CCEA A-Level Further Mathematics AS 2 Applied Mathematics: a complete overview

What this unit demands

AS 2 Applied Mathematics is the AS year's applied half, combining mechanics and statistics in one paper. CCEA tests confident modelling: choosing the right kinematics equation, drawing and resolving force diagrams, handling friction and connected particles, and on the statistics side, selecting sampling methods, computing summary statistics, and applying probability and the binomial distribution. Clear method and correct units carry the marks.

This guide walks through the four dot points, then sets out the exam patterns CCEA repeats. Each topic has a matching dot-point page with practice questions; this overview ties them together.

Kinematics and motion graphs

For constant acceleration, the five suvat equations each omit one of $s, u, v, a, t$, so you pick the one matching your knowns. For vertical motion, choose a positive direction and apply $a = \pm g$ consistently. On a velocity-time graph the gradient is acceleration and the area is displacement; on a displacement-time graph the gradient is velocity.

Forces and Newton's laws

Forces are vectors: resolve them and set the resultant to zero for equilibrium. Newton's laws give constant velocity with no resultant force, $\mathbf{F} = m\mathbf{a}$, and equal-and-opposite forces on different bodies. Friction satisfies $F \leq \mu R$ (limiting at $\mu R$), with $R = mg\cos\theta$ on a slope. For connected particles over a smooth pulley the tension is common and accelerations share a magnitude.

Sampling and data

Sampling estimates a population: simple random, stratified (proportional to subgroup size) and systematic methods, weighed for bias. Summarise the centre with the mean, median or mode and the spread with the range, interquartile range, variance $\frac{\sum x^2}{n} - \bar{x}^2$ and standard deviation, and choose displays such as histograms and box plots.

Probability and the binomial distribution

Probability uses the addition rule, independence ($P(A \cap B) = P(A)P(B)$) and conditional probability $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$. A discrete random variable has $E(X) = \sum xP(X=x)$ and $\operatorname{Var}(X) = E(X^2) - [E(X)]^2$. The binomial $B(n, p)$ models successes in $n$ independent constant-probability trials, with mean $np$ and variance $np(1-p)$.

How this unit is examined

A typical CCEA profile for AS 2 Applied:

• Kinematics. A multi-stage suvat problem or vertical-motion question, and a graph to interpret.
• Forces. A force-diagram problem with friction, a slope, or connected particles.
• Sampling and data. Naming and justifying a sampling method, and computing the mean and standard deviation.
• Probability. Combined events, conditional probability, and a binomial calculation with the mean.

Check your knowledge

A mix of recall and technique questions covering the unit. Attempt them under timed conditions, then check against the solutions.

1. A car accelerates from rest at $3\,\text{m s}^{-2}$ for $5\,\text{s}$. Find its final speed and the distance travelled. (3 marks)
2. What does the gradient of a velocity-time graph represent? (1 mark)
3. A $6\,\text{kg}$ block on a rough horizontal surface has $\mu = 0.2$. Find the limiting friction ($g = 9.8\,\text{m s}^{-2}$). (2 marks)
4. State Newton's third law. (2 marks)
5. Name a sampling method that represents each subgroup in proportion to size. (1 mark)
6. For data with $\sum x = 60$, $\sum x^2 = 800$, $n = 5$, find the standard deviation. (3 marks)
7. For $X \sim B(12, 0.25)$, write down the mean and variance. (2 marks)