A very long wire \(ABDMNDC\) is shown in figure carrying current \(I. AB\) and \(BC\) parts are straight, long and at right angle. At \(D\) wire forms a circular turn \(DMND\) of radius \(R. AB.\) \(\mathrm{BC}\) parts are tangential to circular turn at \(\mathrm{N}\) and \(D\). Magnetic field at the centre of circle is

A cubical block of side \(30\,cm\)  is moving with  velocity \(2\,\,ms^{-1}\) on a smooth horizontal surface. The surface has a bump at a point \(O\) as shown in figure. The angular velocity (in rad/s) of the block immediately after it hits the bump, is

A \(1 \mathrm{~kg}\) mass is suspended from the ceiling by a rope of length \(4 \mathrm{~m}\). A horizontal force ' \(F\) ' is applied at the mid point of the rope so that the rope makes an angle of \(45^{\circ}\) with respect to the vertical axis as shown in figure. The magnitude of \(F\) is _______.

Due to cold weather a \(1\, {m}\) water pipe of cross-sectional area \(1\, {cm}^{2}\) is filled with ice at \(-10^{\circ} {C}\). Resistive heating is used to melt the ice. Current of \(0.5\, {A}\) is passed through \(4\, {k} \Omega\) resistance. Assuming that all the heat produced is used for melting, what is the minimum time required ? (In \({s}\)) (Given latent heat of fusion for water/ice \(=3.33 \times 10^{5}\, {J} {kg}^{-1}\), specific heat of ice \(=2 \times 10^{3}\, {J}\) \({kg}^{-1}\) and density of ice \(=10^{3}\, {kg} / {m}^{3}\)

Three containers \(\mathrm{C}_{1}, \mathrm{C}_{2}\) and \(\mathrm{C}_{3}\) have water at different temperatures. The table below shows the final temperature \(T\) when different amounts of water (given in litres) are taken from each containers and mixed (assume no loss of heat during the process) \(\begin{array}{|c|c|c|c|}\hline \mathrm{C_{1 }} & {\mathrm{C}_{2}} & {\mathrm{C}_{3}} & {\mathrm{T}} \\ \hline {1 l} & {2 l} & {-} & {60^{\circ} \mathrm{C}} \\ \hline {-} & {1 l} & {2 l} & {30^{\circ} \mathrm{C}} \\ \hline  {2 l} & {-} & {1 l} & {60^{\circ} \mathrm{C}} \\ \hline  {1 l} & {1 l} & {1 l} & {\theta} \\ \hline\end{array}\) The value of \(\theta\) (in \(^{\circ} \mathrm{C}\) to the nearest integer) is

A body of mass \(5\, kg\) under the action of constant force \(\overrightarrow F  = {F_x}\hat i + {F_y}\hat j\) has velocity at \(t\,= 0\, s\) as \(\overrightarrow v  = \left( {6\hat i - 2\hat j\,m/s} \right)\)  and at \(t\, = 10\,s\) as \(\overrightarrow v  = +6\hat j\,m/s\). The force \(\overrightarrow F \) is 

An engine approaches a hill with a constant speed. When it is at a distance of \(0.9\, km\), it blows a whistle whose echo is heard by the driver after \(5\, seconds\). If the speed of sound in air is \(330\, m/s\), then the speed of the engine is .... \(m/s\)

Let \(y=y(x)\) be the solution of the differential equation \((x+y+2)^2 d x=d y, y(0)=-2\). Let the maximum and minimum values of the function \(y=y(x)\) in \(\left[0, \frac{\pi}{3}\right]\) be \(\alpha\) and \(\beta\), respectively. If \((3 \alpha+\pi)^2+\beta^2=\gamma+\delta \sqrt{3}, \gamma, \delta \in \mathbb{Z}\), then \(\gamma+\delta\) equals ....................

The figure shows two solid discs with radius \({R}\) and \({r}\) respectively. If mass per unit area is same for both, what is the ratio of \({MI}\) of bigger disc around axis \({AB}\) (Which is \(\perp\) to the plane of the disc and passing through its centre) of MI of smaller disc around one of its diameters lying on its plane? Given \('M'\) is the mass of the larger disc. (\(MI\) stands for moment of inertia)

The mean of the numbers \(a, b, 8,5,10\) is \(6\) and their variance is \(6.8\). If \(M\) is the mean deviation of the numbers about the mean, then \(25\; M\) is equal to

If the equation \(a| z |^{2}+\overline{\bar{\alpha} z +\alpha \overline{ z }}+ d =0\) represents \(a\) circle where \(a,d\) are real constants then which of the following condition is correct ?

Let \(P(\alpha,\beta,\gamma)\) be the point on the line \(\frac{x-1}{2}=\frac{y+1}{-3}=z\) at a distance \(4\sqrt{14}\) from the point (1, -1, 0) and nearer to the origin. Then the shortest distance, between the lines \(\frac{x-\alpha}{1}=\frac{y-\beta}{2}=\frac{z-\gamma}{3}\) and \(\frac{x+5}{2}=\frac{y-10}{1}=\frac{z-3}{1}\), is equal to

An electric dipole is formed by two equal and opposite charges \(q\) with separation \(d\). The charges the same mass \(m\). It is kept in a uniform electric field \(E\). If it slightly rotated from its equilibrium orientation, then its angular frequency \(\omega \) is