If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + 4x + 1 = 0$,then $(\alpha + \beta)^{-1} + (\beta + \gamma)^{-1} + (\gamma + \alpha)^{-1} = $

$A$ thin rectangular magnet suspended freely has a period of oscillation equal to $T.$ Now it is broken into two equal halves (each having half of the original length) and one piece is made to oscillate freely in the same field. If its period of oscillation is $T',$ the ratio $T'/T$ is :-

$A$ vibratory motion is represented by $x = 2A \cos \omega t + A \cos (\omega t + \frac{\pi}{2}) + A \cos (\omega t + \pi) + \frac{A}{2} \cos (\omega t + \frac{3\pi}{2})$. The resultant amplitude of the motion is:

$A$ wire elongates by $l \, mm$ when a load $W$ is hung from it. If the wire goes over a pulley and two weights $W$ each are hung at the two ends,the elongation of the wire will be (in $mm$)

$CH_4$ diffuses two times faster than a gas $X$. The number of molecules present in $32 \ g$ of gas $X$ is ($N$ is Avogadro number).

What do athletes misuse to increase muscle strength?