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The equation of a wave travelling on a string is $y=\left(0.10mm\right)sin\left[\right(31.4m^{-1})x+(314s^{-1}\left)t\right].$ (a) In which direction does the wave travel ? (b) Find the wave speed, the wavelength and the frequency of the wave. (c) What is the maximum displacement and the maximum speed of a portion of the string ?

A hollow cylinder has length $L$ and inner and outer radii $R_1$ and $R_2$ respectively. The cylinder carries a uniform charge density $\rho$. An expression for the magnetic moment as a function of the angular velocity $\omega$ of rotation of the cylinder about its axis is

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 $g=9.8m/s^2$]

The position of the centre of mass of a system consisting of two particles of masses $m_1$ and $m_2$, separated by distance $L$, from $m_2$ is

A company has to construct a flyover. It deployed 10 men who can finish the work 30 days each working 8 hrs a day. After five days of work due to heavy rain $50\%$ of the flyover constructed so far collapsed. After the incident the management decided to stop the construction work during the period it rains. Also the Management deployed 5 more men and 10 women to complete the project faster. From the sixth day onwards it rained every alternative day for 2 hours each. Find the fraction of construction work left after the 17th day. Assume that all the men work with same efficiency, and all women workwith half of the efficiency of men. Also assume that it rains only during the working hours.

Find the total kinetic energy $\left(T\right)$ of the two particles in the reference frame fixed to their centre of inertia having masses $m_1$ and $m_2$ and velocities $\overrightarrow{v_1}$ and $\overrightarrow{v_2}$ (perpendicular to each other) respectively. Here $\mu =\frac{m_1{m}_2}{m_1+m_2}$

A thin liquid film formed between a $U$ - shaped wire and supports a light weight of $1.5\times {10}^{-2}\text{N}$ (see figure). The length of the slider is $30\text{cm}$ and its weight is negligible. The surface tension of the liquid film is

The path difference between the two waves

$y_1=a_1\sin (\omega t-\frac{2\pi x}{\lambda })$ and $y_2=a_2\cos (\omega t-\frac{2\pi x}{\lambda }+\varphi )is$

Let $A=(a_1,a_2)\text{and}B=(b_1,b_2)$ be two points in the plane with integer coordinates. Which one of the following is not a possible value of the distance between $A$ and $B?$

Three rods of equal length $l$ are joined to form an equilateral $\Delta PQR$. $O$ is the midpoint of $QR$. Distance $OP$ remains same for small changes in temperature. If the coefficient of linear expansion for $PQ$ and $PR$ is ${\alpha }_2$ and for $QR$ is ${\alpha }_1$, then

The coefficient of performance of a Carnot refrigerator working between ${30}^{\circ }C$ and $0^{\circ }C$

On applying force on a rod along its length, the rod gets elongated by $0.04\text{m}$. If the length of rod is doubled and diameter is also doubled and same force is applied along the length, and it is found that the rod gets elongated by $x\times {10}^{-2}\text{m}$. Find the value of $x$.

An aircraft moving with a speed of $972\text{km/hr}$ is at a height of $6000\text{m}$, just overhead of an anti-aircraft gun. If the muzzle velocity of the gun is $540\text{m/s}$, then the firing angle $\theta$ with the horizontal for the bullet to hit the aircraft should be

For a sound wave travelling towards +x direction, sinusoidal longitudinal displacement $\xi$ at a certain time is given as a function of x(Figure). If bulk modulus of air is $B=5\times {10}^{5}N/m^2$, the variation of pressure excess will be

The potential energy between two atoms in a molecule is given by $U=a{x}^3-b{x}^2$ where $a$ and $b$ are positive constants and $x$ is the distance between the atoms. The atom is in stable equilibrium when $x$ is equal to

Why is your face upside down in the inward curve of the spoon?

A particle of mass $m=1kg$ executes SHM about mean position 'O' with angular frequency $\omega =1.0rad/s$ and total energy $2J$. $x$ is positive if measured towards right from 'O'. At $t=0$, particle is at 'O' and moving towards right. Then which of the options can be correct.


$\begin{array}{cccc}& Column-I& & Column-II\\ \left(P\right)& Speedofparticleis\sqrt{2}m/sat& \left(1\right)& x=+1m\\ \left(Q\right)& Kineticenergyoftheparticleis1Jat& \left(2\right)& x=-1m\\ \left(R\right)& Att=\frac{\pi }{6}sparticleisat& \left(3\right)& x=+\sqrt{2}m\\ \left(S\right)& Kineticenergyis1.5Jat& \left(4\right)& x=-\sqrt{2}m\end{array}$

Which of these options is the correct tangent to the given curve at the topmost point A

In the $LCR$ circuit, the voltmeter and ammeter readings are

Lenses are constructed by a material of refractive index 1.50.Themagnitude of the radii of curvature are 20 cm and 30 cm. Choose the focal lengths of all the possible lenses with the above specifications:

A $1.0kg$ block collides with a horizontal weightless spring of force constant $2.75N{m}^{–1}$ as shown in figure. The block compresses the spring $4.0m$ from the rest position. If the coefficient of kinetic friction between the block and horizontal surface is $0.25$, the speed of the block at the instant of collision is

A solid uniform ball of volume $V$ floats on the interface of two immiscible liquids (see the figure). The specific gravity of the upper liquid is ${\rho }_1$ and that of lower one is ${\rho }_2$ and the specific gravity of ball is $\rho ({\rho }_1<\rho <{\rho }_2)$. The fraction of the volume of the ball in the upper liquid is

Light is incident at an angle $\varphi$ with the normal to a plane containing two slits of separation $d$. Select the expression that correctly describes the positions of the interference maxima in terms of the incoming angle $\varphi$ and outgoing angle $\theta$.

It is proposed to use the nuclear fusion reaction ${}_1^{2}H+{}_1^{2}H\to {}_2^{4}He$ in a nuclear reactor with an electrical power rating of 200 MW. If the energy from the above reaction is used with 25 percent efficiency in the reactor, how many grams of deuterium fuel will be needed per day? (The masses of ${}_1^{2}H$ and ${}_2^{4}He$ are $2.0141\text{amu}$ and $4.0026\text{amu}$ respectively.)

The Van Der Waal’s equation of state for some gases can be expressed as: $(P+\frac{a}{V^2})(V-b)=RT$
Where $P$ is the pressure, $V$ is the molar volume and $T$ is the absolute temperature of the given sample of gas and $a,b$ and $R$ are constants.

The dimensions of $a$ are

A uniform rod made of material of Young’s modulus of elasticity $Y$ is subjected to forces shown in the diagram. If $A$ is the area of cross section of the rod, then find its extension

A hydrogen atom moving at a speed $v$ absorbs a photon of wavelength $122\text{nm}$ and stops. The value of $v$ is (mass of hydrogen atom $=1.67\times {10}^{-27}\text{kg}$)