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Edexcel International A Level MathematicsP1 · P2 · P3 · P4 · S1 · M1 · one-page formula sheet
★ not in the Pearson Edexcel IAL formulae booklet — learn it. Everything else is printed in the booklet.
P1 · Pure 1
★Quadratic formula: \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
★Discriminant \(b^2-4ac\): \(>0\) two real roots, \(=0\) equal roots, \(<0\) no real roots
★Completing the square: \(x^2+bx+c=\left(x+\tfrac b2\right)^2\) \({}+c-\tfrac{b^2}4\)
★Indices; surds: \(a^{-n}=\tfrac1{a^n},\) \(a^{\frac mn}=\sqrt[n]{a^m};\) \(\frac{1}{\sqrt a}=\frac{\sqrt a}a\)
★Lines: \(y-y_1=m(x-x_1),\) \(m=\frac{y_2-y_1}{x_2-x_1},\) \(m_1m_2=-1\)
★Transformations: \(f(x)+a\) up \(a\); \(f(x+a)\) left \(a\); \(af(x)\) vertical ×\(a\); \(f(ax)\) horizontal ×\(\tfrac1a\)
Mensuration: \(\text{sphere }A=4\pi r^2,\) \(\text{cone curved }A=\pi rl\)
★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): \(\tfrac12ab\sin C;\) \(s=r\theta;\) \(\tfrac12r^2\theta\)
★Differentiate / integrate \(x^n\): \(\tfrac{d}{dx}x^n=nx^{n-1},\) \(\int x^n\,dx=\tfrac{x^{n+1}}{n+1}+c\ (n\ne-1)\)
★Tangent; normal: \(y-y_1=m(x-x_1),\) \(m_\text{n}=-\tfrac1m\)
P2 · Pure 2
★Factor & remainder theorems: \(f(a)=0\iff(x-a)\text{ a factor};\) \(\text{remainder }f(a)\)
★Circle: \((x-a)^2+(y-b)^2=r^2;\) \(\text{tangent}\perp\text{radius}\)
Arithmetic series: \(S_n=\tfrac12n(a+l)=\tfrac12n[2a+(n-1)d]\)
★\(n\)th terms: \(u_n=a+(n-1)d,\) \(u_n=ar^{n-1}\)
Geometric series: \(S_n=\frac{a(1-r^n)}{1-r},\) \(S_\infty=\frac a{1-r}\ (|r|<1)\)
Binomial, \(n\in\mathbb N\): \((a+b)^n=a^n+\tbinom n1a^{n-1}b+\cdots+b^n,\) \(\tbinom nr=\frac{n!}{r!(n-r)!}\)
★Log laws: \(\log_axy=\log_ax+\log_ay,\) \(\log_a\tfrac xy=\log_ax-\log_ay,\) \(\log_ax^k=k\log_ax\)
Change of base: \(\log_ax=\frac{\log_bx}{\log_ba}\)
★Trig: \(\tan\theta=\frac{\sin\theta}{\cos\theta},\) \(\sin^2\theta+\cos^2\theta\) \({}=1\)
Trapezium rule, \(h=\tfrac{b-a}n\): \(\int_a^by\,dx\) \({}\approx\tfrac12h\{(y_0+y_n)\) \({}+2(y_1+\cdots+y_{n-1})\}\)
★Stationary points: \(f'(x)=0;\) \(f''<0\text{ max},\ f''>0\text{ min}\)
★Area under a curve: \(\int_a^by\,dx\)
P3 · Pure 3
Exponential form: \(e^{x\ln a}=a^x\)
★Reciprocal identities: \(\sec^2\theta=1+\tan^2\theta,\) \(\cosec^2\theta=1+\cot^2\theta\)
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: \(\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}\)
★Double angle: \(\sin2A=2\sin A\cos A,\) \(\cos2A=2\cos^2A-1=1-2\sin^2A,\) \(\tan2A=\frac{2\tan A}{1-\tan^2A}\)
★\(R\) form: \(a\sin\theta\pm b\cos\theta\) \({}=R\sin(\theta\pm\alpha),\) \(R=\sqrt{a^2+b^2},\) \(\tan\alpha=\tfrac ba\)
★Standard derivatives: \((e^{kx})'=ke^{kx},\) \((\ln x)'=\tfrac1x,\) \((\sin kx)'=k\cos kx,\) \((\cos kx)'=-k\sin kx\)
Trig derivatives: \((\tan kx)'=k\sec^2kx,\) \((\sec x)'=\sec x\tan x,\) \((\cot x)'=-\cosec^2x,\) \((\cosec x)'=-\cosec x\cot x\)
★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: \(\frac{dx}{dy}=\frac1{dy/dx}\)
★Integrals: \(\int e^{kx}dx=\tfrac1ke^{kx}+c,\) \(\int\tfrac1x\,dx=\ln|x|+c,\) \(\int\cos kx\,dx=\tfrac1k\sin kx+c\)
