2024 IGCSE Additional Mathematics February/March 0606/22 Q4

2024 IGCSE Additional Mathematics February/March 0606/22 Q4

12 marks
    1. Given that \(y=3\sin^2x+\cos x\), show that \(y+\cot x\dfrac{\mathrm{d}y}{\mathrm{d}x}=k(1+\cos^2x)\), where \(k\) is an integer.[4]
    2. Using your value of \(k\), solve the equation \(k(1+\cos^2x)=4\) for \(-\pi\leq x\leq\pi\).[4]
    1. Differentiate \(y=\tan(x-\sqrt x)\) with respect to \(x\).[2]
    2. Hence find \(\displaystyle\int\dfrac{2\sqrt x-1}{\sqrt x\cos^2(x-\sqrt x)}\,\mathrm{d}x\).[2]

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Answer:(a)(i) \(k=3\). (ii) \(x=\pm0.955,\pm2.19\) radians. (b)(i) \(\left(1-\dfrac1{2\sqrt x}\right)\sec^2(x-\sqrt x)\). (ii) \(2\tan(x-\sqrt x)+C\).

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