Iterative methodsEdexcel International A Level Maths: Subtopic test

1. 1
   The equation $x^3-4x-3=0$ has a root $\alpha$ with $2<\alpha<3$. The recurrence relation $x_{n+1}=\sqrt[3]{4x_n+3}$, with $x_0=2$, is used to find $\alpha$.

   (a)

   Find $x_1$, to 3 decimal places.

   [1 mark]

   • A$11$
   • B$5$
   • C$2.224$
   • D$3.317$

   (b)

   The sequence converges to $\alpha$. Which equation must $\alpha$ satisfy?

   [1 mark]

   • A$\alpha^3=4\alpha+3$
   • B$\alpha=4\alpha+3$
   • C$\alpha^3=\sqrt[3]{4\alpha+3}$
   • D$\alpha^3=4\alpha-3$

   (c)

   Find $x_2$ and $x_3$, giving each to 4 decimal places.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The equation $f(x)=x^3+2x-7=0$ has a single real root $\alpha$.

   (a)

   Which interval contains $\alpha$?

   [1 mark]

   • A$1<\alpha<1.5$
   • B$0<\alpha<1$
   • C$2<\alpha<3$
   • D$1.5<\alpha<2$

   (b)

   Which is a correct rearrangement of $f(x)=0$ in the form $x=g(x)$?

   [1 mark]

   • A$x=\sqrt[3]{7+2x}$
   • B$x=\sqrt[3]{7-2x}$
   • C$x=\frac{x^3-7}{2}$
   • D$x=7-x^3$

   (c)

   Use the iteration $x_{n+1}=\sqrt[3]{7-2x_n}$ with $x_0=1.5$ to find $x_2$, giving your answer to 3 decimal places.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $f(x)=\ln x+x-3$, for $x>0$, has a root $\alpha$.

   (a)

   Show that $2<\alpha<2.5$.

   [3 marks]

   (b)

   Show that $f(x)=0$ can be written as $x=3-\ln x$. Using $x_{n+1}=3-\ln x_n$ with $x_0=2$, find $x_1$, $x_2$ and $x_3$ to 3 decimal places.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The equation $x^3-3x^2-2=0$ has a root $\alpha$. Let $f(x)=x^3-3x^2-2$.

   (a)

   (i) Show that $3<\alpha<3.5$.
   (ii) Show that the equation can be written as $x=3+\frac{2}{x^2}$.
   (iii) Using $x_{n+1}=3+\frac{2}{x_n^2}$ with $x_0=3.2$, find $x_1$ to 4 decimal places.

   [6 marks]

   (b)

   Continue the iteration $x_{n+1}=3+\frac{2}{x_n^2}$ from $x_0=3.2$.
   (i) Find $x_2$ and $x_3$ to 4 decimal places.
   (ii) By considering the sign of $f(x)$ at $3.1955$ and $3.1965$, show that $\alpha=3.196$ to 3 decimal places.

   [6 marks]

   Total for question 4: 12 marks

End of questions