Write True or False and justify your answer in each of the following:
If $A, B, C, D$ are four points such that $\angle BAC = 30^{\circ}$ and $\angle BDC = 60^{\circ}$, then $D$ is the centre of the circle through $A, B$ and $C$.

(B) The given statement is False.
Justification:
According to the theorem in geometry, the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
If $D$ were the centre of the circle passing through $A, B,$ and $C$, then the angle subtended by the arc $BC$ at the centre $D$ $(\angle BDC)$ must be twice the angle subtended by the same arc at the circumference $(\angle BAC)$.
Given $\angle BAC = 30^{\circ}$, then $\angle BDC$ should be $2 \times 30^{\circ} = 60^{\circ}$.
However, the condition $\angle BDC = 60^{\circ}$ is necessary but not sufficient for $D$ to be the centre. There can be infinitely many points $D$ such that $\angle BDC = 60^{\circ}$ (for example, any point on the major arc $BC$ of a circle passing through $B, C$ and $D$).
Therefore, $D$ is not necessarily the centre of the circle passing through $A, B,$ and $C$.