$A$ coin and a six-faced die, both unbiased, are thrown simultaneously. The probability of getting a head on the coin and an odd number on the die is:

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and $\lambda$ is a real number, then the vectors $\vec{a}+2 \vec{b}+3 \vec{c}$, $\lambda \vec{b}+4 \vec{c}$, and $(2 \lambda-1) \vec{c}$ are non-coplanar for:

$\int \frac{dx}{(x+1) \sqrt{4x+3}}$ is equal to

$A$ thin circular ring of mass $M$ and radius $r$ is rotating about its axis with a constant angular velocity $\omega$. Four particles,each of mass $m$,are gently placed on the ring at opposite points of two perpendicular diameters. The new angular velocity of the ring will be:

If $\frac{e^x + 2}{(e^x - 1)(2e^x - 3)} = -\frac{3}{e^x - 1} + \frac{B}{2e^x - 3}$,then $B = $

The reagent used for converting acetylene to oxalic acid is