$A$ 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 ......... $\%$

$\cos 36^{\circ} - \cos 72^{\circ}$ is equal to

$A$ tank with vertical walls is mounted so that its base is at a height $H$ above the horizontal ground. The tank is filled with water to a depth $h$. $A$ hole is punched in the side wall of the tank at a depth $x$ below the water surface. To have maximum range of the emerging stream, the value of $x$ is

For the following cell reaction,$Ag|Ag^{+}|AgCl|Cl^{-}|Cl_2, Pt$
$\Delta G_f^{\circ}(AgCl) = -109 \ kJ / mol$
$\Delta G_f^{\circ}(Cl^{-}) = -129 \ kJ / mol$
$\Delta G_f^{\circ}(Ag^{+}) = 78 \ kJ / mol$
The $E^{\circ}$ of the cell is:

Let $a, b$,and $c$ be such that $\frac{1}{(1-x)(1-2x)(1-3x)} = \frac{a}{1-x} + \frac{b}{1-2x} + \frac{c}{1-3x}$. Then $\frac{a}{1} + \frac{b}{3} + \frac{c}{5}$ is equal to

The equation of the lines passing through the origin and having slopes $3$ and $-\frac{1}{3}$ is