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?

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?

A block of mass $$M$$ has a circular cut with a frictionless surface as shown. The block resets on the horizontal frictionless surface of a fixed table. Initially the right edge of the block is at $$x=0,$$ in a co-ordinate system fixed to the table. A point mass $$m$$ is released from rest at the topmost point of the path as shown and it slides down.

When the mass loses contact with the block, its position is $$x$$ and the velocity is $$v.$$ At that instant, which of the following options is/are correct?

A block $$M$$ hangs vertically at the bottom end of a uniform rope of constant mass per unit length. The top end of the rope is attached to fixed rigid support at $$O.$$ A transverse wave pulse (Pulse 1) of wavelength $${\lambda _0}$$ is produced at point $$O$$ on the rope. The pulse takes time $${T_{OA}}$$ to reach point $$A.$$ If the wave pulse of wavelength $${\lambda _0}$$ is produced at point $$A$$ (Pulse 2) without disturbing the position of $$M$$ it takes time $${T_{AO}}$$ to reach point $$O.$$ which of the following options is/are correct?