If the sum of the roots of the equation $x^2 + px + q = 0$ is three times their difference,then which of the following is true?

The angle between the lines joining the origin to the points of intersection of the line $x + 2y + 1 = 0$ and the curve $2x^2 - 2xy + 3y^2 + 2x - y - 1 = 0$ is

Two spheres $A$ and $B$ are kept very near to each other. $A$ is negatively charged and $B$ is earthed. The true statements are:
$(A)$ Charge on $B$ is zero
$(B)$ Potential at $B$ is zero
$(C)$ Charge is uniformly distributed on $A$
$(D)$ Charge is non-uniformly distributed on $A$

If the points $(1,2)$ and $(3,4)$ lie on the same side of the straight line $3x - 5y + a = 0$,then $a$ lies in the set

If $f: R \rightarrow R$ is defined by $f(x)=x-[x]-\frac{1}{2}$ for $x \in R$,where $[x]$ is the greatest integer not exceeding $x$,then $\{x \in R: f(x)=\frac{1}{2}\}$ is equal to :

If $\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z=\frac{\pi}{2}$,then $1-xy-yz-zx$ is equal to