Quadratic formula and discriminant

For \(ax^2 + bx + c = 0\):

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

\[\Delta = b^2 - 4ac\]

The quadratic formula matters whenever factoring is slow, messy or impossible, and the discriminant matters because it tells you the nature of the roots before you solve. That helps with graph sketches, inequalities, intersections and argument-based questions where you need to explain whether a curve crosses, touches or misses the axis.

Common mistake: students calculate the roots correctly but never connect them to the graph. If \(\Delta = 0\), you are looking at a repeated root and therefore a tangent touch, not two separate crossings.

Indices, exponentials and logarithms

\[a^m \cdot a^n = a^{m+n}\]

\[\frac{a^m}{a^n} = a^{m-n}\]

\[(a^m)^n = a^{mn}\]

\[\log_a(xy)=\log_a x+\log_a y\]

\[\log_a\left(\frac{x}{y}\right)=\log_a x-\log_a y\]

\[\log_a(x^k)=k\log_a x\]

\[\log_a x=\frac{\log_b x}{\log_b a}\]

These laws are not only for simplification. They are the grammar of exponential models, growth and decay, solving equations with different bases, and proving identities. At A Level, the marks often go not because the law is unknown, but because the student uses the correct law on the wrong structure.

Arithmetic sequences and sums

\[u_n = a + (n-1)d\]

\[S_n = \frac{n}{2}(2a + (n-1)d)\]

\[S_n = \frac{n}{2}(a+l)\]

This is one of the cleanest formula families in Pure, which is why students sometimes lose easy marks by using it mechanically. Always identify what each symbol stands for in the question before substituting. Confusing the common difference \(d\) with the last term \(l\) is one of the fastest ways to turn a simple sequence question into a wrong answer that still looks neat.

Geometric sequences and sums

\[u_n = ar^{n-1}\]

\[S_n = \frac{a(1-r^n)}{1-r}\quad(r\neq1)\]

\[S_\infty = \frac{a}{1-r}\quad(|r|<1)\]

The sum to infinity condition is where the formula stops being a memory exercise and becomes a reasoning test. If \(|r|\) is not less than 1, the shortcut does not apply. A surprising number of errors happen because students remember the formula but forget the condition that makes it legal.

Straight lines

\[m=\frac{y_2-y_1}{x_2-x_1}\]

\[y-y_1=m(x-x_1)\]

\[y=mx+c\]

\[d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\]

\[M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\]

These are deceptively basic. The marks are rarely in writing the formula down; they are in interpreting slope, choosing the correct form, and avoiding sign errors. Parallel lines share a gradient. Perpendicular lines have gradients whose product is \(-1\), but only when both gradients exist. That final caveat matters in questions involving vertical lines.

Circles

\[(x-a)^2+(y-b)^2=r^2\]

\[x^2+y^2+2gx+2fy+c=0\]

Coordinate geometry questions usually want you to connect algebraic form to geometric meaning. In the standard form, the centre is \((a,b)\) and the radius is \(r\). In the expanded form, you are often expected to complete the square or read the centre carefully after rearrangement. Students lose marks here by expanding correctly but interpreting the signs incorrectly.

Core identities, compound angles and double angles

\[\sin^2\theta + \cos^2\theta = 1\]

\[\tan\theta=\frac{\sin\theta}{\cos\theta}\]

\[\sin(A\pm B)=\sin A\cos B \pm \cos A\sin B\]

\[\cos(A\pm B)=\cos A\cos B \mp \sin A\sin B\]

\[\tan(A\pm B)=\frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}\]

\[\sin 2A = 2\sin A \cos A\]

\[\cos 2A = \cos^2A - \sin^2A = 2\cos^2A - 1 = 1 - 2\sin^2A\]

\[\tan 2A = \frac{2\tan A}{1-\tan^2A}\]

These are the formulas students recognise but still mishandle because the sign structure is easy to flip. The safest revision method is not just rereading them. Rewrite them from memory, then expand one identity into another form so you feel how the structure works.

Sine rule, cosine rule, area and small-angle ideas

\[\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\]

\[a^2=b^2+c^2-2bc\cos A\]

\[\text{Area}=\frac{1}{2}ab\sin C\]

\[\sin\theta \approx \theta,\quad \cos\theta \approx 1-\frac{\theta^2}{2},\quad \tan\theta \approx \theta\]

The triangle formulas are usually about selecting the right model from limited information. The small-angle approximations are different: they are not exact identities, so you must remember they are approximations and that \(\theta\) is measured in radians.

