$A$ bag contains $n$ white and $n$ black balls. Pairs of balls are drawn at random without replacement successively,until the bag is empty. If the number of ways in which each pair consists of one white and one black ball is $14400$,then $n$ is equal to

If $(x, y, z)$ is an arbitrary point lying on a plane $P$ which passes through the points $(42, 0, 0)$,$(0, 42, 0)$,and $(0, 0, 42)$,then the value of the expression $3 + \frac{x-11}{(y-19)^2(z-12)^2} + \frac{y-19}{(x-11)^2(z-12)^2} + \frac{z-12}{(x-11)^2(y-19)^2} - \frac{x+y+z}{14(x-11)(y-19)(z-12)}$ is:

Two resistances of $400\,\Omega$ and $800\,\Omega$ are connected in series with a $6\,V$ battery of negligible internal resistance. $A$ voltmeter of resistance $10,000\,\Omega$ is used to measure the potential difference across the $400\,\Omega$ resistor. The error in the measurement of potential difference (in volts) is approximately:

The thermo emf of a thermocouple is given by $E = aT + bT^2$,where $\frac{a}{b} = -200^{\circ}C$. If the cold junction is kept at $30^{\circ}C$,then the inversion temperature is ($\varepsilon$ in volt,$T$ in centigrade). (in $K$)

$A$ large open tank has two holes in the wall. One is a square hole of side $L$ at a depth $y$ from the top and the other is a circular hole of radius $R$ at a depth $4y$ from the top. When the tank is completely filled with water,the quantities of water flowing out per second from both the holes are the same. Then $R$ is equal to

At $298 \ K$,the molar conductivities at infinite dilution $(\Lambda_m^{\circ})$ of $NH_4Cl$,$KOH$,and $KCl$ are $152.8$,$272.6$,and $149.8 \ S \ cm^2 \ mol^{-1}$ respectively. The $\Lambda_m^{\circ}$ of $NH_4OH$ in $S \ cm^2 \ mol^{-1}$ and $\%$ dissociation of $0.01 \ M \ NH_4OH$ with $\Lambda_m = 25.1 \ S \ cm^2 \ mol^{-1}$ at the same temperature are: