The resistance across $M$ and $N$ in the given figure is

$X$ being a higher category is the assemblage of families which exhibit a few $Y$ characters. The $Z$ characters are less in number as compared to different genera included in a family. Identify $X$,$Y$,and $Z$.

If $f(x)=(x-1)(x-2)(x-3)$ for $x \in[0,4]$,then the value of $c \in(0,4)$ satisfying Lagrange's mean value theorem is:

If $a = \hat{i} + \hat{j} + t \hat{k}$ and $b = \hat{i} + 2 \hat{j} + 3 \hat{k}$,then the values of $t$ for which $(a+b)$ and $(a-b)$ are perpendicular are:

The focal chord to $y^2 = 16x$ is tangent to $(x - 6)^2 + y^2 = 2$. Then the possible values of the slope of this chord are

In each of the following,determine whether the statement is true or false. If it is true,prove it. If it is false,give a counterexample.
If $x \in A$ and $A \in B,$ then $x \in B$.