IGCSE Math Revisionigcsemathrevision.com
Edexcel International GCSE Mathematics A (4MA1)Higher tier · one-page formula sheet · ★ = not given on the paper
★ not on the 4MA1 Higher formulae sheet — learn it. Everything else is printed on the exam paper.
Number
★Indices: \(a^m\times a^n=a^{m+n},\) \(a^m\div a^n=a^{m-n},\) \((a^m)^n=a^{mn},\) \(a^0=1,\) \(a^{-n}=\tfrac1{a^n},\) \(a^{\frac mn}=(\sqrt[n]a)^m\)
★Standard form: \(A\times10^n,\) \(1\le A<10,\ n\in\mathbb Z\)
★Surds: \(\sqrt{ab}=\sqrt a\sqrt b,\) \(\frac{a}{\sqrt b}=\frac{a\sqrt b}{b},\) \(\frac1{c+\sqrt b}=\frac{c-\sqrt b}{c^2-b}\)
★Percentage change: \(\frac{\text{new}-\text{original}}{\text{original}}\times100\%\)
★Reverse percentage: \(\text{original}=\frac{\text{new}}{1\pm\frac r{100}}\)
★Compound growth / decay: \(A=P\left(1\pm\tfrac r{100}\right)^n\)
★Recurring decimal: \(x=0.\dot2\dot7\), then \(100x-x=27\), so \(x=\tfrac{27}{99}=\tfrac3{11}\)
★Bounds: a value to the nearest \(u\) lies in \([x-\tfrac u2,\ x+\tfrac u2)\); for \(a-b\) and \(a\div b\) use max with min
★Compound measures: \(\text{speed}=\frac{\text{distance}}{\text{time}},\) \(\text{density}=\frac{\text{mass}}{\text{volume}},\) \(\text{pressure}=\frac{\text{force}}{\text{area}}\)
★Sets: \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\)
Algebra
★Expanding: \((a+b)^2=a^2+2ab+b^2,\) \((a-b)^2=a^2-2ab+b^2,\) \(a^2-b^2=(a+b)(a-b)\)
Quadratic formula, \(ax^2+bx+c=0\): \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
★Completing the square: \(x^2+bx+c=\left(x+\tfrac b2\right)^2\) \({}-\tfrac{b^2}4+c\)
★Turning point of \(y=(x+p)^2+q\): \((-p,\,q)\)
Arithmetic series, sum of \(n\) terms: \(S_n=\tfrac n2\big[2a+(n-1)d\big]\)
★Arithmetic \(n\)th term: \(u_n=a+(n-1)d\)
★Proportion: \(y\propto x^n\Rightarrow y\) \({}=kx^n;\) \(y\propto\tfrac1{x^n}\Rightarrow y\) \({}=\tfrac k{x^n}\)
★Functions: \(fg(x)=f\big(g(x)\big);\) \(f^{-1}\text{: swap }x,y\text{ and rearrange}\)
★Transformations: \(f(x)+a\) up \(a\); \(f(x+a)\) left \(a\); \(-f(x)\) reflect in \(x\)-axis; \(f(-x)\) reflect in \(y\)-axis
★Inequalities: multiplying or dividing by a negative number reverses the sign
Graphs & calculus
★Straight line: \(y=mx+c,\) \(m=\frac{y_2-y_1}{x_2-x_1}\)
★Parallel; perpendicular: \(m_1=m_2;\) \(m_1\times m_2=-1\)
★Midpoint; length: \(\left(\tfrac{x_1+x_2}2,\tfrac{y_1+y_2}2\right),\) \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
★Differentiation: \(y=ax^n\Rightarrow\frac{dy}{dx}\) \({}=anx^{n-1}\)
★Turning points: \(\frac{dy}{dx}=0\); gradient of the tangent at \(x=a\) is \(\frac{dy}{dx}\) at \(a\)
★Kinematics: \(v=\frac{ds}{dt},\) \(a=\frac{dv}{dt}\)
★Distance–time gradient = speed; velocity–time gradient = acceleration, area = distance
Shape & space
★Pythagoras: \(a^2+b^2=c^2\)
★SOHCAHTOA: \(\sin\theta=\frac{\text{opp}}{\text{hyp}},\) \(\cos\theta=\frac{\text{adj}}{\text{hyp}},\) \(\tan\theta=\frac{\text{opp}}{\text{adj}}\)
Sine rule: \(\frac a{\sin A}=\frac b{\sin B}\) \({}=\frac c{\sin C}\)
Cosine rule: \(a^2=b^2+c^2-2bc\cos A\)
Area of a triangle: \(\tfrac12ab\sin C\)
★Exact values: \(\sin30^\circ=\tfrac12,\) \(\cos60^\circ=\tfrac12,\) \(\tan45^\circ=1,\) \(\sin60^\circ=\tfrac{\sqrt3}2,\) \(\sin45^\circ=\tfrac{\sqrt2}2\)
★Polygons (\(n\) sides): \(\text{interior sum}=(n-2)\times180^\circ,\) \(\text{exterior angle (regular)}\) \({}=\tfrac{360^\circ}n\)
★Areas: \(\text{triangle }\tfrac12bh,\) \(\text{parallelogram }bh\)
Area of a trapezium: \(\tfrac12(a+b)h\)
★Circle: \(C=2\pi r=\pi d,\) \(A=\pi r^2\)
★Arc; sector: \(\text{arc}=\frac{\theta}{360}\times2\pi r,\) \(\text{sector}=\frac{\theta}{360}\times\pi r^2\)
Prism; cylinder: \(V=\text{cross-section}\times\text{length};\) \(V=\pi r^2h,\) \(\text{curved }A=2\pi rh\)
Cone (slant \(l\)): \(V=\tfrac13\pi r^2h,\) \(\text{curved }A=\pi rl\)
Sphere: \(V=\tfrac43\pi r^3,\) \(A=4\pi r^2\)
★Pyramid: \(V=\tfrac13\times\text{base area}\times h\)
★Similar shapes, scale factor \(k\): \(\text{lengths}\times k,\) \(\text{areas}\times k^2,\) \(\text{volumes}\times k^3\)
★Circle theorems: angle at centre = 2 × angle at circumference; angle in a semicircle = 90°; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; tangent ⟂ radius; two tangents from a point are equal; alternate segment theorem; perpendicular from the centre bisects a chord
★Intersecting chords: \(AP\times PB=CP\times PD\)
★Vectors: \(|\binom xy|=\sqrt{x^2+y^2},\) \(\overrightarrow{AB}=\mathbf b-\mathbf a\)
★Bearings: measured clockwise from north, written with 3 figures
Statistics & probability
★Mean from a table: \(\bar x=\frac{\sum fx}{\sum f}\)
★Median position (\(n\) values): \(\tfrac{n+1}2\text{th value}\)
★Interquartile range: \(Q_3-Q_1\)
★Histogram: \(\text{frequency density}=\frac{\text{frequency}}{\text{class width}}\)
★Probability: \(P(\text{not }A)=1-P(A);\) \(\text{expected frequency}\) \({}=n\times P(A)\)
★Mutually exclusive: \(P(A\text{ or }B)=P(A)+P(B)\)
★Any two events: \(P(A\cup B)=P(A)+P(B)\) \({}-P(A\cap B)\)
★Independent: \(P(A\text{ and }B)=P(A)\times P(B)\)
★Conditional: \(P(B\mid A)=\frac{P(A\cap B)}{P(A)}\)
★Tree diagrams: multiply along branches, add the outcomes you want; without replacement, change the second branch