in the upward direction and remains constant. The force of air resistance on the balloon acts opposite to the direction of velocity and is proportional to it. The balloon carries a mass M ans id found to fall down near the earth's surface with a constatnt velocity v. How much mass should be removed from the balloon so that it may rise with a constant velocity v ?

A 0.02 cm liquid column balances the excess pressure inside a soap bubble of radius 7.5 mm. Determine the density of the liquid. Surface tension of soap solution = 0.03 N m^-1.

A wooden plank of mass $20\text{kg}$ and length $10\text{m}$ is resting on a smooth horizontal floor. A man of mass $60\text{kg}$ starts moving from one end of the plank to the other end. Find the magnitude of displacement of the plank over the floor when the man reaches the other end of the plank.

For the arrangement shown in the figure, the force at the bottom of the vessel is [Take $g=10{\text{m/s}}^2$]

An ice block at $0^{\circ }\text{C}$ and of mass $m$ is dropped from height ${}^{\prime }{h}^{\prime }$ such that the loss in gravitational potential energy of the block is exactly equal to the heat required to completely melt the ice. Taking latent heat of fusion of ice, $L=80\text{cal/gm}$, acceleration due to gravity $g=10{\text{m/s}}^2$ and mechanical equivalent of heat $J=4.2\text{J/cal}$. find the value of ${}^{\prime }{h}^{\prime }$.

In the network shown below, points A, B and C are at the potential of 70 V, 0 V and 10 V respectively. The potential of point D is

The displacement of a particle of a string carrying a travelling wave is given by $y=3\sin \left[6.28\right(0.50x-50t\left)\right]$
$(x\to \text{in cm},y\to \text{cm},t\to \text{in second})$.
What would be the speed of the wave?

What should be the acceleration ${}^{\prime }{a}^{\prime }$ of the box shown in figure so that the block of mass $m$ exerts a force $\frac{mg}{4}$ on the floor of the box ? (where $g$ is acceleration due to gravity)

where does the center of mass of a two particle system lie when one particle is more massive than the other?

A car moves on a circular path of radius 10 metre. It complete 5 revolution in 5 minute. What is its average speed?

Calculate the area of the triangle determined by the two vectors $\overrightarrow{A}=3\hat{i}+4\hat{j}$ and $\overrightarrow{B}=-3\hat{i}+7\hat{j}$

Two cells of 1.25 V and 0.75V are connected in parallel. The effective voltage will be

In figure both the pulleys are massless and frictionless. A force $F$ (of any possible magnitude) is applied in horizontal direction. There is no friction between $M$ and the ground. ${\mu }_1$ and ${\mu }_2$ are coefficients of friction as shown between the blocks. Column I gives the different relations between ${\mu }_1$ and ${\mu }_2$ and Column II is regarding the motion of $M$. Match the columns.


$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{i. If}{\mu }_1={\mu }_2=0& \text{a. may accelerate towards right}\\ \text{ii. If}{\mu }_1={\mu }_2\ne 0& \text{b. may accelerate towards left}\\ \text{iii. If}{\mu }_1>{\mu }_2& \text{c. does not accelerate}\\ \text{iv. If}{\mu }_1<{\mu }_2& \text{d. may or may not accelerate}\end{array}$

What does an axiom mean?

A particle is projected at angle $\theta$ with horizontal, with velocity $v_0$ at $t=0$. Find the radius of curvature at $t=0$.

A block hangs from a string wrapped on a disc of radius $20\text{cm}$ free to rotate about its axis which is fixed in a horizontal position. If the angular speed of the disc is $10\text{rad/s}$ at some instant, with what speed is the block going down at that instant?

Choose the correct option for the given vector?

Find the angle of minimum deviation for an equilateral prism made of a material of refractive index 1.732. What is the angle of incidence for this deviation ?