If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ in four points $(x_i, y_i)$,for $i=1, 2, 3, 4$,then $y_1+y_2+y_3+y_4$ equals

$A$ 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......$mV$

$A$ plane mirror is placed at the bottom of a tank containing a liquid of refractive index $\mu$. $P$ is a small object at a height $h$ above the mirror. An observer $O$,vertically above $P$ outside the liquid,sees $P$ and its image in the mirror. The apparent distance between these two will be

$A$ nucleus ruptures into two nuclear parts which have their velocity ratio equal to $2:1$. What will be the ratio of their nuclear size (nuclear radius)?

Two adjacent sides of a parallelogram $ABCD$ are given by $\overline{AB}=2 \hat{i}+10 \hat{j}+11 \hat{k}$ and $\overline{AD}=-\hat{i}+2 \hat{j}+2 \hat{k}$. The side $AD$ is rotated by an acute angle $\alpha$ in the plane of the parallelogram so that $AD$ becomes $AD^{\prime}$. If $AD^{\prime}$ makes a right angle with side $AB$,then the cosine of the angle $\alpha$ is given by

If the direction cosines of two lines are given by $l+m+n=0$ and $l^2-5m^2+n^2=0$,then the angle between them is