$100$ cells each of e.m.f. $5 \ V$ and internal resistance $1 \ \Omega$ are to be arranged so as to produce maximum current in a $25 \ \Omega$ resistance. Each row is to contain an equal number of cells. The number of rows should be

The equation of the hyperbola which passes through the point $(2,3)$ and has the asymptotes $4x+3y-7=0$ and $x-2y-1=0$ is

The equation of the pair of lines passing through the point $(2, 1)$ and perpendicular to the pair of lines $4xy + 2x + 6y + 3 = 0$ is:

$A$ ball is dropped from a height of $20 \ m$ above the surface of water in a lake. The refractive index of water is $4/3$. $A$ fish inside the lake, in the line of fall of the ball, is looking at the ball. At an instant, when the ball is $12.8 \ m$ above the water surface, the fish sees the speed of the ball as ........ $m/s$ $[g = 10 \ m/s^2.]$

$A$ container of mass $m$ is pulled by a constant force $F$. Inside the container,a second block of the same mass $m$ is placed,connected to the wall by a massless spring of constant $k$. Initially,the spring is at its natural length. Find the velocity of the container at the instant when the compression in the spring is maximum for the first time.

$A$ radioactive nucleus (initial mass number $A$ and atomic number $Z$) emits $3$ $\alpha$-particles and $2$ positrons. The ratio of neutrons to protons in the final nucleus will be: