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A bead of mass ‘m’ is located on a parabolic wire with its axis vertical and vertex at the origin as shown in figure and whose equation is $x^2$ =4ay. The wire frame is fixed and the bead can slide on it without friction. The bead is released from the point $y=4a$ on the wire frame from rest. The tangential acceleration of the bead when it reaches the position given by $y=a$ is (Acceleration due to gravity is g)

One end of a spring of length $\text{1}0\text{m}$ is attached to a plate of mass $\text{5}\text{kg}$ which is rotated along a circular path with a linear velocity of $\text{1}\text{m/s}$. If the spring gets extended by a length of $\text{0}.025\text{m}$, find the spring constant.

For a particle moving on a circular path of radius $5cm$, the velocity at an instant ‘$t$’ is $3\hat{i}+4\hat{j}m/s$. Assuming the particle to be in uniform circular motion, the value of centripetal acceleration is

A man of mass $M$ stands at one end of a plank of length $L$ which is at rest on a frictionless horizontal surface. The man walks to the other end of the plank. If mass of the plank is $\frac{M}{3}$, the distance that the man moves relative to the ground is

A cylindrical wire of radius R is carrying current i uniformly distributed over its cross-section. If a circular loop of radius 'r' is taken as an amperian loop, then the variation of $\oint \overrightarrow{B}.\overrightarrow{dl}$ (along y-axis) over this loop, with radius ' r’ of loop will be best represented by:

In a meter – bridge circuit as shown, when one more resistance of $100\Omega$ is connected is parallel with unknown resistance ‘x’, then ratio $\frac{{\ell }_1}{{\ell }_2}$ becomes $‘2^{\prime }.{\ell }_1$ is balance length. AB is a uniform wire. Then value of ‘x’ must be

Two identical thin plano–convex glass lenses(refractive index 1.5) each having radius of curvature of 20 cm are placed with their convex surfaces in contact at the centre. The intervening space is filled with oil of refractive index 1.7.The focal length of the combination is :

A harmonic wave $y_i=0.002cos(2\pi x-50\pi t)$ travels along a string towards a boundary at x = 0, with a second string. The wave speed on the second string is $50m{s}^{-1}$. The expression for transmitted wave is (Assume SI units)

If the refracting angle of a prism is ${60}^{\circ }$ and minimum deviation is ${30}^{\circ }$, the angle of incidence is

The coefficient of static friction between the two blocks shown in the figure is ‘$\mu$’ and the table is smooth. The maximum value of ‘F’ so that both blocks moves together, is

The equation of the line making ${37}^{\circ }$ with $x-axis$ and passing through $(2,-4)$ is
[Hint:- $\tan {37}^{\circ }=\frac{3}{4}$]

Rocky made $12\text{kg}$ of trail mix for his family's hiking trip. His family ate $8600\text{g}$ of the trail mix on the hiking trip. How many grams of trail mix did Rocky have left?

One end of a steel wire is fixed to the ceiling of an elevator moving up with an acceleration $2{\text{m/s}}^2$ and a load of $10\text{kg}$ hangs from other end. Area of cross -section of the wire is $2{\text{cm}}^2$. The longitudinal strain in the wire is $(g=10{\text{m/s}}^2$ and $Y=2\times {10}^{11}{\text{Nm}}^{-2})$

The position vector of a projectile is given by $\overrightarrow{r}=3t\hat{i}+(4t-5t^2)\hat{j}$. Find the ratio of maximum height and initial speed.
($g=10{\text{m/s}}^2$)

A galvanometer of resistance $20\Omega$ is to be converted into an ammeter of range $1A$. If a current of $1mA$ produces full-scale deflection, the shunt required for the purpose is

A linear aperture whose width is $0.02cm$ is placed immediately in front of a lens of focal length $60cm$. The aperture is illuminated normally by a parallel beam of wavelength $5\times {10}^{-5}cm$. The distance of the first dark band of the diffraction pattern from the centre of the screen is

A soap bubble, having radius of $2\text{mm}$, is blown from a detergent solution having a surface tension of $5\times {10}^{-2}\text{N/m}$. Pressure inside the soap bubble is observed to be the same as the pressure at a point $h$ below the free surface of water taken in a container. Find the value of $h$.
[Take $g=10{\text{m/s}}^2$]

A photocell stops emission if it is maintained at 2V negative potential. The energy of most energetic photoelectron is

Charge ‘q’ is uniformly distributed over the surface of an annular-non-conducting disc of inner radius $R_1$ and outer radius $R_2$. The disc is made to rotate about an axis passing through its centre and perpendicular to its plane with a constant frequency v (rotations per second). Magnetic moment of the disc can be expressed as :

Two cylinders of same cross section area and length L but made of two material of densities $d_1$ and $d_2$ are connected together to form a cylinder of length 2L. The combination floats in a liquid of density d with a length $\frac{L}{2}$ above the surface of the liquid. If $d_1>d_2$ then

A body initially moving with a velocity of $4{\text{ms}}^{–1}$, attains a velocity of $20{\text{ms}}^{–1}$ in $4s$ . The acceleration of the body is:
(Assume constant acceleration)

Consider a car moving on a straight road with a speed of $100m{s}^{-1}$. The distance at which car can be stopped is $[{\mu }_k=0.5]$

In a historical experiment to determine Planck's constant, a metal surface was irradiated with light of different wavelengths. The emitted photoelectric energies were measured by applying a stopping potential. The relevant data for the wavelength ($\lambda$) of incident light at the corresponding stopping potential ($V_0$) are given below:

If $y=e^{2^x}-\cos \left(\ln x\right)$, then $\frac{dy}{dx}$ is

A particle has a rectilinear motion and the figure gives its displacement as a function of time.

