2025 TMJC P2 Q2

It is given that \(\mathrm{f}(\theta) = a\theta^2 + b\theta + c\), where \(a\), \(b\) and \(c\) are real constants. The curve with equation \(y = \mathrm{f}(\theta)\) passes through the point with coordinates \((0, 1)\) and has a turning point at \(\left( -\frac{\sqrt{3}}{3}, \frac{1}{2} \right)\).

1. Find the exact values of \(a\), \(b\) and \(c\).[3]
2. In a triangle \(ABC\), \(AB = 1\), \(AC = \sqrt{3}\) and angle \(BAC = \theta + \frac{\pi}{6}\) radians. Given that \(\theta\) is sufficiently small for \(\theta^3\) and higher powers of \(\theta\) to be neglected, show that \(BC^2 \approx \mathrm{f}(\theta)\).[3]
3. Hence, show that \(BC \approx 1 + \frac{\sqrt{3}}{2}\theta + \frac{3}{8}\theta^2\).[3]