Further Pure Math (Binomial Theorem)

1. (2011/june/Paper02/q9)

(a) Expand $\left(1-\displaystyle\frac{3 x}{4}\right)^{\frac{1}{3}}$ in ascending powers of $x$ up to and including the term in $x^{3}$, simplifying your terms as far as possible.

(b) Expand $\left(1+\displaystyle\frac{3 x}{4}\right)^{-\displaystyle\frac{1}{3}}$ in ascending powers of $x$ up to and including the term in $x^{3}$, simplifying your terms as far as possible.

(c) Write down the range of values of $x$ for which both of your expansions are valid.

(d) Expand $\left(\displaystyle\frac{4-3 x}{4+3 x}\right)^{\frac{1}{3}}$ in ascending powers of $x$ up to and including the term in $x^{3}$, simplifying your terms as far as possible.

(e) Hence obtain an estimate, to 3 significant figures, of

$$\displaystyle\int_{0}^{0.5}\left(\displaystyle\frac{4-3 x}{4+3 x}\right)^{\frac{1}{3}} \mathrm{~d} x$$


2. $(2012 / \mathrm{jan} / \mathrm{paper} 01 / \mathrm{q} 4)$

Find the coefficient of $x^{7}$ in the expansion of $\left(1+\displaystyle\frac{x}{\sqrt{3}}\right)^{10}$, giving your answer in the form $a \sqrt{3}$, where $a$ is a rational number.


3. (2012/jan/paper02/q5)

(a) Expand $(1+3 x)^{\frac{1}{5}}$ in ascending powers of $x$ up to and including the term in $x^{3}$, simplifying your terms as far as possible.

(b) By substituting $x=-\displaystyle\frac{1}{8}$ into your expansion, obtain an approximation for $\sqrt[5]{20}$

Write down all the figures on your calculator display.

(c) Explain why you cannot obtain an approximation for $\sqrt[5]{4}$ by substituting $x=1$ into your expansion.


4. (2012/june/paper01/q3)

(a) Find the full binomial expansion of $(1+x)^{5}$, giving each coefficient as an integer. $(2)$

(b) Hence find the exact value of $(1-2 \sqrt{3})^{5}$, giving your answer in the form $a+b \sqrt{3}$, where $a$ and $b$ are integers.


5. (2013/jan/paper02/q3)

(a) Expand $\left(1+3 x^{2}\right)^{-\displaystyle\frac{1}{4}}$ in ascending powers of $x$ up to and including the term in $x^{6}$, giving each coefficient as a fraction in its lowest terms.

(b) Find the range of values of $x$ for which your expansion is valid.

$$\mathrm{f}(x)=\displaystyle\frac{3+k x^{2}}{\left(1+3 x^{2}\right)^{\frac{1}{4}}} \quad k \in \mathbb{R}^{+}$$

(c) Obtain a series expansion for $\mathrm{f}(x)$ in ascending powers of $x$ up to and including the term in $x^{6}$.

Given that the coefficient of $x^{4}$ in the series expansion of $f(x)$ is zero

(d) find the exact value of $k$. (2)


6. (2013/june/paper02/q9)

(a) Expand, in ascending powers of $x$ up to and including the term in $x^{3}$, simplifying each term as far as possible,

(i) $(1+x)^{-1}$

(ii) $(1-2 x)^{-1}$ $(4)$

Given that $\displaystyle\frac{2}{1-2 x}+\displaystyle\frac{1}{1+x}=\displaystyle\frac{A x+B}{(1-2 x)(1+x)}$

(b) find the value of $A$ and the value of $B$. (2)

(c) (i) Obtain a series expansion for $\displaystyle\frac{1}{(1-2 x)(1+x)}$ in ascending powers of $x$ up to and including the term in $x^{2}$

(ii) State the range of values of $x$ for which this expansion is valid.

