Binomial expansion for rational nAQA A-Level Maths: Subtopic test

1. 1
   The function $f(x)=(1+3x)^{-2}$ is expanded in ascending powers of $x$.

   (a)

   Find the coefficient of $x$ in the expansion.

   [1 mark]

   • A$6$
   • B$-6$
   • C$-2$
   • D$-18$

   (b)

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

   [1 mark]

   • A$3$
   • B$54$
   • C$27$
   • D$-27$

   (c)

   Use the first three terms of the expansion, with a suitable value of $x$, to estimate $1.03^{-2}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $g(x)=\sqrt{4+x}$.

   (a)

   Which expression is equal to $g(x)$, in the form needed for the binomial expansion?

   [1 mark]

   • A$4\left(1+\frac{x}{4}\right)^{\frac12}$
   • B$2(1+x)^{\frac12}$
   • C$2\left(1+\frac{x}{2}\right)^{\frac12}$
   • D$2\left(1+\frac{x}{4}\right)^{\frac12}$

   (b)

   For which values of $x$ is the binomial expansion of $g(x)$ valid?

   [1 mark]

   • A$|x|<4$
   • B$|x|<1$
   • C$|x|<2$
   • D$|x|<\frac14$

   (c)

   Find the first three terms of the expansion of $g(x)$ in ascending powers of $x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(x)=\frac{1}{2+5x}$.

   (a)

   Find the expansion of $f(x)$ in ascending powers of $x$, up to and including the term in $x^2$.

   [3 marks]

   (b)

   Explain why $x=0.1$ may be used in the expansion to estimate $\frac{1}{2.5}$. Use it to find an estimate, and find the percentage error in your estimate.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $f(x)=\frac{1+x}{\sqrt{1-2x}}$, for values of $x$ for which the binomial expansion is valid.

   (a)

   (i) Expand $(1-2x)^{-\frac12}$ in ascending powers of $x$ up to and including $x^3$.
   (ii) State the range of values of $x$ for which this expansion is valid.
   (iii) Hence find the expansion of $f(x)$ up to and including $x^3$.

   [6 marks]

   (b)

   Use $x=0.01$ in your expansion of $f(x)$ to estimate $\sqrt2$, giving your answer to 4 decimal places.

   [6 marks]

   Total for question 4: 12 marks

End of questions