Let $Q$ be the cube with the set of vertices $\left\{\left(x_1, x_2, x_3\right) \in \mathbb{R}^3: x_1, x_2, x_3 \in\{0,1\}\right\}$. Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$. Let $S$ be the set of all four lines containing the main diagonals of the cube $Q$; for instance, the line passing through the vertices $(0,0,0)$ and $(1,1,1)$ is in $S$. For lines $\ell_1$ and $\ell_2$, let $d\left(\ell_1, \ell_2\right)$ denote the shortest distance between them. Then the maximum value of $d\left(\ell_1, \ell_2\right)$, as $\ell_1$ varies over $F$ and $\ell_2$ varies over $S$, is :

Consider the rectangles lying the region

$$\left\{ {(x,y) \in R \times R:0\, \le \,x\, \le \,{\pi \over 2}} \right.$$ and $$\left. {0\, \le \,y\, \le \,2\sin (2x)} \right\}$$

and having one side on the X-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is

By approximately matching the information given in the three columns of the following table.

Let f(x) = x + log_e x $$-$$ x log_e x, x$$ \in $$(0, $$\infty $$)

Column 1 contains information about zeroes of f(x), f'(x) and f"(x).

Column 2 contains information about the limiting behaviour of f(x), f'(x) and f"(x) at infinity.

Column 3 contains information about increasing/decreasing nature of f(x) and f'(x).

	Column - 1	Column - 2	Column - 3
(i)	f(x) = 0 for some $$x \in (1,{e^2})$$	(i) $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = 0$$	f is increasing in (0, 1)
(ii)	f'(x) = 0 for some $$x \in (1,e)$$	$$\mathop {\lim }\limits_{x \to \infty } \,f(x) = - \infty $$	f is decreasing in (e, $${e^2}$$)
(iii)	f'(x) = 0 for some $$x \in (0,1)$$	$$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = - \infty $$	f' is increasing in (0, 1)
(iv)	f'(x) = 0 for some $$x \in (1,e)$$	$$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = 0$$	f' is decreasing in (e, $${e^2}$$)

Which of the following options is the only INCORRECT combination?

By approximately matching the information given in the three columns of the following table.

Let f(x) = x + log_e x $$-$$ x log_e x, x$$ \in $$(0, $$\infty $$)

Column 1 contains information about zeroes of f(x), f'(x) and f"(x).

Column 2 contains information about the limiting behaviour of f(x), f'(x) and f"(x) at infinity.

Column 3 contains information about increasing/decreasing nature of f(x) and f'(x).

	Column - 1	Column - 2	Column - 3
(i)	f(x) = 0 for some $$x \in (1,{e^2})$$	(i) $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = 0$$	f is increasing in (0, 1)
(ii)	f'(x) = 0 for some $$x \in (1,e)$$	$$\mathop {\lim }\limits_{x \to \infty } \,f(x) = - \infty $$	f is decreasing in (e, $${e^2}$$)
(iii)	f'(x) = 0 for some $$x \in (0,1)$$	$$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = - \infty $$	f' is increasing in (0, 1)
(iv)	f'(x) = 0 for some $$x \in (1,e)$$	$$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = 0$$	f' is decreasing in (e, $${e^2}$$)

Which of the following options is the only CORRECT combination?