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

1. 1
   $f(x)=(1+4x)^{-\frac12}$

   (a)

   Find the coefficient of $x$ in the binomial expansion of $f(x)$.

   [1 mark]

   • A$2$
   • B$-\frac12$
   • C$-4$
   • D$-2$

   (b)

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

   [1 mark]

   • A$|x|<1$
   • B$|x|<\frac14$
   • C$|x|<4$
   • D$|x|<\frac12$

   (c)

   Find the coefficient of $x^2$ in the expansion of $f(x)$.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

   Which expression is equal to $g(x)$?

   [1 mark]

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

   (b)

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

   [1 mark]

   • A$|x|<1$
   • B$|x|<2$
   • C$|x|<4$
   • D$|x|<16$

   (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
   $h(x)=\frac{1}{\sqrt{9-2x}}$

   (a)

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

   [3 marks]

   (b)

   State the range of values of $x$ for which the expansion is valid. Use $x=0.1$ in your expansion to estimate $\frac{1}{\sqrt{8.8}}$, giving your answer to 4 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   $f(x)=\frac{5-x}{(1+x)(1-2x)}$

   (a)

   (i) Find constants $A$ and $B$ such that $f(x)=\frac{A}{1+x}+\frac{B}{1-2x}$.
   (ii) Hence find the first three terms of the binomial expansion of $f(x)$ in ascending powers of $x$.

   [6 marks]

   (b)

   (i) State the range of values of $x$ for which the expansion of $f(x)$ is valid.
   (ii) Use the first three terms of the expansion to estimate $f(0.1)$. Find the percentage error in this estimate, giving your answer to 2 significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions