Binomial Series (Edexcel IGCSE Further Pure Maths): Exam Questions

Show that

$\frac{a}{\sqrt{4+bx}}=\frac{a}{2}{\left(1+\frac{bx}{4}\right)}^{-\frac{1}{2}}$ where $a$ and $b$are positive integers.

The expansion of $\frac{a}{\sqrt{4+bx}}$ in ascending powers of $x$can be written as

$P+Qx+R{x}^2+S{x}^3$

where $P$, $Q$, $R$ and $S$ are rational numbers.

Show that $Q=-\frac{ab}{16}$ and $S=-\frac{5a{b}^3}{2048}$

and find $P$ and $R$ in terms of $a$ and $b$, as fractions in their lowest terms.

Given that «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«semantics»«mrow»«mi»Q«/mi»«mo»=«/mo»«mfrac»«mn»128«/mn»«mn»5«/mn»«/mfrac»«mi»S«/mi»«/mrow»«annotation encoding=¨application/vnd.wiris.mtweb-params+json¨»{¨fontFamily¨:¨Times New Roman¨,¨fontSize¨:¨18¨,¨autoformat¨:true,¨toolbar¨:¨«toolbar ref=`general`»«tab ref=`general`»«removeItem ref=`setColor`/»«removeItem ref=`bold`/»«removeItem ref=`italic`/»«removeItem ref=`autoItalic`/»«removeItem ref=`setUnicode`/»«removeItem ref=`mtext` /»«removeItem ref=`rtl`/»«removeItem ref=`forceLigature`/»«removeItem ref=`setFontFamily` /»«removeItem ref=`setFontSize`/»«/tab»«/toolbar»¨}«/annotation»«/semantics»«/math» and $R=\frac{9}{256}$

show that $a$ = 3 and $b$ =1

Hence, using an appropriate value of $x$, find, to 3 decimal places, an approximate value for $\frac{\sqrt{6}}{2}$

Expand ${\left(1+\frac{x}{3}\right)}^{-3}$ in ascending powers of $x$ up to and including the term in $x^3$

Where appropriate express each coefficient as an exact fraction in its lowest terms.

Write down the range of values of $x$ for which your expression is valid.

Express ${\left(3+x\right)}^{-3}$ in the form $P{\left(1+Qx\right)}^{-3}$ where $P$ and $Q$ are rational numbers whose values should be stated.

$\mathrm{f}\left(x\right)=\frac{1+4x}{(3+x{)}^3}$

Obtain a series expansion for $\mathrm{f}\left(x\right)$ in ascending powers of $x$ up to and including the term in $x^2$

$\mathrm{f}\left(x\right)=\frac{1+4x}{(3+x{)}^3}$

Hence, using algebraic integration, obtain an estimate of ${\int }_0^{0.2}\mathrm{f}\left(x\right) \mathrm{d}x$

Give your answer to 5 significant figures.

Expand $(1-8x^2{)}^{-\frac{1}{2}}$ in ascending powers of $x$, up to and including the term in $x^6$ giving each coefficient as an integer.

$\mathrm{g}\left(x\right)=\frac{a+bx}{\sqrt{1-8x^2}}$ where $a$ and $b$ are prime numbers

Given that the fourth and fifth terms, in ascending powers of $x$, in the series expansion of $\mathrm{g}\left(x\right)$ are 20$x^3$and 48$x^4$ respectively,

find the value of $a$ and the value of $b$

Using the first five terms, in ascending powers of $x$, in the series expansion of $\mathrm{g}\left(x\right)$

obtain an estimate, to 4 significant figures, of ${\int }_0^{0.2}\mathrm{g}\left(x\right) \mathrm{d}x$

Given that ${\left(8+3x\right)}^{\frac{1}{3}}$ can be expressed in the form ${p\left(1+qx\right)}^{\frac{1}{3}}$ where $p$ and $q$ are constants,

find the value of $p$ and the value of $q$

Hence, expand ${\left(8+3x\right)}^{\frac{1}{3}}$ in ascending powers of $x$ up to and including the term in $x$² , expressing each coefficient as an exact fraction in its lowest terms.

Using the expansion found in part (b) with a suitable value of $x$

show that $\sqrt[3]{9}\approx \frac{599}{288}$