★Iteration; sign change: \(x_{n+1}=g(x_n);\) \(f(a)f(b)<0\Rightarrow\text{root in }(a,b)\)
★Functions: \(fg(x)=f(g(x))\); \(f^{-1}\) reflects \(f\) in \(y=x\); modulus graphs
P4 · Pure 4
Binomial, \(|x|<1\), \(n\in\mathbb Q\): \((1+x)^n=1+nx+\frac{n(n-1)}{1\times2}x^2+\cdots\)
★Partial fractions: \(\frac{\dots}{(x-a)(x-b)}=\frac{A}{x-a}\) \({}+\frac{B}{x-b},\) \(\frac{\dots}{(x-a)^2(x-b)}\) \({}=\frac{A}{x-a}+\frac{B}{(x-a)^2}\) \({}+\frac{C}{x-b}\)
★Parametric; implicit: \(\frac{dy}{dx}=\frac{dy/dt}{dx/dt};\) \(\tfrac{d}{dx}y^2=2y\tfrac{dy}{dx}\)
Integrals: \(\int\sec^2kx\,dx=\tfrac1k\tan kx,\) \(\int\tan x\,dx=\ln|\sec x|,\) \(\int\cot x\,dx=\ln|\sin x|\)
More integrals: \(\int\cosec x\,dx=-\ln|\cosec x+\cot x|,\) \(\int\sec x\,dx=\ln|\sec x+\tan x|\)
By parts: \(\int u\frac{dv}{dx}\,dx=uv\) \({}-\int v\frac{du}{dx}\,dx\)
★Reverse chain: \(\int\frac{f'(x)}{f(x)}\,dx\) \({}=\ln|f(x)|+c\)
★Volume of revolution: \(V=\pi\int_a^by^2\,dx\)
★Separable DE: \(\int\frac{dy}{g(y)}=\int f(x)\,dx\)
★Vectors: \(\mathbf a\cdot\mathbf b=a_1b_1\) \({}+a_2b_2+a_3b_3\) \({}=|\mathbf a||\mathbf b|\cos\theta\)
★Line; perpendicular: \(\mathbf r=\mathbf a+\lambda\mathbf b;\) \(\mathbf a\cdot\mathbf b=0\)
★Magnitude: \(|\mathbf a|=\sqrt{a_1^2+a_2^2+a_3^2}\)
S1 · Statistics 1
★Mean; variance: \(\bar x=\frac{\sum fx}{\sum f},\) \(\sigma^2=\frac{\sum fx^2}{\sum f}-\bar x^2\)
★Coding \(y=\frac{x-a}b\): \(\bar x=a+b\bar y,\) \(\sigma_x=|b|\sigma_y\)
★Median by interpolation: \(Q_2=L+\frac{\frac n2-F}{f}\times w\)
★Outliers (if asked): \(<Q_1-k(Q_3-Q_1)\) \(\text{or}\) \({}>Q_3+k(Q_3-Q_1)\)
Union: \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
Intersection: \(P(A\cap B)=P(A)P(B\mid A)\)
★Conditional; independent: \(P(A\mid B)=\frac{P(A\cap B)}{P(B)};\) \(P(A\cap B)=P(A)P(B)\)
Discrete r.v.: \(E(X)=\sum xP(X=x),\) \(\mathrm{Var}(X)=\sum x^2P(X\) \({}=x)-\mu^2\)
Function of \(X\): \(E\big(g(X)\big)=\sum g(x)P(X\) \({}=x)\)
★Linear: \(E(aX+b)=aE(X)+b,\) \(\mathrm{Var}(aX+b)=a^2\mathrm{Var}(X)\)
★Discrete uniform on \(1,\dots,n\): \(E(X)=\tfrac{n+1}2,\) \(\mathrm{Var}(X)=\tfrac{n^2-1}{12}\)
\(S_{xx}\), \(S_{xy}\): \(S_{xx}=\sum x^2-\frac{(\sum x)^2}n,\) \(S_{xy}=\sum xy-\frac{\sum x\sum y}n\)
PMCC: \(r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}\)
Regression line \(y=a+bx\): \(b=\frac{S_{xy}}{S_{xx}},\) \(a=\bar y-b\bar x\)
★Normal: \(Z=\frac{X-\mu}\sigma\sim N(0,1)\)
M1 · Mechanics 1 (nothing printed)
★suvat: \(v=u+at,\) \(s=ut+\tfrac12at^2,\) \(s=vt-\tfrac12at^2,\) \(s=\tfrac12(u+v)t,\) \(v^2=u^2+2as\)
★Vectors: \(\mathbf v=\mathbf u+\mathbf at,\) \(\mathbf r=\mathbf r_0+\mathbf vt,\) \(\text{speed}=|\mathbf v|\)
★Newton's 2nd law; weight: \(F=ma,\) \(W=mg\ (g=9.8\text{ m s}^{-2})\)
★Friction: \(F\le\mu R;\) \(\text{limiting }F=\mu R\)
★Momentum; impulse: \(p=mv;\) \(I=Ft=mv-mu\)
★Conservation of momentum: \(m_1u_1+m_2u_2=m_1v_1+m_2v_2\)
★Moments: \(\text{moment}=F\times d_\perp;\) \(\text{equilibrium: }\textstyle\sum F\) \({}=0,\ \sum M=0\)
★Slope at \(\theta\): weight component \(mg\sin\theta\) down the slope; \(R=mg\cos\theta\) if nothing else acts perpendicular to it
★Velocity–time graph: gradient = acceleration, area = displacement