(d) Use your series expansion from part (c) to obtain an estimate, to 3 decimal places, of $\displaystyle\int_{0.1}^{0.2} \displaystyle\frac{1}{(1-2 x)(1+x)} \mathrm{d} x$


7. (2014/jan/paper02/q9)

(a) Show that the first four terms of the expansion of $(1-x)^{-k}, k \neq 0$, in ascending powers of $x$ can be written as

$$1+k x+\displaystyle\frac{k(k+1)}{2} x^{2}+\displaystyle\frac{k(k+1)(k+2)}{6} x^{3}$$

(b) Expand $(1+k x)^{\frac{1}{2}}, k \neq 0$, in ascending powers of $x$, up to and including the term in $x^{3}$, simplifying your terms.

Given that the coefficients of $x^{2}$ in the two expansions are equal,

(c) find the value of $k$.

Given that $\sqrt{15}=\lambda \sqrt{\displaystyle\frac{3}{5}}$

(d) find the value of $\lambda$ (2)

(e) Hence, using your value of $k$ and one of your expansions with a suitable value of $x$, obtain an approximation for $\sqrt{15}$


8. (2014/june/paper01/q6)

(a) Expand $\left(1+4 x^{2}\right)^{-\displaystyle\frac{1}{5}}$ in ascending powers of $x$ up to and including the term in $x^{6}$, expressing each coefficient as an exact fraction in its lowest terms.

(b) Find the range of values of $x$ for which your expansion is valid.

$$\mathrm{f}(x)=\displaystyle\frac{1+k x}{\left(1+4 x^{2}\right)^{\frac{1}{5}}} \quad \text { where } k \neq 0$$

(c) Obtain a series expansion for $\mathrm{f}(x)$ in ascending powers of $x$ up to and including the term in $x^{5}$

Given that the coefficients of $x^{2}$ and $x^{5}$ in the expansion of $\mathrm{f}(x)$ are equal,

(d) find the value of $k$.


9. (2015/jan/paper02/q8)

(a) Find the full binomial expansion of $(1-2 x)^{5}$, giving each coefficient as an integer.

(b) Expand $(1+2 x)^{-5}$ in ascending powers of $x$ up to and including the term in $x^{3}$, giving each coefficient as an integer.

(c) Write down the range of values of $x$ for which this expansion is valid.

(d) Expand $\left(\displaystyle\frac{1-2 x}{1+2 x}\right)^{5}$ in ascending powers of $x$ up to and including the term in $x^{2}$, giving each coefficient as an integer.

(e) Find the gradient of the curve with equation $y=\left(\displaystyle\frac{1-2 x}{1+2 x}\right)^{5}$ at the point $(0,1)$.


10. (2015/june/paper01/q7)

(a) Expand $\left(1+\displaystyle\frac{x}{3}\right)^{\frac{1}{4}}$ in ascending powers of $x$ up to and including the term in $x^{3}$, giving each coefficient as an exact fraction.

(b) Expand $\left(1-\displaystyle\frac{x}{3}\right)^{-\displaystyle\frac{1}{4}}$ in ascending powers of $x$ up to and including the term in $x^{3}$, giving each coefficient as an exact fraction.

(c) Write down the range of values of $x$ for which both of your expansions are valid.

(d) Expand $\left(\displaystyle\frac{3+x}{3-x}\right)^{\frac{1}{4}}$ in ascending powers of $x$ up to and including the term in $x^{2}$, giving each coefficient as an exact fraction.

(e) Hence obtain an estimate, to 3 significant figures, of $\displaystyle\int_{0}^{0.6}\left(\displaystyle\frac{3+x}{3-x}\right)^{\frac{1}{4}} \mathrm{~d} x$


11. (2016/jan/paper01/q5)

Given that $\displaystyle\frac{1}{\sqrt{4-x}}$ can be written as $p(1-q x)^{-\displaystyle\frac{1}{2}}$

(a) find the value of $p$ and the value of $q$. (2)

(b) (i) Find the first four terms in the expansion of $\displaystyle\frac{1}{\sqrt{4-x}}$ in ascending powers of $x$, simplifying each term.

(ii) State the range of values of $x$ for which this expansion is valid.

