$A$ candle placed $25 \ cm$ from a lens forms an image on a screen placed $75 \ cm$ on the other side of the lens. The focal length and type of the lens are:

If the angle between the pair of lines $2x^2 + 2hxy + 2y^2 - x + y - 1 = 0$ is $\tan^{-1}(3/4)$ and $h$ is a positive rational number,then the point of intersection of these two lines is

Consider a solid insulating sphere of radius $R$ with charge density varying as $\rho = \rho_0 r^2$ (where $\rho_0$ is a constant and $r$ is measured from the centre). Consider two points $A$ and $B$ at distances $x$ and $y$ respectively from the centre,such that $x < R$ and $y > R$. If the magnitudes of the electric fields at points $A$ and $B$ are equal,then:

$\tan A + 2 \tan 2A + 4 \tan 4A + 8 \cot 8A = $

If $\alpha, \beta$ and $\gamma$ are the lengths of the altitudes of a $\triangle ABC$ with area $\Delta$,then $\frac{\Delta^2}{R^2}\left(\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}\right)$ is equal to

If the extremities of a diagonal of a square are $(1, -2, 3)$ and $(2, -3, 5)$,then the length of its side is