2026 RI Promo Q3

Junior College 1

8 marks

It is given that \(e^x+e^y+4y=2\) and that \(y=0\) when \(x=0\).
1. Show that \(e^x+e^y\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)^2+(4+e^y)\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}=0\).[2]
2. Find in exact form, the first two non-zero terms of the Maclaurin expansion of \(y\).[2]
Given that \(\theta\) is sufficiently small, show that
\[\frac{\sin\left(\frac{\pi}{4}+2\theta\right)}{1-\tan\theta}\approx\frac{1}{\sqrt2}(a+b\theta+c\theta^2),\]
where \(a\), \(b\) and \(c\) are real constants to be determined.[4]

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Answer:(a)(ii) \(y=-\dfrac{x}{5}-\dfrac{13x^2}{125}+\cdots\). (b) \(a=1,\ b=3,\ c=1\).

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