Let set X = {1, 2, 3} and a relation R is defined in X as : R = {(1, 3), (2, 2), (3, 2)}, then minimum ordered pairs which should be added in relation R to make it reflexive and symmetric are

A Linear Programming Problem is as follows:

Minimize z = 2x + y

subject to the constraints

x $$\ge$$ 3, x $$\le$$ 9, y $$\ge$$ 0

x $$-$$ y $$\ge$$ 0, x + y $$\le$$ 14

The feasible region has

The function $$f(x) = \left\{ {\matrix{ {{{{e^{3x}} - {e^{ - 5x}}} \over x},} & {if\,x \ne 0} \cr {k,} & {if\,x = 0} \cr } } \right.$$ is continuous at x = 0 for the value of k, as

If C_ij denotes the cofactor of element p_ij of the matrix $$P = \left[ {\matrix{ 1 & { - 1} & 2 \cr 0 & 2 & { - 3} \cr 3 & 2 & 4 \cr } } \right]$$, then the value of $${C_{31}}\,.\,{C_{23}}$$ is