2026 MSHS P1 Q17

The diagram shows a sector \(ABC\) of a circle with centre at \(A\).

\(PQC\) is a sector of another circle with centre at \(Q\). \(APB\) is tangent to this circle at point \(P\).

Angle \(BAC=\dfrac\pi6\) radian and angle \(PQC=\dfrac{2\pi}3\) radian.

1. Given that the length of arc \(PC\) is \(2\pi\text{ cm}\), find the length of \(PQ\).[1]
2. Calculate the length \(AQ\).[1]
3. Calculate the shaded area.[4]