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A person travels along a straight road for the first half of the time with a velocity $v_1$ and the second half time with a velocity $v_2$. Then the mean velocity $\overline{v}$ is given by

Acceleration of the particle in a rectilinear motion is given as $a=4t+3$ in $\left(m/s^2\right)$. The final position of the particle as a function of time $t$ is

(Given, at $t=0,u=3m/s,x=2m$)

Water $(\rho =1000{\text{kg/m}}^3)$ and kerosene $(\rho =800{\text{kg/m}}^3)$ are filled separately in two identical cylindrical vessels having small heights. Both vessels have small holes at their bottom. The speed of the water and kerosene coming out of their holes are $v_1$ and $v_2$ respectively. Select the correct option.

A source of sound S has a frequency $f$. Wind is blowing from source to observer O with velocity $u$. If speed of sound with respect to air is $C$, the wavelength of sound detected by O is:

A light string fixed at one end to a clamp on the ground passes over a fixed pulley and hangs at the other side. It makes an angle of ${30}^{\circ }$ with the ground. A monkey of mass $5\text{kg}$ climbs up the rope. The clamp can tolerate a vertical force of $40\text{N}$ only. The maximum acceleration in upward direction with which the monkey can climb safely is (neglect friction and take $g=10{\text{m/s}}^2$)

What is the reading of spring balance $D$ ?

(All the springs are ideal and identical)

Between the plates of a parallel plate capacitor a dielectric plate is introduced just to fill the space between the plates. The capacitor is charged and later disconnected from the battery. The dielectric plate is slowly drawn out of the capacitor parallel to the plates. The plot of the potential difference across the plates and the length of the dielectric plate drawn out is

A block $m_2$ is loaded onto another block $m_1$ placed on a spring of stiffness $k$ as shown in the figure. If block $m_2$ is suddenly removed, then magnitude of acceleration of block $m_1$ just after the removal of $m_2$ will be:

An ideal gas has a molar heat capacity $C_v$ at constant volume. It undergoes a process given by $T=T_0{e}^{\frac{\alpha }{V}}$, where $T_0$ and $\alpha$ are constants. Find the molar heat capacity of the gas as a function of $V$.

By photoelectric effect, Einstein, proved

In planetary motion the areal velocity of position vector of a planet depends on angular velocity $\left(\omega \right)$ and the distance of the planet from sun (r). If so the correct relation for areal velocity is

A body of mass 5 kg is moving with a momentum of 10 kg-m/s.A force of 0.2 N acts on it in the direction of motion of the body for 10 seconds. The increase in its kinetic energy is

Magnitude of acceleration of the two blocks is :

(Radius of pulley is $1\text{m}$)

What is the potential difference between points B and D in the given circuit.

A 20 - kg block attached to a spring of spring constant 5 N $m^{-1}$ is released from rest at A. The spring at this instant is having an elongation of 1 m. The block is allowed to move in smooth horizontal slot with the help of a constant force 50 N in the rope as shown. The velocity of the block as it reaches B is (assume the rope to be light)

A concrete sphere of radius R has a cavity of radius r which is packed with sawdust. The specific gravities of concrete and sawdust are respectively 2.4 and 0.3. For this sphere to float with its entire volume submerged under water. Ratio of mass of concrete to mass of sawdust will be

The vibrations of four air columns under identical conditions are represented in the figure below. The ratio of frequencies $f_p:f_q:f_r:f_s$ will be

A circular disc of radius $1\text{m}$ is rotating about an axis passing through its $COM$ and perpendicular to its plane. The moment of inertia about this axis is given by $I_{com}={\text{2 kg-m}}^2$. If the disc is rotated about an axis along its diameter, then the moment of inertia along that axis is $I_{dia}$. The value of $(I_{com}-I_{dia})$ is equal to

Two rods of equal length $\left(L\right)$ and equal mass $\left(M\right)$ are kept along $x$ and $y$ axis respectively such that their centres of mass lie at the origin. The moment of inertia about the line $y=x$, is

A spherical body having radius $0.5\text{m}$ is transfering heat to its surrounding through convection. Find the temperature difference between surrounding and the body surface, if the rate of heat transfer is $1.57\times {10}^5\text{W}$.

