Match the following columns:

Column-$I$	Column-$II$
$A$. Christmas tree	$i$. Bennettites
$B$. Fossil plant	$ii$. Cycas
$C$. Ornamental plant	$iii$. Araucaria
$D$. Tallest tree	$iv$. Sequoia
	$v$. Thuja

Let $a, b, c$ be distinct non-negative numbers. If the vectors $a \hat{i} + a \hat{j} + c \hat{k}$,$\hat{i} + \hat{k}$,and $c \hat{i} + c \hat{j} + b \hat{k}$ lie in a plane,then $c$ is:

The centre of mass of a body

$A$ finite ladder is constructed by connecting several sections of $2\,\mu F$ and $4\,\mu F$ capacitor combinations as shown in the figure. It is terminated by a capacitor of capacitance $C$. What value should be chosen for $C$ such that the equivalent capacitance of the ladder between the points $A$ and $B$ becomes independent of the number of sections in between? (Answer in $\mu F$)

$A$ straight line which makes equal intercepts on positive $X$ and $Y$ axes and which is at a distance of $1$ unit from the origin intersects the straight line $y = 2x + 3 + \sqrt{2}$ at $(x_0, y_0)$. Then $2x_0 + y_0$ is equal to

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is