I found this problem in TIFR-GS paper.I have also solved this problem.Can someone please tell me if there are more interesting facts hidden in this problem that needs my attention or which I have missed.I would also like if someone gives me some reference about more such problems.
Soln. Let $x=\begin{bmatrix} x_1 \\ x_2\\ \vdots\\ x_n\\ \end{bmatrix}$ and $y=\begin{bmatrix} y_1 \\ y_2\\ \vdots\\ y_n\\ \end{bmatrix}$
Notice that $xy^t=\begin{bmatrix} x_1y_1 & x_1y_2 & \dots &x_1y_n\\ x_2y_1 & x_2y_2 & \dots & x_2y_n\\ \vdots & \vdots & \dots &\dots \\ x_ny_1 & x_ny_2 & \dots & x_ny_n\\ \end{bmatrix}$
each column is a multiple of the column matrix $x\neq 0$ and at least one column is non-zero as $y\neq 0$.
Suppose,we take $x,y\in \mathbb R^4, x=\begin{bmatrix} 1 \\ 0\\ 0\\ 0\\ \end{bmatrix}$ and $y=\begin{bmatrix} 0 \\ 1\\ 0\\ 0\\ \end{bmatrix}$
So,$A=xy^t=\begin{bmatrix} 0 & 1 & 0 &0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\\ \end{bmatrix}$
So,rank of $A$ is $1$,so $(d)$ is correct.
Yes what you write is correct and answer is d. For more similar questions you may want to check "Linear Algebra Problem Book" by Paul Halmos.