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A car is moving along a straight line. It's displacement (x) - time (t) graph is shown. Match the entries in column I with points on graph as in Column-II.

Column I

(p) x $\to$ negative,v $\to$ positive,a $\to$ positive (q) x $\to$ positive, v $\to$ negative,a $\to$ negative

(r) x $\to$ negative,v $\to$ negative,a $\to$ positive (s) x $\to$ positive, v $\to$ positive,a $\to$ negative.

Column II

An iceberg is floating in an ocean. What fraction of its volume is above the water? (Given: density of ice$=900{\text{kg/m}}^3$ and density of ocean water $=1030{\text{kg/m}}^3$)

A glass prism of refractive index 1.5 is immersed in water $(\mu =\frac{4}{3})$. A light beam incident normally on the face AB is totally reflected to reach the face BC if

Some parameters are given in Column - I which are to be matched with their correct answers given in Column - II.

Choose the correct option.

$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{a)}& \text{Velocity at}t=0\text{of a particle following}& \text{p)}& \alpha \\ & \text{equation}x=u(t-8)+\alpha (t-2{)}^2& & \\ \text{b)}& \text{Acceleration of a particle moving according to equation}& \text{q)}& 2\alpha \\ & x=4(t-5)+\alpha (t-2{)}^2\text{at}t=0& & \\ \text{c)}& \text{Velocity of the particle moving according to equation}& \text{r)}& u\\ & x=ut+5\alpha t^{2}ln(1+\frac{t}{2})+\alpha t^2\text{at}t=0& & \\ \text{d)}& \text{Acceleration of the particle moving according to the equation}& \text{s)}& u-4\alpha \\ & x=ut+5\alpha t^{2}ln(1+\frac{t}{2})+\alpha t^2\text{at}t=0& & \end{array}$

In the given figure, when temperature is increased then which of the following increases.

Four particles, each of mass $m$ and equidistant from each other, move along a circle of radius $r$ under the action of their mutual gravitational attraction. The speed of each particle is :

Two drops of equal radii coalesce to form a bigger drop. What is ratio of surface energy of bigger drop to smaller one?

In the figure, a ladder of mass m is shown leaning against a wall. It is in static equilibrium making an angle $\theta$ with the horizontal floor. The coefficient of friction between the wall and the ladder is $\mu$ and that between the floor and the ladder is ${\mu }_2.$ The normal reaction of the wall on the ladder is $N_1$ and that of the floor is $N_2.$ If the ladder is about to slip, then

A body of mass $2\text{kg}$ is travelling with uniform velocity from $(1,2,4)\text{m}$ to $(3,4,6)\text{m}$ in $4\text{seconds}$. The kinetic energy of the body is

Which of the following is/are necessary and sufficient condition(s) for pure translation?

A 250-turn rectangular coil of length $2.1cm$ and width $1.25cm$ carries a current of 85 $\mu A$ and subjected to a magnetic field of strength $0.85T$. Work done in rotating the coil by ${180}^0$ against the torque is

A ball rolls off the top of a stairway horizontally with a velocity of $5\text{m/s}$. Each step is $0.1\text{m}$ high and $0.1\text{m}$ wide. If $g=10{\text{m/s}}^2$ and the ball strikes the $n^{th}$ step, then find the value of $n$.

What is the translational kinetic energy of a Helium molecule (He) at 1500 K?

If $|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-\overrightarrow{b}|$, then which of the following options is correct?

A particle starts from rest at $t=0$ and moves in a straight line with acceleration as shown in the following figure.



The velocity of the particle at $t=3$ seconds is

The value of g at a place decreases by 2%. The barometric height of mercury

A spherical ball of radius $5.5\text{cm}$ is oscillating in a hemispherical bowl. If the time taken to complete one oscillation by the ball is $\frac{\pi }{2}s$, then find the radius of hemispherical bowl.

A ray parallel to $x$ - axis is incident at a point $P$ on a parabolic reflecting surface, and the reflected ray becomes parallel to $y$ - axis as shown in the figure. If the equation of parabola is given by $y^2=2x$, then the coordinates of point $P$ are

A point source is producing sinusoidal transverse waves. If the displacement from the equilibrium position of a point $4\text{cm}$ from source is half the amplitude of wave at $t=\frac{T}{6}$ ($T$ being time period). The wavelength of travelling wave is

Two blocks of masses $m_1=m_2=m$ are connected by a string of negligible mass which passes over a frictionless pulley fixed on the top of an inclined plane as shown in the figure. When the angle of inclination $\theta ={30}^{\circ }$, the mass $m_1$ just begins to move up the inclined plane. What is the coefficient of friction between block$m_1$ and the inclined plane?

If $E_1,E_2$ and $E_3$ respectively represent the kinetic energies of an electron, an $\alpha$-particle and a proton each having same de-Broglie’s wavelength, then

A body is dropped from a certain height $h$ ($h$ is very large) and second body is thrown downward from the same height $h$ with velocity of $5\text{m/s}$ simultaneously. What will be difference in heights of the two bodies after $3s$ ?

