(A) The amplitude is $A$ not $2A$ .
(B) The direction of the velocity can be found by required that the argument of the sine function is a constant:
$\begin{align*}\frac{t}{T} - \frac{x}{\lambda} = \mbox{ const}\end{align*}$
Increasing t a little bit requires you to increase x a little bit as well in order to maintain this equality, so the velocity in the positive x-direction.
(C) The period is $T$ .
(D) $x$ and $t$ are variables here, so this cannot be right.
(E) Consider the equation above. Let $\Delta x$ be the change in $x$ during a small time interval $\Delta t$ . Then, in order to maintain that equality we must have that
$\begin{align*}\frac{\Delta t}{T} - \frac{\Delta x}{\lambda} = 0 \Rightarrow \frac{\Delta x}{\Delta t} = \frac{ \lambda}{T}\en...$
So, answer (E) is correct.

Solution to 1986 Problem 4