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A man standing on a road has to hold his umbrella at ${30}^{\circ }$ with the vertical to keep the rain away. He throws the umbrella and starts running at $10km/h$. He finds that raindrops are hitting his head vertically. Find the speed of raindrops with respect to the ground.

A gaseous mixture enclosed in a vessel of volume $2m^3$ consists of $2$ moles of a gas A with ${\gamma }_A=\frac{C_P}{C_V}=\frac{5}{3}$, another gas B with ${\gamma }_B=\frac{7}{5}$ at a temperature ${43}^{\circ }C$. The gram molecular weights of A & B are $32\text{gm}$ and $4\text{gm}$ respectively. Assume gases to be non - reacting. The gaseous mixture follows $P{V}^{20/14}=\text{constant}$ in adiabatic process. Find the number of moles of the gas B in the gaseous mixture.

An object kept at $\text{O}$ is in equilibrium under the action of four coplanar forces shown in the figure. Find the magnitudes of $F_1$ and $F_2$.

A brass rod and a lead rod each $80\text{cm}$ long at $0{}^{\circ }\text{C}$ are clamped together at one end with their free ends coinciding. The separation the of free ends of the rods, if the system is placed in a steam bath is

$[{\alpha }_{\text{brass}}=18\times {10}^{-6}/{}^{\circ }\text{C}$ and ${\alpha }_{\text{lead}}=28\times {10}^{-6}/{}^{\circ }\text{C}\text{and temperature of steam}={100}^{\circ }\text{C}]$

An electric heater of power $1\text{kW}$ emits thermal radiations. The surface area of the heating element of the heater is $200{\text{cm}}^2$. If this heating element is treated like a black body, find the temperature at its surface. Assume its temperature is very much higher than its surroundings.

[$\sigma =5.67\times {10}^{-8}{\text{W m}}^{-2}{\text{K}}^{-4}$]

A boat moves with a speed of $5km/h$ relative to water in a river flowing with a speed of $3km/h$ and having a width of $1km$. The time taken in a round trip, when boat travel shortest distance is

A football player kicked a ball with a velocity of $20{\text{ms}}^{-1}$ in a direction making an angle of ${60}^{\circ }$ with horizontal. Find the velocity with which he should run in the same direction so that the ball remains above this head.

The number of photons emitted by a 5mW laser source operating at 632.8 nm is __$\times {10}^{16}$

In the given figure for a projectile



Which of the following option is correct?

A dipole with dipole moment $3.60nC-m$ is oriented at ${60}^{\circ }$ to an electric field of magnitude $4\times {10}^{9}N/C$, how much work is required to rotate the dipole until it is antiparallel to field.

A rock band gives rise to an average sound level of $100\text{dB}$ at a distance $20\text{m}$ from the centre of the band. As an approximation, assume that the sound radiated by the band spreads out in the shape of a sphere centred at the band. What is the power of sound produced by the band?

A wire of length $L$ and $3$ identical cells of negligible internal resistances are connected in series. Due to this current, the temperature of the wire is raised by $\Delta T$ in a time $t$. A number $N$ of similar cells is now connected in series with a wire of the same material and cross-section but of length $2L$. The temperature of wire is raised by same amount $\Delta T$ in the same time $t$. Find the value of $N$.

A conducting ring is placed around the core of an electromagnet as shown in fig. When key K is pressed, the ring

The refractive index of a prism is 2. This prism can have A mximum refracting angle of

A light rod of length $l=1\text{m}$ is hinged at its centre and two masses of $m_1=5\text{kg}$ and $m_2=2\text{kg}$ are hung from the ends as shown in figure. Find the initial angular acceleration of the rod assuming that it was horizontal in beginning. (Take $g=10m/s^2$)

Two identical particles having the same mass $m$ and charges $+q$ and $-q$, separated by a distance $d$, enter in a uniform magnetic field $B$ directed perpendicular to paper inward with speeds $v_1$ and $v_2$ as shown in the figure. The particles will not collide if

Figure shows a loop – the – loop track of radius $R$. A car (without engine) starts from a platform at a distance $h$ above the top of the loop and goes around the loop without falling off the track. Then the minimum value of $h$ for a successful looping is (neglect friction).

If $E_A,E_B,E_{C}and{E}_D$ represent the magnitudes of electric fields at A, B, C and D. Which of the options are correct?

A ball is projected from an inclined plane having inclination $\theta$, normal to the plane with speed $u$. It strikes the vertical wall horizontally and again it stikes the inclined plane. If all the collisions are elastic, find the time taken by the ball to hit the inclined plane.

Two harmonic waves are represented in SI units by, $y_1(x,t)=0.2sin(x-3.0t)$ and $y_2(x,t)=0.2sin(x-3.0t+\varphi )$. What is the expression for the sum $y=y_1+y_{2}for\varphi =\frac{\pi }{2}$.

What is the phase difference between the two simple harmonic motions whose displacement – time graphs are shown in fig ?

