Read a tangent from the first two coefficients · source reference

Printed question

Find the first three terms in the expansion of $\frac{1+x^2}{\sqrt{3-2x}}$ in ascending powers of $x$.

State the range of values of $x$ for which the expansion is valid. [3]

Deduce that the equation of the tangent to the curve $y=\frac{2+2x^2}{\sqrt{3-2x}}$ at the point where $x=0$, is given by $9y=2\sqrt3x+6\sqrt3$. [2]

[modified DHS/2008/Prelim/I/5]

Printed working

$$\begin{aligned}\frac{1+x^2}{\sqrt{3-2x}}&=(1+x^2)(3-2x)^{-1/2}=\frac1{\sqrt3}(1+x^2)(1-2x/3)^{-1/2}\\&=\frac1{\sqrt3}(1+x^2)[1+(-1/2)(-2x/3)+\tfrac{(-1/2)(-3/2)}{2!}(-2x/3)^2+\cdots]\\&=\frac1{\sqrt3}(1+x^2)(1+x/3+x^2/6+\cdots)\\&=\frac1{\sqrt3}+\frac{x}{3\sqrt3}+\frac{7x^2}{6\sqrt3}+\cdots.\end{aligned}$$

$|-2x/3|<1\Rightarrow|x|<3/2\Rightarrow-3/2<x<3/2$.

$y=2/\sqrt3+2x/(3\sqrt3)+14x^2/(6\sqrt3)+\cdots$; at $x=0$, $y=2/\sqrt3$, $y'=2/(3\sqrt3)$. Tangent $y=2/\sqrt3+2x/(3\sqrt3)\Rightarrow3\sqrt3y=6+2x\Rightarrow9y=6\sqrt3+2\sqrt3x$.

Teacher note 1

Need to factorise out $3$ to make the constant term 1. Use the law of indices $(AB)^n=A^nB^n$. Then use the binomial theorem

$$(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\cdots,$$

where $n=-1/2$ and replace $x$ by $-2x/3$.

Teacher note 2

At $x=0$, $y=2/\sqrt3$ and $y'=2/(3\sqrt3)$. Thus tangent at $x=0$ is $y-2/\sqrt3=(2/(3\sqrt3))(x-0)$. The constant and term in $x$ are the first two terms in the expansion of $y$.

Mathematical review

$(3-2x)^{-1/2}=3^{-1/2}(1+x/3+x^2/6+O(x^3))$. Multiplication gives $(1+x^2)/\sqrt{3-2x}=1/\sqrt3+x/(3\sqrt3)+7x^2/(6\sqrt3)+O(x^3)$ for $|x|<3/2$.

The curve is twice this expression: $y(0)=2/\sqrt3$, $y'(0)=2/(3\sqrt3)$. Its tangent is $y=2/\sqrt3+2x/(3\sqrt3)$, equivalent to $9y=6\sqrt3+2\sqrt3\,x$.

Provenance

TJC JC2 2025 H2 Maths (9758) MSM. Native transcription checked against original visible question, working and sidebars. Question page(s): 5. Working page(s): 5. The two Series source files are byte-identical; their shared visible content is expressed once and mapped to both page sets. Original records remain private for QA; this native article has no PDF or raster worksheet dependency.