2024 IGCSE Additional Mathematics May/June 0606/23 Q5

2024 IGCSE Additional Mathematics May/June 0606/23 Q5

14 marks
  1. The function \(\mathrm{f}\) is defined by \(\mathrm{f}(x)=\dfrac{1+2\sin^2x}{\cos^2x}\) for \(-\dfrac\pi2<x<\dfrac\pi2\).
    1. Show that \(\mathrm{f}(x)\) can be written as \(a\tan^2x+b\), where \(a\) and \(b\) are integers.[2]
    2. Hence solve the equation \(\mathrm{f}(x)=4\).[3]
    3. Hence also find the gradient of the curve \(y=\mathrm{f}(x)\) at each of the points where \(y=4\).[4]
  2. Solve the equation \(50\cos^2\theta=5\sin\theta+47\) for \(0^\circ\leq\theta\leq360^\circ\).[5]

Solution:

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Answer:(a)(i) \(3\tan^2x+1\). (ii) \(x=\pm\dfrac\pi4\). (iii) Gradients \(-12,12\), respectively. (b) \(\theta=11.5^\circ,168.5^\circ,197.5^\circ,342.5^\circ\).

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