Pressure inside two soap bubbles of a liquid solution are $1.01\text{atm}$ and $1.02\text{atm}$. Ratio of their volume is

What is the angle between the electric field vectors at points A and B because of an infinitely long line charge as shown?

Two masses - ' $\sqrt{3}$m' and '$\sqrt{2}$m' are tied by a light string are placed on a wedge of mass '4m'. The wedge is placed on a smooth horizontal surface. Find out the value of $\theta$ such that the wedge does not move even after the system is set free from the state of rest.

A simple pendulum with a brass bob has a time period T. The bob is now immersed in a non-viscous liquid and oscillated. If the density of the liquid is $\frac{1}{8}$ that of brass, the time period of the same pendulum will be

The inductance of a solenoid is $5\text{H}$ and its resistance is $5\Omega$. If it is connected to a $10\text{Volt}$ battery, then time taken by the current to reach ${\frac{9}{10}}^{th}$ of its maximum will be

If the time period $\left(T\right)$ of vibrations of a liquid drop depends on the surface tension $\left(S\right)$ of the fluid over which it is kept, radius $\left(r\right)$ of the drop and density $\left(\rho \right)$ of the liquid then the equation of $T$ is

[here, $k$ is a dimensionless proportionality constant]

Two concentric shells A and B have radii a and b. A is given a charge q and B is grounded. Find the potential difference between the shells.

The acceleration-displacement graph of a particle executing SHM is shown in given figure. The time period of its oscillation in seconds is

The moment of inertia of a thin square plate ABCD, of uniform thickness about an axis passing through the centre O and perpendicular to the plane of the plate is

The distance $x$ moved by a body of mass 0.5 kg due to a force varies with time $t$ as

$x=3t^2+4t+5$

where $x$ is expressed in metre and t in second. What is the work done by the force in the first 2 seconds?

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

Two blocks of masses $2.9\text{kg}$ and $1.9\text{kg}$ are suspended from a rigid support by two inextensible wires each of length $1\text{m}$, the upper wire has negligible mass and the lower wire has a uniform mass of $0.2\text{kg/m}$. The whole system of block, wire and support has an upward acceleration of $0.2{\text{m/s}}^2$.

The tension at the mid-point of lower wire is:

A particle executing simple harmonic motion along y-axis has its motion described by the equation. $y=Asin\left(\omega t\right)+B$. The amplitude of the simple harmonic motion is

A homogenous rod of length ℓ = ηx and mass, M is lying on a smooth horizontal floor. A bullet of mass 'm' hits the rod at a distance 'x' from the middle of the rod at a velocity $v_0$ perpendicular to the rod and comes to rest after collision. If the velocity of the farther end of the rod just after the impact is in the opposite direction of $v_0$, then:

Two liquids of densities ${\rho }_1$ and ${\rho }_2$ are mixed in two different ways. In first case, equal mass of both the liquids were mixed and the mixture density was found to be $3{\text{kg/m}}^3$ . In second case, equal volume of both the liquids were mixed and the mixture density was found to be $5{\text{kg/m}}^3$ . Then product of densities ${\rho }_1$ and ${\rho }_2$ is

A particle is projected vertically upwards with speed $20\text{m/s}$ from top of a tower of height $20\text{m}$ as shown in figure. Given $B$ is top most point of trajectory and $C$ is at same height as $A$ :

$\begin{array}{cccc}& \text{Column - I}& & \text{Column - II}\\ \left(A\right)& \text{ratio of maximum height from ground (BD)}& \left(P\right)& \frac{1}{\sqrt{2}}\\ & \text{to the initial height from ground (AD)is}& & \\ \left(B\right)& \text{ratio of distance travelled in 1}{}^{st}\text{second to}& \left(Q\right)& \frac{1}{\sqrt{3}}\\ & \text{the distance travelled in 2}{}^{nd}\text{second is}& & \\ \left(C\right)& \text{ratio of initial speed at A to the final}& \left(R\right)& 1\\ & \text{just before reaching to ground (D) is}& & \\ \left(D\right)& \text{ratio of time taken from A to C and time}& \left(S\right)& 2\\ & \text{taken from A to B is}& & \\ & & \left(T\right)& 4\\ & & \left(U\right)& 3\end{array}$

Which of the following option has the correct combination considering column-I and column-II

An insulated solid sphere has a volume charge density ρ. ‘r’ is the distance of a point P from the Centre. Which graph represents the correct variation of electric field with respect to r?

Find the angular magnification in this case. (image at infinity)

Compare the radius of curvature at different points on the curve shown in the figure.

Statement I : A plane mirror can never form a real image.

Statement II : The reflected rays from a mirror need to actually intersect to form a real image.

A river flows due south with a speed of 2.0 m/s. A man steers a motorboat across the river; his velocity relative to the water is 4 m/s due east. The river is 800 m wide. What is his velocity relative to the earth? Assume North as positive y-direction.