A circular coil of radius \(2\) cm and \(125\) turns carries a current of \(1\) A. The coil is placed in a uniform magnetic field of magnitude \(0.4\) T. The axis of the coil makes an angle of \(30°\) with the direction of the magnetic field. The torque acting on the coil is \(\alpha \times 10^{-4}\) N.m. The value of \(\alpha\) is ______.
(\(\pi=3.14\))

A body of mass ' \(m\) ' is projected with a speed ' \(u\) ' making an angle of \(45^{\circ}\) with the ground. The angular momentum of the body about the point of projection, at the highest point is expressed as \(\frac{\sqrt{2} \mathrm{mu}^3}{\mathrm{Xg}}\). The value of ' \(\mathrm{X}\) ' is _______.

A body rolls down an inclined plane without slipping. The kinetic energy of rotation is \(50 \,\%\) of its translational kinetic energy. The body is :

A \(12.5\, eV\) electron beam is used to bombard gaseous hydrogen at room temperature. It will emit

A system consists of two types of gas molecules \(A\) and \(B\) having same number density \(2 \times\) \(10^{25}\, / {m}^{3}\). The diameter of \({A}\) and \({B}\) are \(10\, \stackrel{\circ}{{A}}\) and \(5\, \stackrel{\circ}{{A}}\) respectively. They suffer collision at room temperature. The ratio of average distance covererd by the molecule \(A\) to that of \(B\) between two successive collision is \(.....\,\times 10^{-2}\)

A block is simply released from the top of an inclined plane as shown in the figure above. The maximum compression in the spring when the block hits the spring is _______.

Two long parallel conductors \(S_{1}\) and \(S_{2}\) are separated by a distance \(10 \,cm\) and carrying currents of \(4\, A\) and \(2 \,A\) respectively. The conductors are placed along \(x\)-axis in \(X - Y\) plane. There is a point \(P\) located between the conductors (as shown in figure). A charge particle of \(3 \pi\) coulomb is passing through the point \(P\) with velocity \(\overrightarrow{ v }=(2 \hat{ i }+3 \hat{ j }) \,m / s\); where \(\hat{i}\) and \(\hat{j} \quad\) represents unit vector along \(x\) and \(y\) axis respectively. The force acting on the charge particle is \(4 \pi \times 10^{-5}(-x \hat{i}+2 \hat{j}) \,N\). The value of \(x\) is

A person of mass \(M\) is, sitting on a swing of length \(L\) and swinging with an angular amplitude \(\theta_0\). If the person stands up when the swing passes through its lowest point, the work done by him, assuming that his centre of mass moves by a distance \(l\, ( l < < L)\), is close to

A disc of radius \(R\) and mass \(M\) is rolling horizontally without slipping with speed \(v\). It then moves up an inclined smooth surface as shown in figure. The maximum height that the disc can go up the incline is _______.

A regular hexagon is formed by six wires each of resistance \(r \Omega\) and the corners are joined to centre by wires of same resistance. If the current enters at one corner and leaves at the opposite corner, the equivalent resistance of the hexagon between the two opposite corners will be

If the system of equations  \(k x+y+2 z=1\) ; \(3 x-y-2 z=2\) ; \(-2 x-2 y-4 z=3\) has infinitely many solutions, then \(k\) is equal to ..........

Let \( f:R\rightarrow(0,\infty) \) be a twice differentiable function such that \( f(3)=18, \) \( f^{\prime}(3)=0 \) and \( f^{\prime\prime}(3)=4 \). Then \( \lim_{x\rightarrow1}(\log_{e}(\frac{f(x+2)}{f(3)})^{\frac{18}{(x-1)^{2}}}) \) is equal to:

A player kicks a football with an initial speed of \(25\, {ms}^{-1}\) at an angle of \(45^{\circ}\) from the ground. What are the maximum height and the time taken by the football to reach at the highest point during motion ? (Take g \(=10 \,{ms}^{-2}\) )