The length of the $x$-intercept made by the pair of lines $2x^2 + xy - 6y^2 - 2x + 17y - 12 = 0$ is

$A$ coin is tossed $n$ times. The probability of getting a head at least once is greater than $0.8$. Then,the least value of such $n$ is:

The plates of a parallel plate capacitor are charged up to $100\;V$. $A$ $2\;mm$ thick plate is inserted between the plates. Then,to maintain the same potential difference,the distance between the plates is increased by $1.6\;mm$. The dielectric constant of the plate is:

The variance of $20$ observations is $5$. If each observation is multiplied by $2$,then the new variance of the resulting observations is:

$\sec h^{-1}\left(\frac{1}{2}\right)-\operatorname{cosec} h^{-1}\left(\frac{3}{4}\right)$ is equal to

In a meter bridge,a $30 \Omega$ resistance is connected in the left gap and a pair of resistances $P$ and $Q$ in the right gap. Measured from the left,the balance point is $37.5 \text{ cm}$ when $P$ and $Q$ are in series,and $71.4 \text{ cm}$ when they are in parallel. The values of $P$ and $Q$ (in $\Omega$) are: