2026 ZHSS P1 Q12

The diagram shows part of the curve \(y=a\sin kx\) where \(a\) and \(k\) are positive constants and \(x\) is in radians.

The curve crosses the positive \(x\)-axis at \(A\) and \(B\).

Point \(A\) is the second positive \(x\)-intercept and point \(B\) is the next positive \(x\)-intercept after point \(A\).

1. 1. Find, in terms of \(k\), the coordinates of \(A\) and of \(B\).[2]
      Point \(C\) is the maximum point on the curve between \(A\) and \(B\).
      Points \(A\), \(B\) and \(C\) form an isosceles triangle as shown in the diagram where angle \(ABC=\) angle \(BAC=\dfrac\pi6\).
   2. Show that \(a=\dfrac{\sqrt3\pi}{6k}\).[3]
2. Given that the amplitude of the curve \(y=a\sin kx\) is \(3^{3/2}\), find the period of the curve.[3]