Two identical balls $A$ and $B$ are released from the positions shown in the figure. They collide elastically on the horizontal portion $MN$. The ratio of the heights attained by $A$ and $B$ after the collision will be (neglect friction):

The angle between the lines whose direction cosines satisfy the equations $l+m+n=0$ and $l^2+m^2-n^2=0$ is

$A$ satellite is launched into a circular orbit of radius $R$ around the Earth,while a second satellite is launched into an orbit of radius $1.02 R$. The percentage difference in the time periods of the two satellites is ......... $\%$

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 angle $\alpha$ in the plane of the parallelogram so that $AD$ becomes $AD^{\prime}$. If $AD^{\prime}$ makes a right angle with the side $AB$,then the cosine of the angle $\alpha$ is given by

If $x$ is small,so that $x^2$ and higher powers can be neglected,then the approximate value for $\frac{(1-2 x)^{-1}(1-3 x)^{-2}}{(1-4 x)^{-3}}$ is

The cell junctions called tight,adhering,and gap junctions are found in