What is the primary purpose of data entry in the context of mathematical research?

$A$ body is projected from the earth at an angle of $30^{\circ}$ with the horizontal with some initial velocity. If its range is $20 \ m$,the maximum height reached by it is (in meters):

If the angles of a triangle are in the ratio $1: 1: 4$,then the ratio of the perimeter of the triangle to its largest side is

In a series $LCR$ circuit,what will be the nature of the circuit for frequencies higher than the resonance frequency?

Each of the roots of the equation $x^3-6x^2+6x-5=0$ is increased by $h$. If the new transformed equation does not contain the $x^2$ term,then $h$ is equal to:

Two cells $E_1$ and $E_2$ having equal e.m.f '$E$' and internal resistances $r_1$ and $r_2$ $(r_1 > r_2)$ respectively are connected in series. This combination is connected to an external resistance '$R$'. It is observed that the potential difference across the cell $E_1$ becomes zero. The value of $R$ will be