Four particles \(A, B, C, D\) of mass \(\frac{\mathrm{m}}{2}, \mathrm{~m}, 2 \mathrm{~m}, 4 \mathrm{~m}\), have same momentum, respectively. The particle with maximum kinetic energy is _______.

An ideal cell of emf \(10\, V\) is connected in circuit shown in figure. Each resistance is \(2\, \Omega .\) The potential difference (in \(V\)) across the capacitor when it is fully charged is

A bar magnet has total length \(2 l=20\) units and the field point \(P\) is at a distance \(\mathrm{d}=10\) units from the centre of the magnet. If the relative uncertainty of length measurement is \(1 \%\), then uncertainty of the magnetic field at point P is :

The total length of a sonometer wire between fixed ends is \(110\, cm\). Two bridges are placed to divide the length of wire in ratio \(6 : 3 : 2\). The tension in the wire is \(400\, N\) and the mass per unit length is \(0.01\, kg/m\) . What is the minimum common frequency with Which three parts can vibrate ........... \(Hz\) ?

The circuit shown below contains two ideal diodes, each with a forward resistance of \(50\,\Omega \). If the battery voltage is \(6\,V\), the current through the \(100\,\Omega \) resistance (in Amperes) is

An inductor of self inductance 1 H connected in series with a resistor of \(100 \pi \mathrm{ohm}\) and an ac supply of \(100 \pi\) volt, 50 Hz. Maximum current flowing in the circuit is ________ A.

As shown in the figure, a bob of mass \(\mathrm{m}\) is tied by a massless string whose other end portion is wound on a fly wheel (disc) of radius \(\mathrm{r}\) and mass \(m\). When released from rest the bob starts falling vertically. When it has covered a distance of \(h\). the angular speed of the wheel will be

If \(A=\left(\begin{array}{cc}\frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{-2}{\sqrt{5}} & \frac{1}{\sqrt{5}}\end{array}\right), B=\left(\begin{array}{ll}1 & 0 \\ i & 1\end{array}\right), i=\sqrt{-1}\), and \(\mathrm{Q}=\mathrm{A}^{\mathrm{T}} \mathrm{BA}\), then the inverse of the matrix \(\mathrm{A} \mathrm{Q}^{2021} \mathrm{~A}^{\mathrm{T}}\) is equal to :

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

If the coefficient of \(x ^{10}\) in the binomial expansion of \(\left(\frac{\sqrt{x}}{5^{\frac{1}{4}}}+\frac{\sqrt{5}}{x^{\frac{1}{3}}}\right)^{60}\) is \(5^{ k } l\), where \(l, k \in N\) and \(l\) is coprime to \(5\) , then \(k\) is equal to

Thermodynamic process is shown below on a \(P-V\) diagram for one mole of an ideal gas. If \(V _{2}=2 V _{1}\) then the ratio of temperature \(T _{2} / T _{1}\) is ...... .

The sum of all the real roots of the equation \(\left( e ^{2 x }-4\right)\left(6 e ^{2 x }-5 e ^{ x }+1\right)=0\) is

In a class of \(140\) students numbered \(1\) to \(140\), all even numbered students opted Mathematics course, those whose number is divisible by \(3\) opted Physics course and those whose number is divisible by \(5\) opted Chemistry course. Then the number of students who did not opt for any of the three courses is