2025 CGS PRELIMS P2 Q10

2025 CGS PRELIMS P2 Q10

Secondary 4
15 marks
2025 Crescent Girls' School Prelims A Math Paper 2
    1. Show that \(\dfrac{(\sin \theta - \cos \theta)(1 + \sin \theta \cos \theta)}{\cos^3 \theta} = \tan^3 \theta - 1\).[4]
    2. Hence, for \(0 < \theta < 2\pi\), find the reflex angle \(\theta\) such that \((\sin \theta - \cos \theta)(1 + \sin \theta \cos \theta) = \cos^3 \theta\).[3]
  2. Solve the equation \(\sin 3A \cos 2A = \cos 3A \sin 2A\) for \(0 \leq A \leq 180^\circ\).[2]
  3. Given that \(\sin x = -\dfrac{4}{5}\) and \(\tan x > 0\), find, without using a calculator, the value of \(\cos 4x\).[2]
  4. Without using a calculator, show that \(\tan \dfrac{\pi}{12} = 2 - \sqrt{3}\).[4]

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Answer:(a)(i) \(\tan^3\theta-1\) (a)(ii) \(\theta=\pi+\tan^{-1}(\sqrt[3]{2})=4.04\text{ rad}\) (b) \(A=0^\circ,180^\circ\) (c) \(-\dfrac{527}{625}\) (d) \(2-\sqrt3\)

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