A-Level Maths · Pure · Numerical methods and vectors · last updated 21 October 2026

Numerical methods and vectors practice questions (A-Level Maths)

This pure topic covers locating roots by change of sign, fixed-point iteration and its diagrams, the Newton-Raphson method and when it fails, and vectors in two and three dimensions: magnitude, direction, position vectors and vector geometry. There are no free sample questions on it yet, so this page gives 7 key terms with the formulas typeset, and a door to make an A-Level Maths-style paper on this topic from your own notes. The free exams are typeset PDFs with an answer key; the exam downloads without a signup.

Make a practice exam from your notes →Upload your notesA-Level Maths style preselected; tick "Pure · Numerical methods and vectors" on the next screen. Free to try, no card.

Sample questions

No free sample questions on numerical methods and vectors yet. The key terms below cover what A-Level Maths expects you to know for this topic; to practise it, make a paper from your notes or from the A-Level outline with the unit ticked.

Make a practice exam from your notes →Upload your notes

Key terms

The definitions and formulas A-Level Maths expects for numerical methods and vectors, typeset as they appear on the paper.

Change of sign method: If $f$ is continuous and $f(a)$ and $f(b)$ have opposite signs, there is a root between $a$ and $b$ . Showing a root lies in an interval, or that a value is correct to a given accuracy, means evaluating $f$ at both ends and stating the sign change and continuity.
Fixed-point iteration: Rearranging $f(x) = 0$ as $x = g(x)$ and iterating from a starting value. The sequence converges to the root when $|g'(x)| < 1$ near it; staircase and cobweb diagrams show the path.
$x_{n+1} = g(x_n)$
Newton-Raphson method: Follows the tangent to the $x$ -axis at each step and usually converges quickly. It fails when $f'(x_n) = 0$ (a stationary point) or when the starting value is far from the root.
$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$
Vectors: magnitude and direction: A vector has size and direction; its magnitude is found by Pythagoras in two or three dimensions. A unit vector in the direction of $\mathbf{a}$ is $\mathbf{a}$ divided by its magnitude.
$|\mathbf{a}| = \sqrt{x^2 + y^2 + z^2}, \qquad \hat{\mathbf{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}$
Position vectors and the vector between two points: The position vector of $A$ is $\overrightarrow{OA}$ . The vector from $A$ to $B$ is the difference of the position vectors; its magnitude is the distance $AB$ .
$\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$
Parallel vectors and collinearity: Two vectors are parallel when one is a scalar multiple of the other. Points are collinear when the vectors between them are parallel and share a point.
$\mathbf{a} \parallel \mathbf{b} \iff \mathbf{a} = \lambda\mathbf{b}$
Vector geometry: Finding midpoints, points that divide a segment in a given ratio, and proving geometric facts (a quadrilateral is a parallelogram, a point lies on a line) by writing each vector in terms of a few base vectors.
$\text{midpoint of } AB: \; \frac{1}{2}(\mathbf{a} + \mathbf{b})$

Frequently asked questions

Are there free numerical methods and vectors practice questions for A-Level Maths?

Not as a ready-made sample yet. The key terms on this page cover what A-Level Maths expects for numerical methods and vectors. To practise it, open the generator with the A-Level Maths style, tick "Pure · Numerical methods and vectors" and the questions are written from the A-Level outline for that topic; or upload your own notes on it.

How do I make an A-Level Maths practice paper on numerical methods and vectors from my notes?

Open the create page with the A-Level Maths style preselected, tick "Pure · Numerical methods and vectors" under the units to cover (or upload your notes and slides on it), choose the question mix and time limit, and generate. You get a typeset PDF in the A-Level format with a separate answer key, in about a minute. It is free to try, no card needed.

What does A-Level Maths expect you to know about numerical methods and vectors?

This pure topic covers locating roots by change of sign, fixed-point iteration and its diagrams, the Newton-Raphson method and when it fails, and vectors in two and three dimensions: magnitude, direction, position vectors and vector geometry. The terms to have cold: Change of sign method, Fixed-point iteration, Newton-Raphson method, Vectors: magnitude and direction, Position vectors and the vector between two points. Each is defined below with its formula typeset where one applies.

What is the A-Level Maths exam format?

The A-Level Maths guide explains the papers, their timing and mark allocation, with a sample question and mark scheme and the common mistakes, and links to the generator with the A-Level Maths style preselected.

Make an A-Level Maths paper on numerical methods and vectors from your notes

Open "Pick your exam" with the A-Level Maths style preselected and tick "Pure · Numerical methods and vectors", or upload your notes on it. You get a typeset paper in the A-Level format with a separate answer key.