2025 MGS PRELIMS P1 Q9

2025 MGS PRELIMS P1 Q9

Secondary 4
9 marks
    1. Sketch the graph of \(y=3e^x\).[1]
    2. Find the equation of a suitable straight line that can be inserted in part (a)(i) to solve the equation \(e^x-x=6\).[2]
    1. Without using a calculator, show that \(\tan 75^\circ=a+b\sqrt{3}\), where \(a\) and \(b\) are integers.[3]
    2. Hence, find the exact value of \(\cot^2 75^\circ\).[3]

Solution:

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Answer:(a)(i) Increasing exponential curve through \((0,3)\) with horizontal asymptote \(y=0\) | (a)(ii) \(y=3x+18\) | (b)(i) \(\tan75^\circ=2+\sqrt3\), \(a=2\), \(b=1\) | (b)(ii) \(\cot^2 75^\circ=7-4\sqrt3\)

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