Let a , b and c are distinct integers. Let A be a matrix

​$$A = \begin{pmatrix}a^2 & b^2 & c^2 \\a^5 & b^5 & c^5 \\a^{11} & b^{11} & c^{11}\end{pmatrix}$$

Which among the following is the set of all possible ranks of A ? 

Correct option is B

Let ,

​$$A = \begin{pmatrix}a^2 & b^2 & c^2 \\a^5 & b^5 & c^5 \\a^{11} & b^{11} & c^{11}\end{pmatrix}$$;

Rank of A cannot be zero because all the integers should be zero

to achieve that , but we have to take a,b,c distinct ,

Hence Option D is incorrect.

Now, 

(i) Let a , b , c = 1,0,-1 , Then:

$$A = \begin{pmatrix}1^2 & 0^2 & (-1)^2 \\1^5 & 0^5 & (-1)^5 \\1^{11} & 0^{11} & (-1)^{11}\end{pmatrix}= \begin{pmatrix}1 & 0 & 1 \\1 & 0 & -1 \\1 & 0 & -1\end{pmatrix} \implies \rho(A)=2$$​  Hence, Option A is incorrect. 

(ii) Let a,b,c = 1,2,3 Then A becomes:

​$$A = \begin{pmatrix}1^2 & 2^2 & 3^2 \\1^5 & 2^5 & 3^5 \\1^{11} & 2^{11} & 3^{11}\end{pmatrix} \text{This matrix is non-singular}\\[10pt]\implies \rho(A)=3\\[10pt]$$​​

According to given condition it is impossible to find a matrix A of rank 1. 

​$$\textbf{Hence, Option B is correct.}$$​​