Water rises in a capillary tube to a certain height such that the upward force due to surface tension is balanced by $75\times {10}^{-4}\text{N}$ force due to the weight of the liquid. If the surface tension of water is $6\times {10}^{-2}{\text{Nm}}^{-1}$, the inner circumference of the capillary must be

(Assume angle of contact=$0$)

At $t=0$ a projectile is fired from a point $O$ (taken as origin) on the ground with a speed of $50m/$s at an angle of ${53}^{\circ }$ with the horizontal. It just passes two points $A$ and $B$ each at height $75m$ above horizontal as shown.



The horizontal separation between the points $A$ and $B$ is

(Take $g=10m/s^2$)

A point $P$ having the coordinate $(2,3)$. Find the position vector of $P$ ?

Two simple harmonic motions are represented by the equations $y_1=0.1\sin (100\pi t+\frac{\pi }{3})$ and $y_2=0.1\cos \pi t$. The phase difference of the velocity of particle $1$ with respect to the velocity of particle $2$ is

Polarised glass is used in sun glasses because

[CPMT 1981]

If y=lnx, find $\frac{dy}{dx}$

Two trains one of length 100 m and another of length 125 m, are moving in mutually opposite directions along parallel lines, meet each other, each with speed 10 m/s. If their accelerations are $0.3m/s^{2}and0.2m/s^2$ respectively, then the time they take to pass each other will be

A boy whose mass is $30\text{kg}$ climbs with a constant speed on a vertical rope of $6\text{m}$ length in $10\text{s}$. The power of the boy during the climb is

(Take $g=10{\text{m/s}}^2)$

A block of mass m is placed on a smooth wedge of inclination $\theta$. The whole system is accelerated horizontally, so that the block does not slip on the wedge. The force exerted by the wedge on the block ($g$ is acceleration due to gravity) will be:

Two blocks of mass 2 kg and 4 kg are kept in contact with each other on a smooth horizontal surface. A horizontal force of 12 N is applied on the first block due to which they move with certain constant acceleration. Calculate the force between the blocks.

One mole of an ideal monatomic gas has initial temperature $T_0$, is made to go through the cycle $abca$ as shown in fig. If $U$ denotes the internal energy, then choose the correct alternative.

Two rods, one of aluminium and the other of steel, having initial length $l_1$ and $l_2$ respectively are connected together to form a single rod of length $l_1+l_2$. The coefficients of linear expansion for aluminium and steel are ${\alpha }_a$ and ${\alpha }_s$ respectively. If the length of each rod increases by the same amount when their temperatures are raised by $t{}^{\circ }\text{C}$, then find the ratio $l_1/(l_1+l_2)$

A body of certain mass is tied to one end of a light string of length $l$, whose other end is fixed at point $\text{O}$. The body is given a speed of $\sqrt{10gl}$ from the lowermost position $\text{A}$. Find the magnitude of change in velocity of body, while moving from position $\text{A}$ to $\text{B}$ as shown in figure.

A beam of light converges to a point $P$. A concave lens of focal length $16cm$ is placed in the path of the convergent beam $12cm$ from $P$. How far from the lens does the beam converge?

A boat floating in a tank of water is carrying passengers. If the passengers drink some of the water surrounding the boat in the tank , how will it affect the water level of the tank?

Two strings $A$ and $B$ made of same material are stretched by same tension. The radius of cross-section of string $A$ is double the radius of cross-section of string $B$. A transverse wave travels through string $A$ with speed $v_A$ and through string $B$ with speed $v_B$. The ratio $\frac{v_A}{v_B}$ is

Energy required to remove an electron from aluminium surface is 4.2 eV. If light of wavelength $2000\stackrel{\circ }{A}$ falls on the surface, the velocity of the fastest electron ejected from the surface will be

The approximate depth of an ocean is $2700\text{m}$. The compressibility of water is $50\times {10}^{-11}{\text{Pa}}^{-1}$ and density of water is ${10}^3{\text{kg/m}}^3$. What fractional compression of water will be obtained at the bottom of the ocean?

Why don't we get maximum current (saturation current) at 0 potential

Equations $y_1=0.15m(100\pi t+\frac{\pi }{3})$ and y =0.1 cosπ t. The phase difference of the velocity of particle 1 with respect to the velocity of particle 2 is:

If velocity of a particle is $v\left(t\right)=t^2-3t\text{m/s}$, the average speed and magnitude of average velocity of the particle in $6\text{sec}$ respectively are

Find the integral of the given function w.r.t - x

$y=x-\frac{1}{x}+\frac{1}{x^2}$

A uniform magentic field exists in space. A particle of mass m and charges q is projected towards negative x-axis with speed v from a point (d, 0, 0). The maximum value of v for which the particle does not hit the y-z plane is

I cut the rope of an elevator and it begins to fall freely. A person inside (aware of his imminent doom) tries to make a last minute contribution to science by dropping a ball. Then what will he observe?

A circular loop of rope of length L rotates with uniform angular velocity ω about an axis through its centre on a horizontal smooth platform. Velocity of pulse (with respect to rope) produced due to slight radial displacement is given by