1. Write the coordinates of the points

From the given position vectors:

$A=(2,-1,1),\quad B=(1,-3,-5),\quad C=(a,-3,1)$

We are told that the triangle is right-angled at $C$, i.e. $\angle C=\dfrac{\pi}{2}$.

So the vectors $\overrightarrow{CA}$ and $\overrightarrow{CB}$ must be perpendicular.

2. Find the vectors $\overrightarrow{CA}$ and $\overrightarrow{CB}$

$\overrightarrow{CA}=A-C=(2-a,-1-(-3),1-1)=(2-a,2,0)$

$\overrightarrow{CB}=B-C=(1-a,-3-(-3),-5-1)=(1-a,0,-6)$

3. Use the perpendicular condition

For perpendicular vectors,

$\overrightarrow{CA}\cdot \overrightarrow{CB}=0$

So,

$(2-a)(1-a)+2\cdot 0+0\cdot(-6)=0$

$(2-a)(1-a)=0$

Thus,

$a=2 \quad \text{or} \quad a=1$

4. Match with the options

The values are:

$2 \text{ and } 1$

So the correct option is A.

5. Comparison with stored correct answer

Stored correct answer: A

Our derived answer: A

Hence, the derived answer agrees with the stored correct answer.