Standard derivatives

\[\frac{d}{dx}(x^n)=nx^{n-1}\]

\[\frac{d}{dx}(e^x)=e^x\]

\[\frac{d}{dx}(e^{kx})=ke^{kx}\]

\[\frac{d}{dx}(\ln x)=\frac{1}{x}\]

\[\frac{d}{dx}(\sin x)=\cos x\]

\[\frac{d}{dx}(\cos x)=-\sin x\]

\[\frac{d}{dx}(\tan x)=\sec^2x\]

Students often think the derivative table is the end of the story. It is actually the start. The exam skill is recognising which outer structure the table sits inside, especially when exponentials, trig and algebraic powers are mixed together.

Product, quotient and chain rules

\[\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}\]

\[\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\]

\[\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\]

The chain rule is one of the biggest hidden separators between average and strong Pure performance. Students who treat it as a formal trick often miss constants, powers or inner derivatives. Students who see it as "outside times inside derivative" tend to move faster and make fewer errors.

Standard integrals

\[\int x^n\,dx=\frac{x^{n+1}}{n+1}+C \quad (n\neq -1)\]

\[\int \frac{1}{x}\,dx=\ln|x|+C\]

\[\int e^x\,dx=e^x+C\]

\[\int e^{kx}\,dx=\frac{1}{k}e^{kx}+C\]

\[\int \sin x\,dx=-\cos x+C\]

\[\int \cos x\,dx=\sin x+C\]

\[\int \sec^2x\,dx=\tan x+C\]

The constant \(C\) is a small symbol with huge importance. In indefinite integration it must be there. In definite integration it disappears because the limits handle the subtraction. Forgetting which world you are in is a classic avoidable mistake.

Methods and applications

\[\int u\frac{dv}{dx}\,dx = uv - \int v\frac{du}{dx}\,dx\]

\[\int f(g(x))g'(x)\,dx = \int f(u)\,du\]

\[\int_a^b f(x)\,dx = [F(x)]_a^b = F(b)-F(a)\]

\[A=\int_a^b y\,dx\]

\[V=\pi\int_a^b y^2\,dx\]

Integration by parts and substitution are easier when you stop thinking of them as formulas and start seeing them as pattern recognition. One reverses a product-style structure. The other rewrites a composite structure into something simpler.

Newton-Raphson and trapezium rule

\[x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}\]

\[\int_a^b y\,dx \approx \frac{1}{2}h\{(y_0+y_n)+2(y_1+y_2+\cdots+y_{n-1})\}\]

Numerical methods are precision tests disguised as formula questions. The formulas are not especially long, but the notation is unforgiving. Students lose marks by misreading table values, rounding too early, or forgetting what each iteration is supposed to achieve.

Vectors

\[|\mathbf{a}|=\sqrt{a_1^2+a_2^2+a_3^2}\]

\[\hat{\mathbf{a}}=\frac{\mathbf{a}}{|\mathbf{a}|}\]

\[\mathbf{a}\cdot\mathbf{b}=a_1b_1+a_2b_2+a_3b_3\]

\[\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos\theta\]

\[\mathbf{r}=\mathbf{a}+\lambda\mathbf{b}\]

Vector questions reward calm structure. Define the vectors clearly, write the relation you need, and keep the geometry in sight. Parallel vectors are scalar multiples. Perpendicular vectors have zero dot product. Those two ideas unlock a large share of vector marks.

Mean and estimated central tendency

\[\bar{x}=\frac{\sum x}{n}\]

\[\bar{x}=\frac{\sum fx}{\sum f}\]

\[\text{Grouped median} = L+\frac{\frac{n}{2}-F}{f}\times w\]

The mean formulas look simple because they are simple. The real test is whether you know which one matches the data you are given. Raw data, frequency tables and grouped data do not all ask for the same treatment. Students often rush because the arithmetic feels easy, then lose marks by using the wrong form.

