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A Level Mathematics (UK)Edexcel 9MA0 · AQA 7357 · OCR A H240 · one-page formula sheet
E A O ticks = printed in that board's formulae booklet (Edexcel 9MA0 · AQA 7357 · OCR A H240). No tick: learn it.
Algebra & functions
···Quadratic formula: \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
···Discriminant \(b^2-4ac\): \(>0\) two real roots, \(=0\) one repeated root, \(<0\) no real roots
···Completing the square: \(ax^2+bx+c=a\left(x+\tfrac b{2a}\right)^2\) \({}+c-\tfrac{b^2}{4a}\)
···Indices: \(a^ma^n=a^{m+n},\) \((a^m)^n=a^{mn},\) \(a^{-n}=\tfrac1{a^n},\) \(a^{\frac mn}=\sqrt[n]{a^m}\)
···Surds: \(\sqrt{ab}=\sqrt a\sqrt b,\) \(\frac1{\sqrt a+\sqrt b}=\frac{\sqrt a-\sqrt b}{a-b}\)
···Factor theorem: \(f(a)=0\iff(x-a)\text{ is a factor}\)
···Partial fractions: \(\frac{px+q}{(x-a)(x-b)}=\frac{A}{x-a}\) \({}+\frac{B}{x-b},\) \(\frac{\dots}{(x-a)^2(x-b)}\) \({}=\frac{A}{x-a}+\frac{B}{(x-a)^2}\) \({}+\frac{C}{x-b}\)
···Modulus; composite; inverse: \(|x|<a\iff-a<x<a;\) \(fg(x)=f(g(x));\) \(ff^{-1}(x)=x\)
···Transformations: \(f(x)+a\) up \(a\); \(f(x+a)\) left \(a\); \(af(x)\) vertical ×\(a\); \(f(ax)\) horizontal ×\(\tfrac1a\)
Coordinate geometry
···Line; gradient: \(y-y_1=m(x-x_1),\) \(m=\frac{y_2-y_1}{x_2-x_1}\)
···Perpendicular gradients: \(m_1m_2=-1\)
···Midpoint; distance: \(\left(\tfrac{x_1+x_2}2,\tfrac{y_1+y_2}2\right),\) \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
···Circle, centre \((a,b)\), radius \(r\): \((x-a)^2+(y-b)^2=r^2\)
···Circles: tangent ⟂ radius; angle in a semicircle is 90°; perpendicular from centre bisects a chord
···Parametric → Cartesian: eliminate \(t\), e.g. with \(\sin^2t+\cos^2t=1\)
Sequences & series
···Arithmetic \(n\)th term: \(u_n=a+(n-1)d\)
✓✓✓Arithmetic series: \(S_n=\tfrac12n(a+l)=\tfrac12n\big[2a+(n-1)d\big]\)
···Geometric \(n\)th term: \(u_n=ar^{n-1}\)
✓✓✓Geometric series: \(S_n=\frac{a(1-r^n)}{1-r},\) \(S_\infty=\frac a{1-r}\ (|r|<1)\)
✓✓✓Binomial, \(n\in\mathbb N\): \((a+b)^n=a^n+\tbinom n1a^{n-1}b\) \({}+\tbinom n2a^{n-2}b^2+\cdots+b^n\)
✓✓✓Binomial coefficient: \(\binom nr={}^nC_r=\frac{n!}{r!(n-r)!}\)
✓✓✓Binomial, \(|x|<1\), \(n\in\mathbb Q\): \((1+x)^n=1+nx+\frac{n(n-1)}{1\times2}x^2+\cdots\)
···\((a+bx)^n=a^n(1+\tfrac bax)^n\), valid for \(|x|<\left|\tfrac ab\right|\)
···Sigma; recurrence: \(\sum_{r=1}^nr=\tfrac12n(n+1);\) \(u_{n+1}=f(u_n)\)
Exponentials & logarithms
···Definition: \(a^x=b\iff x=\log_ab\)
···Laws: \(\log_axy=\log_ax+\log_ay,\) \(\log_a\tfrac xy=\log_ax-\log_ay,\) \(\log_ax^k=k\log_ax\)
✓··Change of base: \(\log_ax=\frac{\log_bx}{\log_ba}\)
✓··Exponential form: \(a^x=e^{x\ln a}\)
···Inverses: \(e^{\ln x}=x,\) \(\ln e^x=x\)
···Linearising: \(y=ax^n\Rightarrow\log y\) \({}=\log a+n\log x;\) \(y=kb^x\Rightarrow\log y\) \({}=\log k+x\log b\)
Trigonometry
···Sine rule; cosine rule: \(\frac a{\sin A}=\frac b{\sin B},\) \(a^2=b^2+c^2-2bc\cos A\)
···Area; arc; sector (radians): \(\tfrac12ab\sin C;\) \(s=r\theta;\) \(A=\tfrac12r^2\theta\)
···Identities: \(\tan\theta=\frac{\sin\theta}{\cos\theta},\) \(\sin^2\theta+\cos^2\theta\) \({}=1\)
···Reciprocal: \(\sec^2\theta=1+\tan^2\theta,\) \(\cosec^2\theta=1+\cot^2\theta\)
···Double angle: \(\sin2A=2\sin A\cos A,\) \(\cos2A=\cos^2A-\sin^2A\) \({}=2\cos^2A-1=1-2\sin^2A\)
···Double angle (tan): \(\tan2A=\frac{2\tan A}{1-\tan^2A}\)
✓✓✓Compound sin: \(\sin(A\pm B)=\sin A\cos B\) \({}\pm\cos A\sin B\)
✓✓✓Compound cos: \(\cos(A\pm B)=\cos A\cos B\) \({}\mp\sin A\sin B\)
✓✓✓Compound tan: \(\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}\)
···Harmonic form, \(R>0\): \(a\sin\theta+b\cos\theta=R\sin(\theta+\alpha),\) \(R=\sqrt{a^2+b^2},\) \(\tan\alpha=\tfrac ba\)
✓✓✓Small angles (radians): \(\sin\theta\approx\theta,\) \(\cos\theta\approx1-\tfrac{\theta^2}2,\) \(\tan\theta\approx\theta\)
···Exact values: \(\sin30^\circ=\tfrac12,\) \(\sin45^\circ=\tfrac{\sqrt2}2,\) \(\sin60^\circ=\tfrac{\sqrt3}2,\) \(\tan30^\circ=\tfrac1{\sqrt3},\) \(\tan60^\circ=\sqrt3\)
···Inverse trig ranges: \(\arcsin\in[-\tfrac\pi2,\tfrac\pi2],\) \(\arccos\in[0,\pi],\) \(\arctan\in(-\tfrac\pi2,\tfrac\pi2)\)
···Radians: \(\pi\text{ rad}=180^\circ\)
Differentiation
✓·✓First principles: \(f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\)
···Standard: \((x^n)'=nx^{n-1},\) \((e^{kx})'=ke^{kx},\) \((\ln x)'=\tfrac1x,\) \((\sin kx)'=k\cos kx,\) \((\cos kx)'=-k\sin kx\)
···Exponential base \(a\): \((a^{kx})'=ka^{kx}\ln a\)
✓✓✓Trig: \((\tan kx)'=k\sec^2kx,\) \((\sec kx)'=k\sec kx\tan kx,\) \((\cot kx)'=-k\cosec^2kx,\) \((\cosec kx)'=-k\cosec kx\cot kx\)
···Product; chain: \((uv)'=uv'+vu',\) \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\)
✓✓✓Quotient: \(\frac{d}{dx}\Big(\frac{f}{g}\Big)\) \({}=\frac{f'g-fg'}{g^2}\)
···Inverse; parametric: \(\frac{dy}{dx}=\frac1{dx/dy};\) \(\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\)
···Implicit: \(\tfrac{d}{dx}f(y)=f'(y)\tfrac{dy}{dx},\) \(\tfrac{d}{dx}(xy)=x\tfrac{dy}{dx}+y\)
···Stationary: \(f'=0\); max if \(f''<0\), min if \(f''>0\). Convex \(f''>0\), concave \(f''<0\); inflection: \(f''\) changes sign
···Tangent; normal: \(y-y_1=f'(x_1)(x-x_1);\) \(m_\text{n}=-\tfrac1{f'(x_1)}\)
Integration
···Standard: \(\int x^n\,dx=\tfrac{x^{n+1}}{n+1}+c,\) \(\int\tfrac1x\,dx=\ln|x|+c,\) \(\int e^{kx}\,dx=\tfrac1ke^{kx}+c\)
···Trig: \(\int\cos kx\,dx=\tfrac1k\sin kx+c,\) \(\int\sin kx\,dx=-\tfrac1k\cos kx+c\)
✓··More trig: \(\int\sec^2kx\,dx=\tfrac1k\tan kx+c,\) \(\int\tan kx\,dx=\tfrac1k\ln|\sec kx|+c,\) \(\int\cot kx\,dx=\tfrac1k\ln|\sin kx|+c\)
··✓Reverse chain: \(\int\frac{f'(x)}{f(x)}\,dx\) \({}=\ln|f(x)|+c,\) \(\int f'(x)[f(x)]^n\,dx=\frac{[f(x)]^{n+1}}{n+1}+c\)
···Linear inside: \(\int f(ax+b)\,dx=\tfrac1aF(ax+b)+c\)
✓✓✓By parts: \(\int u\frac{dv}{dx}\,dx=uv\) \({}-\int v\frac{du}{dx}\,dx\)
···Substitution: \(u=g(x)\), replace \(dx\) by \(\frac{du}{g'(x)}\), change the limits
···Area: \(\int_a^by\,dx;\) \(\text{between curves }\int_a^b(y_\text{top}-y_\text{bottom})\,dx;\) \(\text{parametric }\int y\tfrac{dx}{dt}\,dt\)