Given that the first three terms of the expansion of $\displaystyle\frac{2(1+x)}{\sqrt{4-x}}$ are $a+b x+c x^{2}$

(c) find the exact value of

(i) $a$

(ii) $b$

(iii) $c$


12. (2016/june/paper01/q2)

(a) Expand $\left(1+3 x^{2}\right)^{-\displaystyle\frac{1}{3}}, 3 x^{2}<1$, in ascending powers of $x$, up to and including the term in $x^{6}$, simplifying each term as far as possible.

$$\mathrm{f}(x)=\displaystyle\frac{1-k x^{2}}{\left(1+3 x^{2}\right)^{\frac{1}{3}}} \text { where } k \text { is a constant }$$

(b) Obtain a series expansion for $\mathrm{f}(x)$ in ascending powers of $x$ up to and including the term in $x^{4}$.

Given that the coefficient of $x^{2}$ in the expansion of $f(x)$ is $-5$

(c) find the value of $k$. (1)


13. (2017/jan/paper01/q8)

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

(ii) Find the range of values for which your expression is valid.

(b) Express $(2+x)^{-3}$ in the form $A(1+B x)^{-3}$ where $A$ and $B$ are rational numbers whose values should be stated.

$$\mathrm{f}(x)=\displaystyle\frac{(1+4 x)}{(2+x)^{3}}$$

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

(d) Hence obtain an estimate, to 3 significant figures, of $\displaystyle\int_{0}^{0.2} \displaystyle\frac{(1+4 x)}{(2+x)^{3}} d x$


14. (2017/june/paper02/q6)

$$\mathrm{f}(x)=(p+q x)^{6} \text { where } p \neq 0 \text { and } q \neq 0$$

(a) Find the expansion of $f(x)$ in ascending powers of $x$ up to and including the term in $x^{4}$, simplifying each term as far as possible.

In the expansion of $\mathrm{f}(x), 4$ times the coefficient of $x^{4}$ is equal to 9 times the coefficient of $x^{2}$ Given that $(p+q)>0$ and $\mathrm{f}(1)=15625$

(b) find the possible pairs of values of $p$ and $q$.


15. (2018/jan/paper01/q7)

(a) Expand $\left(1-4 x^{2}\right)^{-\displaystyle\frac{1}{2}}$ in ascending powers of $x$, up to and including the term in $x^{6}$, giving each coefficient as an integer.

(b) Write down the range of values of $x$ for which your expansion is valid.

(c) Expand $\displaystyle\frac{3+x}{\sqrt{\left(1-4 x^{2}\right)}}$ in ascending powers of $x$ up to and including the term in $x^{4}$, giving each coefficient as an integer.

(d) Hence, use algebraic integration to obtain an estimate, to 3 significant figures, of

$$\displaystyle\int_{0}^{0.3} \displaystyle\frac{3+x}{\sqrt{\left(1-4 x^{2}\right)}} \mathrm{d} x$$


16. (2018/june/paper02/q7)

(a) Expand $\left(1+\displaystyle\frac{2 x}{5}\right)^{\frac{1}{2}}$ in ascending powers of $x$ up to and including the term in $x^{3}$, giving each coefficient as an exact fraction in its lowest terms.

(b) Expand $\left(1-\displaystyle\frac{2 x}{5}\right)^{-\displaystyle\frac{1}{2}}$ in ascending powers of $x$ up to and including the term in $x^{3}$, giving each coefficient as an exact fraction in its lowest terms.

(c) Write down the range of values of $x$ for which both of your expansions are valid.

(d) Expand $\left(\displaystyle\frac{5+2 x}{5-2 x}\right)^{\frac{1}{2}}$ in ascending powers of $x$ up to and including the term in $x^{2}$, giving each coefficient as an exact fraction in its lowest terms.

(e) Hence use algebraic integration to obtain an estimate of

$$\displaystyle\int_{0.1}^{0.3}\left(\displaystyle\frac{5+2 x}{5-2 x}\right)^{\frac{1}{2}} \mathrm{~d} x$$

Give your answer to 4 significant figures.


17. (2019/june/paper01/q10)

10 (a) Expand $\left(1+2 x^{2}\right)^{-\displaystyle\frac{1}{3}}$ in ascending powers of $x$ up to and including the term in $x^{6}$, expressing each coefficient as an exact fraction in its lowest terms.

(b) State the range of values of $x$ for which your expansion is valid.

$$\mathrm{f}(x)=\displaystyle\frac{2+k x^{2}}{\left(1+2 x^{2}\right)^{\frac{1}{3}}} \quad \text { where } k \neq 0$$

(c) Obtain a series expansion for $\mathrm{f}(x)$ in ascending powers of $x$ up to and including the term in $x^{6}$

Give each coefficient in terms of $k$ where appropriate.

Given that the coefficient of $x^{4}$ in the series expansion of $f(x)$ is zero

(d) find the value of $k$.

(e) Hence use algebraic integration to obtain an estimate, to 4 decimal places, of

$$\displaystyle\int_{0}^{0.5} f(x) \mathrm{d} x$$


18. (2019/juneR/paper02/q6)

Given that $\sqrt{9-x}$ can be expressed in the form $p(1+q x)^{\frac{1}{2}}$ where $p$ and $q$ are constants

(a) find the value of $p$ and the value of $q$.$(2)$

(b) Hence expand $\sqrt{9-x}$ in ascending powers of $x$ up to and including the term in $x^{3}$ expressing each coefficient as an exact fraction in its lowest terms.

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

(c) find an estimate to 5 decimal places for the value of $\sqrt{\displaystyle\frac{31}{4}}$





Answer

1.(a) $1-\displaystyle\frac{x}{4}-\displaystyle\frac{x^{2}}{16}-\displaystyle\frac{5 x^{3}}{192}$ (b) $1-\displaystyle\frac{x}{4}+\displaystyle\frac{x^{2}}{8}-\displaystyle\frac{7 x^{3}}{96}$ (c) $|x|<\displaystyle\frac{4}{3}$ (d) $1-\displaystyle\frac{x}{2}+\displaystyle\frac{x^{2}}{8}-\displaystyle\frac{11 x^{3}}{96}$ (e) $0.441$

2. $\displaystyle\frac{40}{27} \sqrt{3}$

3.(a) $\quad 1+\displaystyle\frac{3}{5} x-\displaystyle\frac{18}{25} x^{2}+\displaystyle\frac{162}{125} x^{3}+\cdots$ (b) $0.91121875 \ldots$ (c) $|x|<\displaystyle\frac{1}{3}$

4.(a) $1+5 x+10 x^{2}+10 x^{3}+5 x^{4}+x^{5}$ (b) $841-538 \sqrt{3}$

5.(a) $1-\displaystyle\frac{3}{4} x^{2}+\displaystyle\frac{45}{32} x^{4}-\displaystyle\frac{405}{128} x^{6}$ (b) $|x|<\displaystyle\frac{1}{\sqrt{3}}$ (c) $3+\left(k-\displaystyle\frac{9}{4}\right) x^{2}+\left(\displaystyle\frac{135}{32}-\displaystyle\frac{3 k}{4}\right) x^{4}+\left(\displaystyle\frac{45 k}{32}-\displaystyle\frac{1215}{128}\right) x^{6}$ (d) $k=\displaystyle\frac{45}{8}$

6.(a)(i) $\quad 1-x+x^{2}-x^{3} \ldots$ (ii) $1+2 x+4 x^{2}+8 x^{3}+\cdots$ (b) $A=0, B=3$ (c) $1+x+3 x^{2},|x|<\displaystyle\frac{1}{2}$ (d) $0.122$

7.(a) Show (b) $1+\displaystyle\frac{1}{2} k x-\displaystyle\frac{1}{8} k^{2} x^{2}+\displaystyle\frac{1}{16} k^{3} x^{3}$ (c) $k=-\displaystyle\frac{4}{5}$ (d) $\lambda=5$ (e) $3.88$

8.(a) $1-\displaystyle\frac{4}{5} x^{2}+\displaystyle\frac{48}{25} x^{4}-\displaystyle\frac{704}{125} x^{6}$ (b) $|x|<\displaystyle\frac{1}{2}$ (c) $1+k x-\displaystyle\frac{4}{5} x^{2}-\displaystyle\frac{4 k}{5} x^{3}+\displaystyle\frac{48}{25} x^{4}+\displaystyle\frac{48 k}{25} x^{5}$ (d) $k= -\displaystyle\frac{5}{12}$

