A proton moving with one tenth of velocity of light has a certain de Broglie wavelength of \(\lambda\). An alpha particle having certain kinetic energy has the same de-Brogle wavelength \(\lambda\). The ratio of kinetic energy of proton and that of alpha particle is:

A coil of circular cross-section having \(1000\, turns\) and \(4\, cm^2\) face area is placed with its axis parallel to a magnetic field which decreases by \(10^{-2}\, Wb\, m^{-2}\) in \(0.01\, s\). The \(e.m.f.\) induced in the coil is....\(mV\)

Given below are two statements: One is labelled as Assertion \((A)\) and the other is labelled as Reason \((R).\) Assertion \((A)\) : In an uniform magnetic field, speed and energy remains the same for a moving charged particle. Reason \((R)\) : Moving charged particle experiences magnetic force perpendicular to its direction of motion.

A thin disc of mass \(M\) and radius \(R\) has mass per unit area \(\sigma (r) = kr^2\) where \(r\) is the distance from its centre. Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is

If two vectors \(\vec{A}\) and \(\vec{B}\) having equal magnitude \(\mathrm{R}\) are inclined at an angle \(\theta\), then...

A particle \((\mathrm{m}=1\; \mathrm{kg})\) slides down a frictionless track \((AOC)\) starting from rest at a point \(A\) (height \(2\; \mathrm{m}\) ). After reaching \(\mathrm{C}\), the particle continues to move freely in air as a projectile. When it reaching its highest point \(P\) (height \(1 \;\mathrm{m}\) ). the kinetic energy of the particle (in \(\mathrm{J}\) ) is : (Figure drawn is schematic and not to scale; take \(\left.g=10 \;\mathrm{ms}^{-2}\right)\)

When a long glass capillary tube of radius \(0.015\;cm\) is dipped in a liquid, the liquid rises to a height of \(15\, cm\) within it. If the contact angle between the liquid and glass to clase to \(0^{\circ},\) the surface tension of the liquid, in \(milli\,Newton \;m ^{-1}\) is \(.....\) \(\left[\rho_{\text {(liquid) }}=900\; kg\,m ^{-3}, g =10\, ms ^{-2}\right]\) (Give answer in closest integer)

Let \(f (\theta)=\sin \theta+\int_{-\pi / 2}^{\pi / 2}(\sin \theta+ t \cos \theta) f ( t ) dt\). Then the value of \(\left|\int_{0}^{\pi / 2} f (\theta) d \theta\right|\) is

A ball of mass \(10\, kg\) moving with a velocity \(10 \sqrt{3} m / s\) along the \(x-\)axis, hits another ball of mass \(20\, kg\) which is at rest. After the collision, first ball comes to rest while the second ball disintegrates into two equal pieces. One piece starts moving along \(y-\)axis with a speed of \(10\) \(m / s\). The second piece starts moving at an angle of \(30^{\circ}\) with respect to the \(x-\)axis. The velocity of the ball moving at \(30^{\circ}\) with \(x-\) axis is \(x\, m / s\). The configuration of pieces after collision is shown in the figure below. The value of \(x\) to the nearest integer is

The value of \(\cot \left( {\sum\limits_{n = 1}^{19} {{{\cot }^{ - 1}}\left( {1 + \sum\limits_{p = 1}^n {2p} } \right)} } \right)\) is

The current in a coil of self inductance \(2.0 \,H\) is increasing according to \(I =2 \sin \left( t ^{2}\right) A\). The amount of energy spent during the period when current changes from \(0\) to \(2\, A\) is......\(J.\)

Let \(f\) be a real valued continuous function defined on the positive real axis such that \(g(x)=\int_0^x \mathrm{t} f(\mathrm{t}) \mathrm{dt}\). If \(\mathrm{g}\left(x^3\right)=x^6+x^7\), then value of \(\sum_{r=1}^{15} f\left(\mathrm{r}^3\right)\) is :

An alternating voltage of amplitude \(40 \mathrm{~V}\) and frequency \(4 \mathrm{kHz}\) is applied directly across the capacitor of \(12 \mu \mathrm{F}\). The maximum displacement current between the plates of the capacitor is nearly _______.