$A$ magnet of magnetic moment $50\,\hat{i}\,\text{A}\,\text{m}^2$ is placed along the $x$-axis in a magnetic field $\vec{B} = (0.5\,\hat{i} + 3.0\hat{j})\,\text{T}$. The torque acting on the magnet is:

$A$ cylindrical vessel open at the top is $20 \ cm$ high and $10 \ cm$ in diameter. $A$ circular hole whose cross-sectional area is $1 \ cm^2$ is cut at the centre of the bottom of the vessel. Water flows from a tube above it into the vessel at the rate of $100 \ cm^3 s^{-1}$. The height of water in the vessel under steady state is ....... $cm$ (Take $g = 1000 \ cm s^{-2}$)

If $a, b, c$ and $d \in \mathbb{R}$ such that $a^2+b^2=4$ and $c^2+d^2=2$ and if $(a+ib)^2=(c+id)^2(x+iy)$,then $x^2+y^2$ is equal to

$A$ particle $A$ is projected vertically upwards. Another identical particle $B$ is projected at an angle of $45^{\circ}$. Both reach the same maximum height. The ratio of the initial kinetic energy of $A$ to that of $B$ is

The origin is translated to $(1,2)$. The point $(7,5)$ in the old system undergoes the following transformations successively.
$I$. Moves to the new point under the given translation of origin.
$II$. Translated through $2$ units along the negative direction of the new $X$-axis.
$III$. Rotated through an angle $\frac{\pi}{4}$ about the origin of the new system in the clockwise direction. The final position of the point $(7,5)$ is

$\int [\sin (\log x) + \cos (\log x)] \;dx = $