Which of the following statement(s) is/are true with respect to the motion ?

An ac source of angular frequency $\omega$ is fed across a resistor R and a capacitor C in series. The current registered is I. If now the frequency of source is changed to $\frac{\omega }{3}$ (but maintaining the same voltage), the current in the circuit is found to be halved. Then the ratio of reactance to resistance at the original frequency \omega is

The total current supplied to the circuit by the battery is

A square platform of side length 8 $m$ is situated in x-z plane such that it is at 16 $m$ from the x-axis and 8 $m$ from the z-axis as shown in figure. A particle is projected from origin with velocity $\overrightarrow{v}=v_2\hat{i}+25\hat{j}m/s$ relative to wind which is blowing with velocity $v_1\hat{k}$. And at the same instant the platform starts with acceleration $\overrightarrow{a}=2\hat{i}+2.5\hat{j}m/s^2$. [Take $g=10m/s^2$]

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(P)}& \text{Least possible values of}{v}_2\text{(in}m/s\text{) so that}& \text{(1)}& 4\\ & \text{particle hits the platform or edge of platform is}& & \\ \text{(Q)}& \text{Least possible value of}{v}_1\text{(in}m/s\text{) so that particle hits}& \text{(2)}& 6\\ & \text{the platform or edge of platform is}& & \\ \text{(R)}& \text{If}t\text{is the time (in second) after particle hits the platform}& \text{(3)}& 8\\ & \text{then}2t\text{is equal to}& & \\ \text{(S)}& \text{Value of displacement with respect to ground (in}m\text{) of the}& \text{(4)}& 20\\ & \text{particle in y - direction, when}{v}_2\text{has its minimum possible}& & \\ & \text{value is (till particle hits the platform or edge of platform)}& & \end{array}$

The ball is thrown vertically up (taken $+z-$ axis) from the ground. The correct momentum-height $(p-h)$ diagram is:

Due to the flow of current in a circular loop of radius $R$, the magnetic field produced at the centre of the loop is $B$. The magnetic moment of the loop is:-

Two objects of masses $2\text{kg}$ and $3\text{kg}$ are dropped from a tower of height of $60\text{m}$. The ratio of their kinetic energy when they reach the ground is

Two ice skaters A and B approach each other at right angles. Skater A has a mass $30\text{kg}$ and velocity $1\text{m/s}$ skater B has a mass $20\text{kg}$ and velocity $2\text{m/s}$. They meet and cling together. The final velocity of the couple is

Blocks B and C are connected by a single inextensible cable, with this cable being wrapped around pulleys at D and E. In addition, the cable is wrapped around a pulley attached to block A as shown. Assume the radii of the pulleys to be small. Blocks B and C move downward with speeds of $V_B=6ft/s$ and $V_C=18ft/s,$ respectively. Determine the velocity of block A when $S_A=4ft.$

A stone falls from a balloon that is descending at a uniform rate of $12\text{m/s}$. The displacement of the stone from the point of release after $10\text{sec}$ is (take $g=9.8m/s^2$)

A thin ring of radius $r$ carries a uniformly distributed charge. The ring rotates at a constant angular speed of $n$ revolutions per second about an axis passing through centre and perpendicular to its plane. If $B$ is the magnitude of magnetic field at the centre of ring, then the charge carried by the ring is

The height of a mercury barometer is $75\text{cm}$ at sea level and $50\text{cm}$ at the top of a hill. Ratio of density of mercury to that of air is ${10}^4$. The height of the hill is

What will happen to the rise of liquid in a capillary tube if its top end is closed?

At a distance of $10\text{m}$ from a point source, loudness is found to be $20\text{dB}$. The distance from the source where loudness is $0\text{dB}$ is

A ball is moving with an initial velocity $30\hat{i}\text{m/s}$ towards a wall oriented at an angle such that the velocity of the ball after collision with the wall becomes $-30\hat{j}\text{m/s}$. If mass of the ball is $0.5\text{kg}$, then the magnitude of the impulse imparted by the wall is

A line makes the same angle $\theta$ with each of the $x$ and $z$ axes. If the angle $\beta$ which it makes with $y$-axis, is such that ${\sin }^2\beta =3{\sin }^2\theta$, then ${\cos }^2\theta$ is-

A current I flows in circular arc of wire which subtends an angle ${\theta }^{\circ }$ at the centre. If the radius of the circle is r then the magnetic field B at center is:

A particle is suspended from a fixed point by a string of length $10\text{m}$. The particle is given a horizontal velocity at the lowest point. The string slacks after the particle have reached a height of $16\text{m}$ above the lowest point. What is the velocity of the particle just before the string slacks?

Show that the four points A, B, C and D with position vectors $4\hat{i}+5\hat{j}+\hat{k},-\hat{j}-\hat{k},3\hat{i}+9\hat{j}+4\hat{k}$ and $4(-\hat{i}+\hat{j}+\hat{k})$ respectively are coplannar.

OR

The scalar product of the vector $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$ with a unit vector along the sum of vector $\overrightarrow{b}=2\hat{i}+4\hat{j}+5\hat{k}$ and $\overrightarrow{c}=\lambda \hat{i}+2\hat{j}+3\hat{k}$ is equal to one. Find the value of $\lambda$ and hence find the unit vector along $\overrightarrow{b}+\overrightarrow{c}$.

If $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$, then show that $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{c}\times \overrightarrow{a}$. Interpret the result geometrically.

A car is travelling on a track which has radius of curvature $50\text{m}$. What is the maximum safe speed with which the car can travel on this track? Take $g=10{\text{m/s}}^2$. Assume coefficient of friction to be 0.8.