GCE Advanced Level (A-Levels) Mathematics Syllabus

State the formula for integration by parts.

$\int u\,\frac{dv}{dx}\,dx=uv-\int v\,\frac{du}{dx}\,dx$. Choose $u$ as the part that simplifies on differentiating (use LIATE as a guide).

How does integration by substitution work for $\int f(g(x))g'(x)\,dx$?

Let $u=g(x)$ so $\frac{du}{dx}=g'(x)$, i.e. $du=g'(x)\,dx$. The integral becomes $\int f(u)\,du$. For definite integrals, also convert the limits to $u$-values.

What is the integral $\int\frac{f'(x)}{f(x)}\,dx$?

$\int\frac{f'(x)}{f(x)}\,dx=\ln|f(x)|+c$. When the numerator is the derivative of the denominator, the result is the natural log of the denominator.

How do you find the area between a curve $y=f(x)$ and the $x$-axis between $x=a$ and $x=b$?

$\text{Area}=\int_{a}^{b}y\,dx$. Where the curve dips below the axis the integral is negative, so split at the roots and take the modulus of each part.

State the Fundamental Theorem of Calculus linking definite integrals to antiderivatives.

If $F'(x)=f(x)$, then $\int_{a}^{b}f(x)\,dx=F(b)-F(a)=\big[F(x)\big]_{a}^{b}$.

How do you solve a first-order differential equation by separation of variables?

For $\frac{dy}{dx}=g(x)h(y)$, rearrange to $\frac{1}{h(y)}\,dy=g(x)\,dx$, integrate both sides, then add a constant and apply any boundary condition.

How do you differentiate parametric equations $x=f(t)$, $y=g(t)$?

$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{g'(t)}{f'(t)}$, provided $\frac{dx}{dt}\neq 0$.

How do you find a Cartesian equation from parametric equations?

Eliminate the parameter $t$ by solving one equation for $t$ and substituting into the other, or by using an identity (e.g. $\sin^{2}t+\cos^{2}t=1$) to combine them.

State the Pythagorean trigonometric identities.

$\sin^{2}\theta+\cos^{2}\theta=1$; $1+\tan^{2}\theta=\sec^{2}\theta$; $1+\cot^{2}\theta=\csc^{2}\theta$.

State the definitions of $\sec\theta$, $\csc\theta$, $\cot\theta$ and $\tan\theta$ in terms of $\sin$ and $\cos$.

$\sec\theta=\frac{1}{\cos\theta}$; $\csc\theta=\frac{1}{\sin\theta}$; $\cot\theta=\frac{1}{\tan\theta}=\frac{\cos\theta}{\sin\theta}$; $\tan\theta=\frac{\sin\theta}{\cos\theta}$.

State the double angle formulas for $\sin 2\theta$, $\cos 2\theta$ and $\tan 2\theta$.

$\sin 2\theta=2\sin\theta\cos\theta$; $\cos 2\theta=\cos^{2}\theta-\sin^{2}\theta=2\cos^{2}\theta-1=1-2\sin^{2}\theta$; $\tan 2\theta=\frac{2\tan\theta}{1-\tan^{2}\theta}$.

State the compound angle (addition) formulas for sine and cosine.

$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$; $\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$; $\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}$.

How do you express $a\sin\theta+b\cos\theta$ in the form $R\sin(\theta+\alpha)$?

$R=\sqrt{a^{2}+b^{2}}$ and $\tan\alpha=\frac{b}{a}$, with $R>0$ and $\alpha$ acute. This is useful for finding maxima/minima and solving equations.

State the key laws of logarithms.

$\log_{a}(xy)=\log_{a}x+\log_{a}y$; $\log_{a}\!\left(\frac{x}{y}\right)=\log_{a}x-\log_{a}y$; $\log_{a}(x^{k})=k\log_{a}x$; $\log_{a}a=1$; $\log_{a}1=0$.

What is the relationship between exponentials and logarithms, including the natural log?

$y=a^{x}\Leftrightarrow x=\log_{a}y$. The natural log is base $e$: $y=e^{x}\Leftrightarrow x=\ln y$, and $e\approx 2.718$.

How can a model $y=ax^{n}$ or $y=ab^{x}$ be linearised using logarithms?

For $y=ax^{n}$: $\log y=\log a+n\log x$ (plot $\log y$ vs $\log x$). For $y=ab^{x}$: $\log y=\log a+x\log b$ (plot $\log y$ vs $x$). The gradient and intercept give the constants.

Give the formulas for the distance, midpoint and gradient between points $(x_{1},y_{1})$ and $(x_{2},y_{2})$.

Distance $=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$; midpoint $=\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\right)$; gradient $m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$.

What is the gradient condition for perpendicular lines, and the equation of a circle?

Perpendicular lines have gradients with product $m_{1}m_{2}=-1$. A circle of centre $(a,b)$ and radius $r$ has equation $(x-a)^{2}+(y-b)^{2}=r^{2}$.

What geometric property links the tangent to a circle and the radius at the point of contact?

The tangent to a circle is perpendicular to the radius drawn to the point of contact. Also, the angle in a semicircle is a right angle, and the perpendicular from the centre bisects a chord.

How do you find the magnitude of a vector $\vec{a}=\begin{pmatrix}x\\y\\z\end{pmatrix}$ and a unit vector in its direction?

Magnitude $|\vec{a}|=\sqrt{x^{2}+y^{2}+z^{2}}$. A unit vector is $\hat{a}=\frac{1}{|\vec{a}|}\vec{a}$.

What is the scalar (dot) product of two vectors and how does it determine the angle between them?

$\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}$. So $\cos\theta=\frac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}$; vectors are perpendicular when $\vec{a}\cdot\vec{b}=0$.