(Convection coefficient,$h_c=1570{\text{W/m}}^2\text{-K}$)

In an elastic solid, the work done by an external force to produce strain is $800\text{J}$. If the external force is removed, then the elastic potential energy of the solid will be -

According to Joule's law of heating, heat produced $H=I^{2}Rt,$ where $I$ is current, $R$ is resistance and $t$ is time. If the errors in the measurements of $I$, $R$ and $t$ are 3%, 4% and 6% respectively, then error in the measurement of H is

Three particles start simultaneously from a point on a horizontal smooth plane. First particle moves with speed $v_1$ towards east, second particle moves towards north with speed $v_2$ and third-one moves towards north east. The velocity of the third particle, so that the three always lie on a line, is

A particle starts from rest. Its acceleration $\left(A\right)$ versus time $\left(t\right)$ is as shown in the figure. The maximum speed of the particle will be

The variation of acceleration due to gravity $g$ with distance $d$ from the center of the earth is best represented by ($R=$ Earth’s radius):

Two blocks of masses $m_1$ and $m_2$ are joined by a wire of Young’s modulus Y via a massless pulley. The area of cross-section of the wire is S and its length is L. When the system is released, increase in length of the wire is

A clock with a metal pendulum beating seconds keeps correct time at $0^{\circ }C$. If it loses 12.5 s a day at ${25}^{\circ }C$, the coefficient of linear expansion of metal of pendulum is

An ideal voltmeter is connected in the figure. The current in circuit is

The Martians use force(F) , acceleration (A) and time (T) as their fundamental physical quantities. The dimensions of length on Martians system are

Consider the following data for a vernier calliper:

10 main scale division = 1 cm.

10 vernier division matches with 9 Main scale division.



While measuring the zero error of the vernier we see that zero of vernier scale is to the right of the zero marking of the main scale with $6^{th}$ vernier division coinciding with main scale divisions.



The actual reading for length measurement is 4.3 cm with 2nd vernier division coinciding with main scale graduation.



Estimate the length.

The brakes applied to a bike produce retardation of $5{\text{m/s}}^2$. If the bike takes $4\text{s}$ to stop after the application of brakes, calculate the distance it travels during this time.

What minimum angular velocity should be given to the rod to complete the circle ?

A diver makes $4\text{revolutions}$ on the way from a $4.9\text{m}$ platform above the water surface. Assuming zero initial vertical velocity, calculate the average angular velocity of the diver. [Take $g=9.8m/s^2$]

A coil of resistance $300\Omega$ and inductance $1.0H$ is connected across an alternating voltage of peak value $300V$ of frequency $\frac{300}{2\pi }Hz.$ Calculate the peak potential difference across inductor.

In the first experiment, two identical conducting rods are joined one after the other and this combination is connected to two vessels, one containing water at ${100}^{\circ }\text{C}$ and the other containing ice at $0^{\circ }\text{C}$ (see Fig). In the second experiment, the two rods are placed one on top of the other and ends are connected to the same vessels. If $q_1$ and $q_2$ (in gram per second) are the respective rates of melting of ice in the two cases, then the ratio $\frac{q_1}{q_2}$ is

If $\frac{d\tan x}{dx}={\sec }^{2}x$, then $\int \frac{2}{1-{\sin }^{2}x}dx$ is equal to:

A block of mass $M$ is pulled along a horizontal surface by applying a force at an angle $\theta$ with the horizontal. The friction coefficient between the block and the surface is $\mu$. If the block travels at a uniform velocity, then the work done by this applied force during a displacement $d$ of the block is

A car is moving on a circular road of radius of curvature $500\text{m}$. If the coefficient of friction is $0.8$ and $g=10{\text{m/s}}^2$, the maximum speed of the car can have without slipping is in $\left(\text{km/hr}\right)$.

A scooter accelerates from rest for time $t_1$ at a constant rate ${\alpha }_1$ and then retards at constant rate ${\alpha }_2$ for time $t_2$ and comes to rest. The correct value of $\frac{t_1}{t_2}$ will be -

The bob of a pendulum at rest is given a hit to impart a horizontal velocity $\sqrt{gl}\text{m/s}$ at the bottommost position of the bob, where $l$ is the length of the string in $\text{metres}$. Find the tangential acceleration at the point where the velocity of the bob is zero. Take $g=10{\text{m/s}}^2$

An observer moves towards a stationary source of sound with a speed (1/5)th of the speed of sound. The wavelength and frequency of the source emitted are λ and f, respectively. The apparent frequency and wavelength recorded by the observer are, respectively,

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

A harmonic source $\left(S\right)$ is driving a taut string. The other end of the string is tied to a wall that is not so rigid. It is observed that standing waves are formed in the string with ratio of amplitudes at the antinodes to that at the nodes equal to $8$. What percentage of wave energy is transmitted to the wall?

A heat flux of $4000\text{J/s}$ is to be passed through a copper rod of length $10\text{cm}$ and area of cross section $100{\text{cm}}^2$. The thermal conductivity of copper is $400{\text{W/m}}^{\circ }C$. The two ends of the rod must be kept at temperature difference of

Find the least count of a vernier caliper whose 50 vernier scale divisions match with 49 main scale divisions?

Given: least count of main scale is 1 mm.