Variance and standard deviation

\[\sigma^2=\frac{\sum(x-\bar{x})^2}{n}=\frac{\sum x^2}{n}-\bar{x}^2\]

\[\sigma=\sqrt{\frac{\sum(x-\bar{x})^2}{n}}\]

\[S_{xx}=\sum x^2-\frac{(\sum x)^2}{n}\]

Spread formulas are important because many Statistics questions are really comparisons in disguise. The student who only reports a mean may miss the fact that one data set is far less consistent than the other. Standard deviation turns vague description into mathematical evidence.

Binomial distribution

\[P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}\]

\[E(X)=np\]

\[\text{Var}(X)=np(1-p)\]

The binomial model only works when the conditions work: a fixed number of trials, two outcomes, constant probability, and independence. Students sometimes memorise the formula but forget to check the conditions, which means the calculation can be beautifully executed and still be wrong.

Normal distribution and standardisation

\[Z=\frac{X-\mu}{\sigma}\]

\[\bar{X}\sim N\left(\mu,\frac{\sigma^2}{n}\right)\]

Normal questions reward careful reading of mean, spread and direction. Most mistakes are simple but costly: wrong tail, wrong inequality sign, forgetting to standardise, or reporting a probability that does not match the context of the question.

SUVAT equations

\[v=u+at\]

\[s=ut+\frac{1}{2}at^2\]

\[v^2=u^2+2as\]

\[s=\frac{1}{2}(u+v)t\]

\[s=vt-\frac{1}{2}at^2\]

SUVAT is only for motion in a straight line with constant acceleration. Students know the formulas, but the condition gets forgotten. As soon as acceleration varies, you move into calculus-based motion instead. That one detail decides whether the method is correct.

Variable acceleration and graph links

\[v=\frac{ds}{dt}\]

\[a=\frac{dv}{dt}=\frac{d^2s}{dt^2}\]

\[s=\int v\,dt\]

Mechanics questions often move between words, graphs and formulas. Velocity is gradient on a displacement-time graph. Acceleration is gradient on a velocity-time graph. Displacement is area under a velocity-time graph. When students connect those three ideas, graph questions become much more predictable.

Forces and friction

\[F=ma\]

\[W=mg\]

\[F\le \mu R\]

\[F_{\max}=\mu R\]

Mechanics force questions are won by diagram discipline. Weight acts vertically downward. Normal reaction is perpendicular to the surface. Friction opposes impending or actual motion. Once those directions are secure, the algebra becomes far less stressful.

Equilibrium and moments

\[M=Fd\]

\[\sum F_x=0,\qquad \sum F_y=0,\qquad \sum M=0\]

Moments reward students who define a pivot clearly and keep clockwise and anticlockwise consistently signed. The formula itself is short. The real method is setting up the geometry and distances correctly.

Work, energy and power

\[W=Fs\cos\theta\]

\[KE=\frac{1}{2}mv^2\]

\[PE=mgh\]

\[P=\frac{W}{t}=\frac{E}{t}=Fv\]

Energy methods are powerful because they can replace longer force-based solutions. If the situation is well suited to conservation ideas, energy can be the fastest route. But you still need to interpret the system carefully and check where energy is gained, lost or transformed.

Momentum, impulse and collisions

\[p=mv\]

\[I=Ft=\Delta p=m(v-u)\]

\[m_1u_1+m_2u_2=m_1v_1+m_2v_2\]

\[e=\frac{v_2-v_1}{u_1-u_2}\]

Collision questions are usually about direction and sequence. Write velocities with signs, not just magnitudes, and decide carefully whether you are using conservation of momentum, the coefficient of restitution, or both.

Usually worth knowing cold

• Exact trig values for key angles and the signs in different quadrants.
• Core trig identities and the meaning of radians in small-angle work.
• Standard derivative and integral patterns.
• Straight-line and circle interpretation, not just the formulas themselves.
• When an arithmetic or geometric model is appropriate.
• Probability language such as complement, independent and conditional.
• The conditions behind binomial, normal and SUVAT models.

Common mistakes even strong students make

• Using the wrong board assumptions, especially for OCR versus Edexcel.
• Forgetting that approximations are not exact identities.
• Dropping the constant of integration in indefinite integrals.
• Mixing up \(P(A|B)\) and \(P(B|A)\).
• Applying SUVAT when acceleration is not constant.
• Using formula memory instead of diagram setup in Mechanics.
• Recognising a formula without knowing what quantity the question is asking for.