Year 13 numerical methods test to download as PDF

This Year 13 numerical methods test lasts 40 minutes and is marked out of 20. It covers the whole chapter for pupils aged 17-18, and the full answers are on a separate page.

⏱️ Duration : 40 min
🎯 Total : 20 marks
❓ 5 questions

Mark/20

Question 1 – Change of sign
\(f(x)=x^3-x-4\).
1. Show that \(f(x)=0\) has a root in \([1,2]\). [2 marks]
2. Show that the root lies in \([1.7,1.8]\). [2 marks]
Question 2 – Iteration and Newton-Raphson
The equation \(x^3-x-4=0\) has a root \(\alpha\) near 1.8.
1. Use \(x_{n+1}=\sqrt[3]{x_n+4}\) with \(x_0=1.5\) to find \(x_1,x_2,x_3\) to 4 decimal places. [2 marks]
2. Use Newton-Raphson on \(f(x)=x^3-x-4\) with \(x_0=1.8\) to find \(x_1\) and \(x_2\) to 5 decimal places. [3 marks]
Question 3 – The trapezium rule
Use the trapezium rule with 4 strips to estimate \(\displaystyle\int_0^1\sqrt{1+x^2}\,dx\) to 4 decimal places, and say whether your estimate is too large or too small.
Question 4 – A failure of Newton-Raphson
\(f(x)=x^3-2x+2\). Newton-Raphson is applied with \(x_0=0\). Write down \(x_1,x_2,x_3\) and explain what happens.
Question 5 – Why does the iteration converge?
The iteration \(x_{n+1}=g(x_n)\) with \(g(x)=\sqrt[3]{x+4}\) is used to solve \(x^3-x-4=0\) with root \(\alpha=1.796\).
1. Find \(g\prime(x)\). [2 marks]
2. Evaluate \(g\prime(\alpha)\) and explain why the iteration converges and what pattern the values follow. [2 marks]

How to use this Year 13 numerical methods test

First, revise with the numerical methods lesson and the worksheet. Then sit the test in one go, with a timer set to 40 minutes. After that, mark your work with the numerical methods test answers. If you lose marks on a question, redo it the next day without help.

Finally, you can download this Year 13 numerical methods test as a PDF and print it like a real exam paper. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday.