Binomial series for any rational powerEdexcel International A Level Maths: Subtopic test

1. 1
   The function $f(x)=(1+4x)^{-\frac12}$ is expanded as a series in ascending powers of $x$.

   (a)

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

   [1 mark]

   • A$-\frac12$
   • B$2$
   • C$-2$
   • D$-\frac18$

   (b)

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

   [1 mark]

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

   (c)

   Find the term in $x^3$ in the expansion of $f(x)$.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

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

   [1 mark]

   • A$|x|<1$
   • B$|x|<\frac12$
   • C$|x|<4$
   • D$|x|<2$

   (b)

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

   [1 mark]

   • A$\frac{3}{16}$
   • B$\frac34$
   • C$3$
   • D$\frac{3}{64}$

   (c)

   Use the expansion of $g(x)$ up to and including the term in $x^2$, with $x=0.1$, to estimate $\frac{1}{2.1^2}$ to 4 decimal places.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $h(x)=\sqrt[3]{8+3x}$ is expanded as a series in ascending powers of $x$.

   (a)

   Find the expansion of $h(x)$ in ascending powers of $x$ up to and including the term in $x^2$, giving each coefficient as a fraction in its simplest form.

   [3 marks]

   (b)

   (i) State the range of values of $x$ for which the expansion is valid.
   (ii) Use the expansion with $x=0.1$ to estimate $\sqrt[3]{8.3}$ to 4 decimal places.
   (iii) Explain why $x=5$ cannot be used with this expansion to estimate $\sqrt[3]{23}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Binomial expansions of rational functions are found by expanding a power directly, or by first writing the function in partial fractions.

   (a)

   Given $f(x)=\frac{10+5x}{(1+2x)(2-x)}$:
   (i) express $f(x)$ in partial fractions;
   (ii) hence find the expansion of $f(x)$ in ascending powers of $x$ up to and including the term in $x^2$;
   (iii) state the range of values of $x$ for which the expansion is valid.

   [6 marks]

   (b)

   In the expansion of $(2+ax)^{-3}$, where $a$ is a positive constant, the coefficient of $x^2$ is $\frac34$.
   (i) Find the value of $a$.
   (ii) Find the coefficient of $x^3$.
   (iii) State the range of values of $x$ for which the expansion is valid.

   [6 marks]

   Total for question 4: 12 marks

End of questions