9.(a) $1-10 x+40 x^{2}-80 x^{3}+80 x^{4}-32 x^{5}$ (b) $1-10 x+60 x^{2}-280 x^{3}+\cdots$ (c) $|x|<\displaystyle\frac{1}{2}$ (d) $1-20 x+200 x^{2}+\cdots$ (e) $-20$

10.(a) $1+\displaystyle\frac{x}{12}-\displaystyle\frac{x^{2}}{96}+\displaystyle\frac{7}{3456} x^{3}+\cdots$ (b) $1+\displaystyle\frac{x}{12}+\displaystyle\frac{5}{288} x^{2}+\displaystyle\frac{5}{1152} x^{3}+\cdots$ (c) $|x|<3$ (d) $1+\displaystyle\frac{x}{6}+\displaystyle\frac{x^{2}}{72}+\cdots$ (e) $0.631$

11.(a) $p=\displaystyle\frac{1}{2}, q=\displaystyle\frac{1}{4}$ (b)(i) $\displaystyle\frac{1}{2}+\displaystyle\frac{x}{16}+\displaystyle\frac{3 x^{2}}{256}+\displaystyle\frac{5 x^{3}}{2048}$ (ii) $|x|<4$ (c) (i) $a=1$ (ii) $b=\displaystyle\frac{9}{8}$ (iii) $c=\displaystyle\frac{19}{128}$

12.(a) $1-x^{2}+2 x^{4}-\displaystyle\frac{14}{3} x^{6}+\cdots$ (b) $1-(1+k) x^{2}+(2+k) x^{4}+\cdots$ (c) $k=4$

13.(a)(i) $1-\displaystyle\frac{3 x}{2}+\displaystyle\frac{3 x^{2}}{2}-\displaystyle\frac{5 x^{3}}{3}($ (ii) $-2<x<2$(b) $(2+x)^{-3}=2^{-3}\left(1+\displaystyle\frac{x}{2}\right)^{-3}, A=\displaystyle\frac{1}{8}$(c) $\displaystyle\frac{1}{8}+\displaystyle\frac{5 x}{16}-\displaystyle\frac{9 x^{2}}{16}+\cdots \quad$ (d) $0.0298$

14.(a) $p^{6}+6 p^{5} q x+15 p^{4} q^{2} x^{2}+20 p^{3} q^{3} x^{3}+15 p^{2} q^{4}x^4+\cdots$ (b) $\quad p=2, q=3$ or $p=-10, q=15$

15.(a) $1+2x^2+6x^4+20x^6+\cdots$ (b)$|x|<\displaystyle\frac{1}{2}$ (c) $3+x+6x^2+2x^3+18x^4$ (d) 1.01

16.(a) $1+\displaystyle\frac{x}{5}-\displaystyle\frac{x^2}{50}+\displaystyle\frac{x^3}{250}-$ (b)$1+\displaystyle\frac{x}{5}+\displaystyle\frac{3x^2}{50}+\displaystyle\frac{x^3}{50}+$  (c) $|x|<\displaystyle\frac{5}{2}$ (d) $1+\displaystyle\frac{2x}{5}+\displaystyle\frac{2x^2}{25}+$ (e) 0.2167

17.(a) $\quad 1-\displaystyle\frac{2 x^{2}}{3}+\displaystyle\frac{8 x^{4}}{9}-\displaystyle\frac{112 x^{6}}{81}$ (b) $|x|<\displaystyle\frac{1}{\sqrt{2}}$ (c) $2+\left(k-\displaystyle\frac{4}{3}\right) x^{2}+\left(\displaystyle\frac{16}{9}-\displaystyle\frac{2 k}{3}\right) x^{4}+\left(\displaystyle\frac{8 k}{9}-\displaystyle\frac{224}{81}\right) x^{6}$ (d) $k=\displaystyle\frac{8}{3}$ (e) $1.0551$

18.(a) $p=3,q=-\displaystyle\frac 19$ (b) $3-\displaystyle\frac x6-\displaystyle\frac{x^2}{216}-\displaystyle\frac{x^3}{3888}$ (c) $2.78393$


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