If $f(1)=1$ and $f^{\prime}(1)=3$,then the derivative of $f(f(f(x)))+(f(x))^2$ at $x=1$ is

Two nuclei have their mass numbers in the ratio of $1 : 3.$ The ratio of their nuclear densities would be

$A$ radioactive material decays by simultaneous emission of two particles with respective half-lives $1620 \ years$ and $810 \ years$. The time (in years) after which one-fourth of the material remains is

The electrons,identified by quantum numbers $n$ and $l,$ $(i)$ $n = 4, l = 1$ $(ii)$ $n = 4, l = 0$ $(iii)$ $n = 3, l = 2$ $(iv)$ $n = 3, l = 1$ can be placed in order of increasing energy,from the lowest to highest,as

If the line $x+3y=0$ is the tangent at $(0,0)$ to the circle of radius $1$,then the centre of one such circle is

$A$ river is flowing from east to west at a speed of $5\ m/min$. $A$ man on the south bank of the river,capable of swimming $10\ m/min$ in still water,wants to swim across the river in the shortest time. He should swim: