Skip to the formulas

Edexcel IAL Maths formulas not in the formula booklet

The exam gives you a formula booklet, but it leaves these 22 out. Learn them by heart. Each has a short worked example: tap “Worked example” to open it.

Checked line by line against Pearson's booklet, units P1 to P4, S1 and M1. It prints no formulae for M1. Checked against Pearson's IAL formulae booklet (PDF) (Issue 2, January 2021) on 5 October 2026.

Test yourself with flashcards One-page formula sheet

P1

Quadratic formula and discriminant

\(\displaystyle x\)​\(\displaystyle {}=\frac{-b\pm\sqrt{b^2-4ac}}{2a};\) \(\displaystyle b^2-4ac>0\text{ two roots},\ \)​\(\displaystyle {}=0\text{ one},\ <0\text{ none}\)

Use it for: Solving quadratics that do not factorise, and "for what \(k\)" root questions.

Worked example

Find the exact roots of \(3x^2-7x+1=0\).

  1. \(b^2-4ac=49-12=37>0\): two real roots
  2. \(x=\frac{7\pm\sqrt{37}}6\)

Answer: \(x=\frac{7\pm\sqrt{37}}6\)

Sine rule

\(\displaystyle \frac a{\sin A}\)​\(\displaystyle {}=\frac b{\sin B}\)​\(\displaystyle {}=\frac c{\sin C}\)

Use it for: A side and its opposite angle are known.

The IAL P1 booklet prints the cosine rule but not the sine rule.

Worked example

In triangle \(ABC\), \(A=40^\circ\), \(B=65^\circ\) and \(a=7\) cm. Find \(b\).

  1. \(b\)​\({}=\frac{7\sin65^\circ}{\sin40^\circ}\)

Answer: \(b\approx9.87\) cm

Area of a triangle

\(\displaystyle \text{Area}\)​\(\displaystyle {}=\tfrac12ab\sin C\)

Use it for: Two sides and the included angle.

Worked example

Two sides of a triangle are 8 cm and 5 cm, and the angle between them is \(150^\circ\). Work out its area.

  1. \(\tfrac12\times8\times5\times\sin150^\circ\)​\({}=20\times\tfrac12\)

Answer: 10 cm²

Arc length and sector area (radians)

\(\displaystyle s=r\theta,\) \(\displaystyle A=\tfrac12r^2\theta\)

Use it for: Any circle-sector question with the angle in radians.

Worked example

A sector has radius 6 cm and angle 1.2 radians. Find its arc length and area.

  1. \(s=6\times1.2=7.2\)
  2. \(A\)​\({}=\tfrac12\times36\times1.2\)​\({}=21.6\)

Answer: Arc 7.2 cm, area 21.6 cm²

Differentiating and integrating xⁿ

\(\displaystyle \frac{d}{dx}x^n=nx^{n-1},\) \(\displaystyle \int x^n\,dx\)​\(\displaystyle {}=\frac{x^{n+1}}{n+1}+c\ \ (n\ne-1)\)

Use it for: Every P1 calculus question: write roots and fractions as powers first.

Worked example

Differentiate \(y=4x^3-\frac6x\), and find \(\int\big(6x^2-\frac2{x^2}\big)\,dx\).

  1. \(y\)​\({}=4x^3-6x^{-1}\)​\({}\Rightarrow\frac{dy}{dx}\)​\({}=12x^2+6x^{-2}\)
  2. \(\int(6x^2-2x^{-2})\,dx\)​\({}=2x^3+2x^{-1}+c\)

Answer: \(12x^2+\frac6{x^2}\); \(2x^3+\frac2x+c\)

P2

Laws of logarithms

\(\displaystyle \log_axy=\log_ax+\log_ay,\) \(\displaystyle \log_a\frac xy\)​\(\displaystyle {}=\log_ax-\log_ay,\) \(\displaystyle \log_ax^k=k\log_ax\)

Use it for: Solving log equations and linearising data.

Worked example

Solve \(\log_2x+\log_2(x-2)=3\).

  1. \(\log_2\big(x(x-2)\big)\)​\({}=3\)​\({}\Rightarrow x^2-2x\)​\({}=8\)
  2. \((x-4)(x+2)=0\); reject \(x=-2\) (log of a negative)

Answer: \(x=4\)

Equation of a circle

\(\displaystyle (x-a)^2+(y-b)^2=r^2\) \(\displaystyle \text{centre }(a,b),\ \text{radius }r\)

Use it for: Circle geometry: centres, radii, tangents and intersections.

Worked example

Complete the square to find where the circle \(x^2-10x+y^2+2y-23=0\) is centred, and how big its radius is.

  1. \(x^2-10x=(x-5)^2-25\) and \(y^2+2y=(y+1)^2-1\)
  2. \((x-5)^2+(y+1)^2\)​\({}=25+1+23\)​\({}=49\)

Answer: Centre \((5,-1)\), radius 7

P3

Double-angle formulae

\(\displaystyle \sin2A=2\sin A\cos A,\) \(\displaystyle \cos2A\)​\(\displaystyle {}=2\cos^2A-1\)​\(\displaystyle {}=1-2\sin^2A,\) \(\displaystyle \tan2A\)​\(\displaystyle {}=\frac{2\tan A}{1-\tan^2A}\)

Use it for: Equations with \(2x\) and \(x\) together, and integrating \(\sin^2x\).

Worked example

\(x\) is acute and \(\cos x=\frac13\). Find the exact values of \(\cos2x\) and \(\sin2x\).

  1. \(\cos2x\)​\({}=2\big(\tfrac13\big)^2-1\)​\({}=-\tfrac79\)
  2. \(\sin x\)​\({}=\sqrt{1-\tfrac19}\)​\({}=\tfrac{2\sqrt2}3\)
  3. \(\sin2x\)​\({}=2\times\tfrac{2\sqrt2}3\times\tfrac13\)​\({}=\tfrac{4\sqrt2}9\)

Answer: \(\cos2x=-\frac79\), \(\sin2x=\frac{4\sqrt2}9\)

Harmonic form (R-form)

\(\displaystyle a\sin\theta+b\cos\theta=R\sin(\theta+\alpha),\) \(\displaystyle R=\sqrt{a^2+b^2},\) \(\displaystyle \tan\alpha=\frac ba\)

Use it for: Maximum and minimum values, and solving \(a\sin\theta+b\cos\theta=c\).

Worked example

Write \(3\sin\theta+4\cos\theta\) as \(R\sin(\theta+\alpha)\) and state its maximum value.

  1. \(R=\sqrt{9+16}=5\)
  2. \(\tan\alpha\)​\({}=\frac43\)​\({}\Rightarrow\alpha\)​\({}\approx53.1^\circ\)

Answer: \(5\sin(\theta+53.1^\circ)\); maximum 5

Standard derivatives

\(\displaystyle \frac{d}{dx}e^{kx}=ke^{kx},\) \(\displaystyle \frac{d}{dx}\ln x\)​\(\displaystyle {}=\frac1x,\) \(\displaystyle \frac{d}{dx}\sin kx=k\cos kx,\) \(\displaystyle \frac{d}{dx}\cos kx\)​\(\displaystyle {}=-k\sin kx\)

Use it for: Every calculus question with exponentials, logs or trig.

Worked example

Differentiate \(y=e^{3x}+\ln(2x)-4\sin\frac x2\).

  1. \(\frac{d}{dx}e^{3x}\)​\({}=3e^{3x}\)
  2. \(\ln(2x)=\ln2+\ln x\), derivative \(\frac1x\)
  3. \(\frac{d}{dx}\big(-4\sin\frac x2\big)\)​\({}=-2\cos\frac x2\)

Answer: \(\frac{dy}{dx}\)​\({}=3e^{3x}+\frac1x-2\cos\frac x2\)

Product rule and chain rule

\(\displaystyle \frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx},\) \(\displaystyle \frac{dy}{dx}\)​\(\displaystyle {}=\frac{dy}{du}\times\frac{du}{dx}\)

Use it for: Differentiating products and functions of functions (the booklet gives only the quotient rule).

The IAL P3 booklet prints the quotient rule only.

Worked example

Differentiate \(y=x^2(2x-1)^3\) and factorise your answer.

  1. \(u=x^2\), \(v=(2x-1)^3\), \(v'=3(2x-1)^2\times2\)
  2. \(\frac{dy}{dx}\)​\({}=2x(2x-1)^3+6x^2(2x-1)^2\)
  3. \(=2x(2x-1)^2\big((2x-1)+3x\big)\)

Answer: \(2x(2x-1)^2(5x-1)\)

Standard integrals

\(\displaystyle \int e^{kx}dx\)​\(\displaystyle {}=\tfrac1ke^{kx}+c,\) \(\displaystyle \int\tfrac1x\,dx\)​\(\displaystyle {}=\ln|x|+c,\) \(\displaystyle \int\cos kx\,dx\)​\(\displaystyle {}=\tfrac1k\sin kx+c,\) \(\displaystyle \int\sin kx\,dx\)​\(\displaystyle {}=-\tfrac1k\cos kx+c\)

Use it for: Reversing the standard derivatives.

The IAL booklet prints the sec², tan, cot, cosec and sec integrals, not these.

Worked example

Find \(\int\big(e^{2x}+\frac1x+\cos3x\big)\,dx\).

  1. Integrate term by term, dividing by the coefficient of \(x\) inside.

Answer: \(\frac12e^{2x}+\ln|x|+\frac13\sin3x+c\)

P4

Partial fractions

\(\displaystyle \frac{px+q}{(x-a)(x-b)}\)​\(\displaystyle {}=\frac A{x-a}+\frac B{x-b}\)

Use it for: Integration and series expansions of rational functions.

Worked example

Split \(\frac{3x+4}{(x+3)(x-2)}\) into partial fractions.

  1. \(3x+4=A(x-2)+B(x+3)\)
  2. \(x=2\): \(10=5B\Rightarrow B=2\); \(x=-3\): \(-5=-5A\Rightarrow A=1\)

Answer: \(\frac1{x+3}+\frac2{x-2}\)

Volume of revolution

\(\displaystyle V=\pi\int_a^by^2\,dx\)

Use it for: Rotating a region through \(360^\circ\) about the \(x\)-axis.

Worked example

Rotate the area under \(y=\sqrt x\) for \(0\le x\le4\) a full turn about the \(x\)-axis. What volume does it sweep out?

  1. \(V\)​\({}=\pi\int_0^4x\,dx\)​\({}=\pi\big[\tfrac{x^2}2\big]_0^4\)

Answer: \(8\pi\)

Scalar product and angle between vectors

\(\displaystyle \mathbf a\cdot\mathbf b=a_1b_1+a_2b_2+a_3b_3\)​\(\displaystyle {}=|\mathbf a||\mathbf b|\cos\theta\)

Use it for: Angles between lines, and testing for perpendicular vectors (\(\mathbf a\cdot\mathbf b=0\)).

Worked example

Work out the angle between the vectors \(2\mathbf i+\mathbf j-2\mathbf k\) and \(3\mathbf i+4\mathbf k\).

  1. Scalar product: \(6+0-8=-2\); magnitudes \(3\) and \(5\)
  2. \(\cos\theta=\frac{-2}{15}\)

Answer: \(\theta\approx97.7^\circ\)

S1

Standardising a normal variable

\(\displaystyle X\)​\(\displaystyle {}\sim N(\mu,\sigma^2)\ \)​\(\displaystyle {}\Rightarrow\ Z\)​\(\displaystyle {}=\frac{X-\mu}\sigma\)​\(\displaystyle {}\sim N(0,1)\)

Use it for: Using the normal tables, and finding an unknown mean or standard deviation.

Worked example

\(X\sim N(50,\,4^2)\). Find \(P(X>56)\).

  1. \(z=\frac{56-50}4=1.5\)
  2. \(P(Z>1.5)\)​\({}=1-\Phi(1.5)\)​\({}=1-0.9332\)

Answer: \(0.0668\)

Expectation and variance of aX + b

\(\displaystyle E(aX+b)=aE(X)+b,\) \(\displaystyle \mathrm{Var}(aX+b)=a^2\,\mathrm{Var}(X)\)

Use it for: Coding and linear functions of a random variable.

Worked example

A random variable has mean 4 and variance 3. Work out the mean and variance of \(Y=7-3X\).

  1. \(E(Y)=7-3\times4=-5\)
  2. \(\mathrm{Var}(Y)\)​\({}=(-3)^2\times3\)​\({}=27\) (the 7 does not change the spread)

Answer: \(E(Y)=-5\), \(\mathrm{Var}(Y)=27\)

Conditional probability and independence

\(\displaystyle P(A\mid B)\)​\(\displaystyle {}=\frac{P(A\cap B)}{P(B)};\) \(\displaystyle A,B\text{ independent}\)​\(\displaystyle {}\iff P(A\cap B)=P(A)P(B)\)

Use it for: Testing independence and reading two-way tables and Venn diagrams.

Worked example

The probability that both \(A\) and \(B\) happen is 0.28. On their own, \(A\) has probability 0.7 and \(B\) has probability 0.4. Show that \(A\) and \(B\) are independent and work out \(P(A\mid B)\).

  1. \(P(A)\times P(B)\)​\({}=0.7\times0.4\)​\({}=0.28=P(A\cap B)\), so they are independent.
  2. \(P(A\mid B)\)​\({}=\frac{0.28}{0.4}\)

Answer: \(P(A\mid B)=0.7\) (equal to \(P(A)\), as independence says)

Coding

\(\displaystyle y\)​\(\displaystyle {}=\frac{x-a}b\ \)​\(\displaystyle {}\Rightarrow\ \bar x=a+b\bar y,\) \(\displaystyle \sigma_x=|b|\,\sigma_y\)

Use it for: Getting back to the original data from coded summaries.

Worked example

Data are coded with \(y=\frac{x-100}5\). The coded mean is 2.4 and the coded standard deviation is 1.2. Find the mean and standard deviation of \(x\).

  1. \(\bar x=100+5\times2.4=112\)
  2. \(\sigma_x=5\times1.2=6\)

Answer: Mean 112, standard deviation 6

M1 (the booklet prints no M1 formulae)

Constant acceleration (suvat)

\(\displaystyle v=u+at,\) \(\displaystyle s=ut+\tfrac12at^2,\) \(\displaystyle s=vt-\tfrac12at^2,\) \(\displaystyle v^2=u^2+2as,\) \(\displaystyle s=\tfrac12(u+v)t\)

Use it for: Straight-line motion with constant acceleration.

Worked example

A car slows uniformly from 12 m s⁻¹ to rest over 30 m. Find its deceleration and the time taken.

  1. \(0\)​\({}=144+2a(30)\)​\({}\Rightarrow a\)​\({}=-2.4\)
  2. \(0=12-2.4t\Rightarrow t=5\)

Answer: Deceleration 2.4 m s⁻², 5 s

Newton's second law, weight and friction

\(\displaystyle F=ma,\) \(\displaystyle W=mg,\) \(\displaystyle F\le\mu R\ \ (\text{limiting: }F\)​\(\displaystyle {}=\mu R)\)

Use it for: Every dynamics question.

Worked example

A 4 kg box on a rough horizontal floor (\(\mu=0.5\)) is pulled by a horizontal force of 30 N. Take \(g=9.8\) m s⁻². Find its acceleration.

  1. \(R=4g=39.2\) N, so friction \(=0.5\times39.2=19.6\) N
  2. \(30-19.6=4a\)

Answer: \(a=2.6\) m s⁻²

Momentum and impulse

\(\displaystyle m_1u_1+m_2u_2=m_1v_1+m_2v_2,\) \(\displaystyle I=Ft=mv-mu\)

Use it for: Collisions and impulses.

Worked example

A 2 kg ball moving at 5 m s⁻¹ hits a 3 kg ball at rest and they move off together. Find their speed and the impulse on the 3 kg ball.

  1. \(2\times5\)​\({}=5v\)​\({}\Rightarrow v\)​\({}=2\)
  2. Impulse \(=3\times2-0=6\) N s

Answer: 2 m s⁻¹; 6 N s

Keep going

Questions

Which Edexcel IAL Maths formulas are not in the formula booklet?

The ones to learn by heart are: Quadratic formula and discriminant; Sine rule; Area of a triangle; Arc length and sector area (radians); Differentiating and integrating xⁿ; Laws of logarithms; Equation of a circle; Double-angle formulae; Harmonic form (R-form); Standard derivatives; Product rule and chain rule; Standard integrals; Partial fractions; Volume of revolution; Scalar product and angle between vectors; Standardising a normal variable; Expectation and variance of aX + b; Conditional probability and independence; Coding; Constant acceleration (suvat); Newton's second law, weight and friction; Momentum and impulse. Each one is on this page with a worked example.

Do I get a formula booklet in the exam?

Yes. Pearson's IAL formulae booklet is provided in every unit exam. It prints nothing for M1, so all M1 formulas must be learnt.

What is the best way to memorise them?

Cover the formula, write it from memory, then try the worked example without looking. Come back to the ones you missed the next day. Our flashcards do the spacing for you.

Is this page a copy of the official booklet?

No. We link to the official Pearson Edexcel IAL Mathematical Formulae and Statistical Tables; we do not reproduce it. This page lists only what the booklet leaves out, in our own words, with our own examples.

Our own list, wording and examples, written and checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.