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If $\int {\sin }^{13}x{\cos }^{3}xdx=A{\sin }^{14}x+B{\sin }^{16}x+C$, then $A+B=$_______.

Which of the following graphs correctly represent the process represented in the given $P-V$ graph?

When is an electric dipole in equilibrium in uniform electric field.

A massless platform is kept on a light elastic spring as shown in fig. When a particle of mass 0.1 kg is dropped on the pan from a height of 0.24 m, the particle strikes the pan, and the spring is compressed by 0.01 m. From what height should the particle be dropped to cause a compression of 0.04 m?

An ideal liquid of density $\rho$ is pushed with velocity v through the central limb of the tube shown in fig. What force does the liquid exert on the tube? The cross sectional areas of the three limbs are equal to A each. Assume stream-line flow.

.A capacitor is charged to potential V and is disconnected . A dielectric of dielectric
constant 4 is inserted filling the whole space between the plates. How do the following
change? –Capacitance, potential difference, Field between the plates, energy stored by
the capacitors.

A car is going towards North with a velocity 20m/s.after some time it starts to go towards south with velocity 20m/s Determine change in velocity of the car

A block of mass $10\text{kg}$ is slowly moved up on a smooth inclined plane of inclination $\theta$ by a person. Work done by the person to move the block through a distance of $2m$ is $120J$. If force is applied in horizontal direction. Find out inclination angle $\theta$. (Take $g=10m/s^2$)

A wooden cube of side 10 cm and specific gravity 0.8 floats in water with its upper surface horizontal. What depth of the cube remains immersed? what mass of aluminum of specific gravity 2.7 must be attached to

(i)The upper surface

(ii) The lower surface so that the cube will be just immersed

A hammer of mass $1\text{kg}$ having a speed of $50\text{m/s}$ hits an iron nail of mass $200\text{gm}$. If the specific heat of iron is $0.105{\text{cal/gm}}^{\circ }\text{C}$ and half the energy of the hammer is converted into heat, then the rise in the temperature of the nail is

A part of a long wire carrying a current I is bent into a circle of radius r as shown in Fig. The net magnetic field at the centre O of the corcular loop is

Two rods of different materials having coefficients of thermal expansion ${\alpha }_1$ and ${\alpha }_2$ and Young's modulii $Y_1$ and $Y_2$ are fixed between two rigid and massive walls. The rods are heated to the same temperature. If there is no bending of the rods, the thermal stresses developed in them are equal provided

A body of mass $m_1$ moving at a constant speed undergoes an elastic collision with a body of mass $m_2$ initially at rest., the ratio of the kinetic energies of masses $m_2$ and $m_1$ after the collision is

Two point charges of + 10 microcoulomb and + 40 microcoulomb respectively are placed 12 CM apart find the position of the point where electric field is zero

A tunnel has been dug through the centre of the earth and a ball is released in it. It will reach the other end of the tunnel after

A vertical column 50 cm long at ${50}^{o}C$ balances another column of same liquid 60 cm long at ${100}^{o}C$. The coefficient of absolute expansion of the liquid is

A particle moving with a uniform acceleration along a straight line covers distances $a$ and $b$ in successive intervals of $p$ and $q$ second. The acceleration of the particle is

A black body is at a temperature of 2880 K. The energy of radiation emitted by this body between wave lengths 499 nm and 500 nm is $U_1$, between 999 nm and 1000 nm is $U_2$ and between 1499 nm and 1500 nm is $U_3$. The Wien’s constant b = 2.88 $\times$ 106 nm K. Then

In a constant volume gas thermometer, the pressure of the working gas is measured by the difference in the levels of mercury in the two arms of a U-tube connected to the gas at one end. When the bulb is placed at the room temperature 27.0℃, the mercury column in the arm open to atmosphere stands 5.00 cms above the level of mercury in the other arm. When the bulb is placed in a hot liquid, the difference of mercury levels becomes 45.0 cms. Calculate the temperature of the liquid. (Atmospheric pressure = 75.0 cm of mercury.)

