The minimum number of elements that must be added to the relation $$ \mathrm{R}=\{(\mathrm{a}, \mathrm{b}),(\mathrm{b}, \mathrm{c})\}$$ on the set $$\{a, b, c\}$$ so that it becomes symmetric and transitive is :

Let the system of linear equations

$$x+y+kz=2$$

$$2x+3y-z=1$$

$$3x+4y+2z=k$$

have infinitely many solutions. Then the system

$$(k+1)x+(2k-1)y=7$$

$$(2k+1)x+(k+5)y=10$$

has :

If [t] denotes the greatest integer $$\le \mathrm{t}$$, then the value of $${{3(e - 1)} \over e}\int\limits_1^2 {{x^2}{e^{[x] + [{x^3}]}}dx} $$ is :

If $$\mathrm{P}(\mathrm{h}, \mathrm{k})$$ be a point on the parabola $$x=4 y^{2}$$, which is nearest to the point $$\mathrm{Q}(0,33)$$, then the distance of $$\mathrm{P}$$ from the directrix of the parabola $$\quad y^{2}=4(x+y)$$ is equal to :