···Identities to integrate: \(\sin^2x=\tfrac12(1-\cos2x),\) \(\cos^2x=\tfrac12(1+\cos2x)\)
···Separable DE: \(\frac{dy}{dx}=f(x)g(y)\) \({}\Rightarrow\int\frac{dy}{g(y)}\) \({}=\int f(x)\,dx\)
Numerical methods & vectors
✓✓✓Trapezium rule, \(h=\tfrac{b-a}n\): \(\int_a^by\,dx\) \({}\approx\tfrac12h\big\{(y_0+y_n)\) \({}+2(y_1+\cdots+y_{n-1})\big\}\)
✓✓✓Newton–Raphson: \(x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}\)
···Sign change: continuous \(f\) with \(f(a)f(b)<0\) has a root in \((a,b)\). Iteration \(x_{n+1}=g(x_n)\): staircase/cobweb
···Magnitude; unit vector: \(|x\mathbf i+y\mathbf j+z\mathbf k|\) \({}=\sqrt{x^2+y^2+z^2},\) \(\hat{\mathbf a}=\frac{\mathbf a}{|\mathbf a|}\)
···\(\overrightarrow{AB}\); distance \(AB\): \(\mathbf b-\mathbf a;\) \(|\mathbf b-\mathbf a|=\sqrt{\Delta x^2+\Delta y^2+\Delta z^2}\)
Mechanics
✓✓✓Constant acceleration: \(v=u+at,\) \(s=ut+\tfrac12at^2,\) \(s=vt-\tfrac12at^2,\) \(s=\tfrac12(u+v)t,\) \(v^2=u^2+2as\)
·✓✓… in vectors: \(\mathbf v=\mathbf u+\mathbf at,\) \(\mathbf r=\mathbf ut\) \({}+\tfrac12\mathbf at^2\)
···Variable acceleration: \(v=\frac{dr}{dt},\) \(a=\frac{dv}{dt}=\frac{d^2r}{dt^2},\) \(r=\int v\,dt,\) \(v=\int a\,dt\)
···Forces: \(F=ma,\) \(W=mg,\) \(F\le\mu R\ (\text{limiting: }F\) \({}=\mu R)\)
···Moments: \(\text{moment}=F\times d_\perp;\) \(\text{equilibrium: }\textstyle\sum F\) \({}=0,\ \sum M=0\)
···Projectiles (speed \(U\), angle \(\alpha\)): \(x=Ut\cos\alpha,\) \(y=Ut\sin\alpha-\tfrac12gt^2\)
···Slope at \(\theta\): weight component \(mg\sin\theta\) down the slope; \(R=mg\cos\theta\) if nothing else acts perpendicular to it
···Speed = \(|\mathbf v|\); velocity–time graph: gradient = acceleration, area = displacement
Statistics & probability
···Mean: \(\bar x=\frac{\sum x}n\) \({}=\frac{\sum fx}{\sum f}\)
✓·✓Standard deviation: \(\sigma=\sqrt{\frac{S_{xx}}n}\) \({}=\sqrt{\frac{\sum x^2}n-\bar x^2}\)
···Outliers (common rule): \(<Q_1-1.5(Q_3-Q_1)\) \(\text{or}\) \({}>Q_3+1.5(Q_3-Q_1)\)
···Coding \(y=\frac{x-a}b\): \(\bar x=a+b\bar y,\) \(\sigma_x=|b|\,\sigma_y\)
✓✓✓Union: \(P(A\cup B)=P(A)+P(B)\) \({}-P(A\cap B)\)
✓✓✓Intersection: \(P(A\cap B)=P(A)\times P(B\mid A)\)
✓·✓Conditional: \(P(A\mid B)=\frac{P(A\cap B)}{P(B)}\)
···Independent: \(P(A\cap B)=P(A)P(B),\) \(P(A\mid B)=P(A)\)
✓✓✓Binomial \(X\sim B(n,p)\): \(P(X=x)=\tbinom nxp^x(1-p)^{n-x}\)
✓✓✓Binomial mean, variance: \(E(X)=np,\) \(\mathrm{Var}(X)=np(1-p)\)
···Standard normal: \(Z=\frac{X-\mu}\sigma\sim N(0,1)\)
···Normal: points of inflection at \(\mu\pm\sigma\); about 68%, 95%, 99.7% within 1, 2, 3 s.d.
···Normal approx. to \(B(n,p)\) (\(n\) large, \(p\) near ½): \(N\big(np,\ np(1-p)\big)\)
···Use a continuity correction, e.g. \(P(X\le4)\approx P(Y<4.5)\)
✓✓·Sample mean: \(\bar X\sim N\!\left(\mu,\tfrac{\sigma^2}n\right),\) \(Z=\frac{\bar X-\mu}{\sigma/\sqrt n}\)
···Tests: reject \(H_0\) if \(p\)-value \(<\alpha\); two-tailed: \(\alpha/2\) each tail; PMCC test \(H_0\!:\rho=0\)