A particle is moving along the positive $x$-axis and at $t=0$, the particle is at $x=0$. The acceleration of the particle is a function of time. The acceleration at any time $t$ is given by $a=2(1–[t\left]\right)$ where $\left[t\right]$ is the greatest integer function. Assuming that the particle is at rest initially, the displacement of the particle in $4s$ is

Energy of a quanta of frequency ${10}^{15}Hz$ and $h=6.6\times {10}^{-34}J-sec$ will be

A photon of wavelength $4400\stackrel{\circ }{A}$ is passing through vacuum. The effective mass and momentum of the photon are respectively

Blocks $A$ and $B$ of mass $m$ are placed inside an elevator which is going upward with an acceleration of $2{\text{m/s}}^2$ and a convex mirror having focal length $12\text{cm}$ is placed below block $B$. All the surfaces are smooth and the pulley is light. The mass pulley system is released from rest with respect to the elevator at time $t=0\text{s}$, when the distance of the block $B$ from the mirror is $42\text{cm}$. Find the distance between the image of the block $B$ and mirror at $t=0.2\text{s}$.

Two rods A and B of different materials are welded together as shown in figure. Their thermal conductivities are $K_1$ and $K_2$. The thermal conductivity of the composite rod will be

A string of length L is stretched by $\frac{L}{20}$ and speed of transverse wave along it is v. The speed of wave when it is stretched by $\frac{L}{10}$ will be (assume that Hooke’s law is applicable)

In an experiment to find the loss of energy w.r.t. time in simple pendulum the correct graph between $(Amplitude{)}^2$ and time is :

A particle of mass m is executing SHM about the origin on x-axis with frequency $\sqrt{\frac{ka}{\pi m}}$, where k is a constant and a is the amplitude. Find its potential energy, if x is the displacement at time t:

The first excitation potential of an atom is V. The ionization potential of the atom $V_1$ is

A balloon mass M is descending at a constant acceleration $\alpha$. When a mass m is released from the balloon. It starts rising with the same acceleration $\alpha$. What is the value of m?

Two masses $m_1$ and $m_2$ are connected by a massless spring of constant $k$ and unstretched length $L$. The masses are placed on a frictionless straight channel, which is considered as x -axis. They are initially at rest at $x=0$ and $x=L$ respectively. At $t=0$, a velocity of $v_0$ is suddenly imparted to the first particle. At a later time $t_0$, the centre of mass of the two masses is at $x=$

Find the potential difference between $A$ and $B$

A vector $\overrightarrow{A}=\hat{i}+\sqrt{3}\hat{j}$ is rotated clockwise through an angle ${30}^{\circ }$ about one of its ends. The unit vector along the new vector is

Starting from rest, a body slides down a ${45}^{\circ }$ rough inclined plane in twice the time it takes to slide down the same distance in the absence of friction (with same inclination). The coefficient of friction between the body and the inclined plane is

A body is moving with velocity $30\text{m/s}$. A variable force $a\text{N}$ is applied at $t=0\text{s}$ on the body such that it increases it's velocity to $90\text{m/s}$ at $t=4\text{s}$ in same direction. The variation of force with time is shown in graph.

If mass of body is $10kg$, find the value of $a$ (shown in graph)

An insulating rod of length $l$ carries a charge $q$ distributed uniformly on it. The rod is pivoted at its mid-point and is rotated at a frequency $f$ about a fixed axis perpendicular to the rod and passing through the pivot. The magnetic moment of the rod system is

An eye specialist prescribes spectacles having combination of convex lens of focal length $40\text{cm}$ in contact with a concave lens of focal length $25\text{cm}$. The power of this lens combination in diopters is

$\begin{array}{cc}ColumnI& ColumnII\\ \text{(A) If v (frequency) is increased keeping}\text{I (intensity) and}\Phi \text{(work function) constant}& \text{(P) Stoping potential increases}\\ \left(B\right)\text{If I is increased keeping v and}\Phi \text{constant}& \text{(Q) Saturation photocurrent increases}\\ \text{(C) If the distance between anode and cathode increases.}& \text{(R) Maximum KE of the photoelectrons increase}\\ \text{(D) If}\Phi \text{is decreased keeping v and I constant}& \text{(S) Stopping potential remains the same}\end{array}$

An ideal monoatomic gas at temperature ${27}^{\circ }\text{C}$ and pressure $100\text{bar}$ occupies $5\text{L}$ volume. $5000\text{cal}$ of heat is added to the system without changing the volume. Calculate the change in temperature of the gas. (Given, $R=8.31\text{J/mol K}$)

Two thin long parallel wires separated by a distance b are carrying a current i amp each. The magnitude of the force per unit length exerted by one wire on the other is

The coefficient of volume expansion of glycerin is $49\times {10}^{-5}{{}^{\circ }\text{C}}^{-1}$. The fractional change in the density on a $30{}^{\circ }\text{C}$ rise in temperature is