If $m$ and $\sigma ^2$ are the mean and variance of random variable $X$,whose distribution is given by:

$X=x$	$0$	$1$	$2$	$3$	$4$
$P(X=x)$	$\frac{1}{3}$	$\frac{1}{2}$	$0$	$\frac{1}{6}$	$0$

then:

If $PM$ is the perpendicular from $P(2, 3)$ onto the line $x+y=3$,then the coordinates of $M$ are

$A$ gas mixture consists of $2$ moles of oxygen and $4$ moles of argon at temperature $T$. Neglecting all vibrational modes,the total internal energy of the system is: (in $RT$)

The average marks of $10$ students in a class was $60$ with a standard deviation of $4$,while the average marks of another $10$ students was $40$ with a standard deviation of $6$. If all the $20$ students are taken together,their standard deviation will be:

$A$ narrow tube is bent in the form of a circle of radius $R,$ as shown in the figure. Two small holes $S$ and $D$ are made in the tube at positions right-angled to each other. $A$ source placed at $S$ generates a wave of intensity $I_0$ which is equally divided into two parts: one part travels along the longer path,while the other travels along the shorter path. Both the waves meet at the point $D$ where a detector is placed. The maximum value of $\lambda$ to produce a maxima at $D$ is given by

The incentre of the triangle formed by the lines $x+y=1$,$x=1$,and $y=1$ is