P4: Binomial expansionEdexcel International A Level Maths: Topic test

1. 1
   The function $f(x)=\frac{1}{(1+2x)^3}$ is expanded as a series in ascending powers of $x$.

   (a)

   Find the coefficient of $x^2$.

   [1 mark]

   • A$6$
   • B$12$
   • C$-24$
   • D$24$

   (b)

   For which values of $x$ is the expansion valid?

   [1 mark]

   • A$|x|<2$
   • B$|x|<\frac12$
   • C$|x|<1$
   • D$|x|<\frac13$

   (c)

   Find the coefficient of $x^3$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $g(x)=\sqrt{9-6x}$ is expanded as a series in ascending powers of $x$.

   (a)

   Find the coefficient of $x$.

   [1 mark]

   • A$-3$
   • B$-\frac13$
   • C$-1$
   • D$1$

   (b)

   For which values of $x$ is the expansion valid?

   [1 mark]

   • A$|x|<\frac32$
   • B$|x|<\frac23$
   • C$|x|<1$
   • D$|x|<9$

   (c)

   Given that $g(x)=3-x-\frac{x^2}{6}+\dots$, use $x=0.06$ to estimate $\sqrt{8.64}$, giving your answer to $4$ decimal places.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $h(x)=(4-x)^{-\frac12}$ is expanded as a series in ascending powers of $x$.

   (a)

   Find the first three terms of the expansion, in ascending powers of $x$.

   [3 marks]

   (b)

   Hence find the first three terms in the expansion of $(1+2x)(4-x)^{-\frac12}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The binomial expansion of $(1+ax)^{-3}$, where $a$ is a negative constant, is $1+bx+54x^2+cx^3+\dots$

   (a)

   Find the values of $a$, $b$ and $c$, and state the range of values of $x$ for which the expansion is valid.

   [6 marks]

   (b)

   (i) Using the value of $a$ found in part (a), find the first three terms in the expansion of $(2+x)(1+ax)^{-3}$.
   (ii) Use your expansion with $x=0.01$ to estimate the value of $\frac{2.01}{0.97^3}$, giving your answer to $4$ decimal places.
   (iii) Explain why $x=0.4$ cannot be used in the expansion to estimate $\frac{2.4}{(-0.2)^3}$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The function $p(x)=(1+3x)^{\frac23}$ is expanded as a series in ascending powers of $x$.

   (a)

   Find the coefficient of $x^2$.

   [1 mark]

   • A$-1$
   • B$1$
   • C$-\frac19$
   • D$-\frac13$

   (b)

   Find the coefficient of $x^3$.

   [1 mark]

   • A$\frac{4}{81}$
   • B$-\frac43$
   • C$\frac43$
   • D$\frac49$

   (c)

   Use the first three terms of the expansion with $x=0.01$ to estimate $1.03^{\frac23}$, giving your answer to $4$ decimal places.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The first two terms in the binomial expansion of $(1+kx)^{\frac12}$, in ascending powers of $x$, are $1+3x$, where $k$ is a constant.

   (a)

   Find the value of $k$.

   [1 mark]

   • A$3$
   • B$6$
   • C$\frac32$
   • D$12$

   (b)

   Find the coefficient of $x^2$ in the expansion.

   [1 mark]

   • A$\frac92$
   • B$-\frac98$
   • C$-\frac18$
   • D$-\frac92$

   (c)

   Find the range of values of $x$ for which the expansion is valid.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The function $m(x)=(1-x)^{-\frac13}$ is expanded as a series in ascending powers of $x$, for $|x|<1$.

   (a)

   Find the first four terms in the expansion of $m(x)$.

   [3 marks]

   (b)

   Hence find the first four terms in the expansion of $(2-x)(1-x)^{-\frac13}$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The function $h(x)=\frac{5-3x}{(1+x)(1-3x)}$ is defined for $|x|<\frac13$.

   (a)

   Express $h(x)$ in partial fractions and hence find the expansion of $h(x)$ in ascending powers of $x$, up to and including the term in $x^3$.

   [6 marks]

   (b)

   (i) Explain why the expansion of $h(x)$ is valid for $|x|<\frac13$.
   (ii) Show that the coefficient of $x^n$ in the expansion is $2(-1)^n+3^{n+1}$.
   (iii) Hence find the coefficient of $x^4$.

   [6 marks]

   Total for question 8: 12 marks

End of questions