State the iterative formula for the Newton-Raphson method.

$x_{n+1}=x_{n}-\frac{f(x_{n})}{f'(x_{n})}$. It converges rapidly to a root provided the starting value is close enough and $f'(x_{n})\neq 0$.

How does the sign-change method locate a root of $f(x)=0$, and when can it fail?

If $f(a)$ and $f(b)$ have opposite signs and $f$ is continuous on $[a,b]$, a root lies between them. It can fail if there is a discontinuity or an even number of roots in the interval.

State the trapezium rule for estimating $\int_{a}^{b}y\,dx$ with $n$ strips of width $h$.

$\int_{a}^{b}y\,dx\approx\frac{h}{2}\left[y_{0}+y_{n}+2(y_{1}+y_{2}+\cdots+y_{n-1})\right]$, where $h=\frac{b-a}{n}$.

How do you calculate the mean, median and quartiles from grouped or listed data?

Mean $\bar{x}=\frac{\sum fx}{\sum f}$. Median is the middle value (the $\frac{n+1}{2}$th in order). Lower/upper quartiles $Q_{1},Q_{3}$ split the data at one-quarter and three-quarters; interquartile range $=Q_{3}-Q_{1}$.

Give the formulas for variance and standard deviation of a data set.

Variance $=\frac{\sum x^{2}}{n}-\bar{x}^{2}=\frac{\sum f x^{2}}{\sum f}-\bar{x}^{2}$. Standard deviation is the square root of the variance, $\sigma=\sqrt{\text{variance}}$.

How are outliers commonly identified using the interquartile range or standard deviation?

An outlier lies more than $1.5\times\text{IQR}$ beyond $Q_{1}$ or $Q_{3}$, or alternatively more than $2$ standard deviations from the mean.

State the addition and multiplication laws of probability, including the conditions.

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$ (subtract overlap). For independent events $P(A\cap B)=P(A)P(B)$. Conditional: $P(A\cap B)=P(A)P(B\mid A)$.

Define mutually exclusive and independent events, and the complement rule.

Mutually exclusive: cannot occur together, $P(A\cap B)=0$. Independent: one does not affect the other, $P(A\mid B)=P(A)$. Complement: $P(A')=1-P(A)$.

State the conditions, formula, mean and variance of the binomial distribution $X\sim B(n,p)$.

Conditions: fixed $n$ trials, two outcomes, constant $p$, independent. $P(X=r)=\binom{n}{r}p^{r}(1-p)^{n-r}$; mean $=np$; variance $=np(1-p)$.

Describe the key properties of the Normal distribution $X\sim N(\mu,\sigma^{2})$ and standardisation.

It is symmetric and bell-shaped about the mean $\mu$; about $68\%$, $95\%$, $99.7\%$ of data lie within $1$, $2$, $3$ standard deviations. Standardise using $Z=\frac{X-\mu}{\sigma}\sim N(0,1)$.

Outline the steps of a hypothesis test for a population proportion (binomial model).

State $H_{0}$ and $H_{1}$, assume $H_{0}$ true, choose significance level, model under $H_{0}$ as $B(n,p)$, find the probability of the observed (or more extreme) result, compare with the level, then reject $H_{0}$ or not and conclude in context.

What is the difference between a one-tailed and a two-tailed hypothesis test, including critical regions?

One-tailed tests a directional alternative ($p>p_{0}$ or $p<p_{0}$) with the whole significance level in one tail. Two-tailed tests $p\neq p_{0}$, splitting the level between both tails (e.g. $2.5\%$ each at the $5\%$ level).

What is the Large Data Set in A-Level Mathematics and why is it used?

It is a real prescribed data set (e.g. UK and overseas weather station data from the Met Office) that students explore in advance, so exam questions can test understanding of context, cleaning anomalies, sampling, and realistic distributions.

State the relationships between displacement, velocity and acceleration using calculus.

$v=\frac{ds}{dt}$ and $a=\frac{dv}{dt}=\frac{d^{2}s}{dt^{2}}$. Conversely $s=\int v\,dt$ and $v=\int a\,dt$.

State the SUVAT equations of motion for constant acceleration.

$v=u+at$; $s=ut+\tfrac{1}{2}at^{2}$; $s=\tfrac{1}{2}(u+v)t$; $v^{2}=u^{2}+2as$; $s=vt-\tfrac{1}{2}at^{2}$.

State Newton's three laws of motion.

1st: a body stays at rest or moves at constant velocity unless acted on by a resultant force. 2nd: $F=ma$. 3rd: every action has an equal and opposite reaction.

Give the formulas for weight, the model for friction, and the relationship at limiting equilibrium.

Weight $W=mg$ (with $g\approx 9.8\,\text{m s}^{-2}$). Friction $F\leq\mu R$, where $R$ is the normal reaction and $\mu$ the coefficient of friction. At the point of slipping (limiting), $F=\mu R$.

What is the moment of a force about a point, and the principle of moments for equilibrium?

Moment $=$ force $\times$ perpendicular distance from the point, measured in $\text{N m}$. For equilibrium of a rigid body, total clockwise moments equal total anticlockwise moments about any point.

How do you analyse projectile motion launched at speed $u$ and angle $\theta$ (ignoring air resistance)?

Resolve into independent components: horizontal $u\cos\theta$ (constant velocity) and vertical $u\sin\theta$ (acceleration $-g$). Apply SUVAT separately to each direction, linked by the common time $t$.

Give the formulas for the time of flight, maximum height and range of a projectile on level ground.

Time of flight $T=\frac{2u\sin\theta}{g}$; maximum height $H=\frac{u^{2}\sin^{2}\theta}{2g}$; range $R=\frac{u^{2}\sin 2\theta}{g}$, which is greatest when $\theta=45^{\circ}$.