2025 PHS PRELIMS P2 Q9

2025 PHS PRELIMS P2 Q9

Secondary 4
12 marks
2025 Presbyterian High School Prelims A Math Paper 2
  1. State
    1. the principal value of \( \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) \),[1]
    2. the values between which the principal value of \( \tan^{-1} x \) must lie,[1]
    3. the range of values of \( k \) for which the equation \( 4\sin^2 x = k \) has a solution.[1]
    1. Prove the identity \( \cot x - \cot x \tan^2 x + \tan x = \frac{1+\cos 2x}{\sin 2x} \).[4]
    2. Hence solve the equation \( \cot x - \cot x \tan^2 x + \tan x = \operatorname{cosec} 2x - 3 \) for \( 0^\circ \leq x \leq 180^\circ \).[5]

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Answer:(a)(i) \(-60^\circ\), (ii) \(-90^\circ<\tan^{-1}x<90^\circ\), (iii) \(0\leq k\leq 4\) (b)(i) proven, (ii) \(x=80.8^\circ,170.8^\circ\)

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