The Young's modulus and Poisson's ratio of a material are respectively Y and \(\sigma\). The force required to decrease the area of cross-section of a wire made of this material by \(\Delta \mathrm{A}\) is

Statement (I) : The slope of kinetic energy-displacement curve of a body in motion will be directly proportional to its acceleration. Statement (II) : From a height of \(15 \mathrm{~m}\) a ball is projected vertically upwards with a velocity of \(30 \mathrm{~m} / \mathrm{s}\). If the ball rises to the same height after hitting the ground, the loss of its energy on hitting the ground is \(30 \%\). Statement (III) : The velocity acquired by a body of mass ' \(m\) ' after travelling a fixed distance from rest under the action of a constant force is directly proportional to mass ' \(\mathrm{m}\) '. Which of the following is correct?

The de-Broglie wavelength of an electron having kinetic energy $100 \mathrm{eV}$ is,
[Use $h=4.14\times {10}^{-15} \mathrm{eVs},$ mass of electron $=\frac{0.5\times {10}^6}{c^2}\mathrm{eV}, 1\mathrm{pm}={10}^{-12} \mathrm{m}$]

Consider the motion of a particle described by \(x=a \cos t, y=a \sin t\) and \(z=t\). The trajectory traced by the particle as a function of time is

A particle of mass \(1 \mathrm{~kg}\) is projected with an initial velocity \(10 \mathrm{~ms}^{-1}\) at an angle of projection \(45^{\circ}\) with the horizontal. The average torque acting on the projectile, between the time at which it is projected and the time at which it strikes the ground, about the point of projection in newton-metre is

Ratio of translational degrees of freedom to rotational degrees of freedom of a polyatomic linear gas molecule is

An infinite number of electric charges each equal to 5 nano-coulomb (magnitude) are placed along \(X\)-axis at \(x=1 \mathrm{~cm}, x=2 \mathrm{~cm}\), \(x=4 \mathrm{~cm}, x=8 \mathrm{~cm} . . .\). and so on. In this set up if the consecutive charges have opposite sign, then the electric field in newton/coulomb at \(x=0\) is : \(\left(\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{~N}-\mathrm{m}^2 / \mathrm{C}^2\right)\)

A point \(P\) is moving in uniform circular motion with radius \(3 \mathrm{~m}\). Let at some instant the acceleration of the point is \(\mathbf{a}=(6 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}^2\), the position vector is \(\mathbf{r}\) and velocity vector is \(\mathbf{v}\). Choose the correct statement.

A wheel of 20 metallic spokes each \(40 \mathrm{~cm}\) long is rotated with a speed of \(180 \mathrm{rev} / \mathrm{min}\) in a plane normal to the horizontal domponent of earth's magnetic field \(\mathrm{H}_{\mathrm{e}}\) at a place. If \(\mathrm{H}_{\mathrm{e}}=0.4 \mathrm{G}\) (Gauss) at that place, the induced emf between the axle and the rim of the wheel is

The radius of the first orbit of hydrogen is \(\mathrm{r}_{\mathrm{H}}\), and the energy in the ground state is - \(13.6 \mathrm{eV}\). Considering a \(\mu^{-}\)-particle with a mass \(207 m_e\) revolving round a proton as in hydrogen atom, the energy and radius of proton and \(\mu^{-}\)- combination respectively in the first orbit are (assume nucleus to be stationary)

An observer is standing \(500 \mathrm{~m}\) away from a vertical hill. Starting between the observer and the hill, a police van sounding a siren of frequency \(1000 \mathrm{~Hz}\) moves towards the hill with a uniform speed. If the frequency of the sound heard directly from the siren is \(970 \mathrm{~Hz}\), the frequency of the sound heard after reflection from the hill (in Hz) is about, (velocity of sound \(=330 \mathrm{~ms}^{-1}\) ) :

Consider the following two circuits $\left[A\right] :$ $20$ bulbs are connected in series to a power supply line $\left[B\right] :$ $20$ bulbs identical to $\left[A\right]$ are connected in a parallel circuit to an identical power supply line Identify which of the following is not true

The centre of mass of three particles of masses \(1 \mathrm{~kg}, 2 \mathrm{~kg}\) and \(3 \mathrm{~kg}\) is at \((2,2,2)\). The position of the fourth mass of \(4 \mathrm{~kg}\) to be placed in the system as that the new centre of mass is at \((0,0,0)\) is.