Square of a sum
(a+b)^2 = a^2 + 2ab + b^2
Expansion of a binomial squared.
Reference · Classes 6–12
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Mathematics
Algebraic Identities
Square of a difference
(a-b)^2 = a^2 - 2ab + b^2
Expansion of a binomial difference squared.
Difference of two squares
a^2 - b^2 = (a+b)(a-b)
Factorise a difference of squares.
Cube of a sum
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
Expansion of a binomial cubed.
Cube of a difference
(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
Expansion of a binomial difference cubed.
Sum of two cubes
a^3 + b^3 = (a+b)(a^2 - ab + b^2)
Factorise a sum of cubes.
Difference of two cubes
a^3 - b^3 = (a-b)(a^2 + ab + b^2)
Factorise a difference of cubes.
Quadratic Equations
Quadratic formula
x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Roots of ax² + bx + c = 0, a ≠ 0.
Discriminant
D = b^2 - 4ac
D > 0 two real roots, D = 0 one, D < 0 none.
Sum & product of roots
\alpha + \beta = -\dfrac{b}{a}, \quad \alpha\beta = \dfrac{c}{a}
For ax² + bx + c = 0 with roots α, β.
Sequences & Series
Arithmetic progression — nth term
a_n = a + (n-1)d
First term a, common difference d.
Arithmetic progression — sum
S_n = \dfrac{n}{2}\,[\,2a + (n-1)d\,] = \dfrac{n}{2}(a + l)
Sum of n terms; l is the last term.
Geometric progression — nth term
a_n = a\,r^{\,n-1}
First term a, common ratio r.
Geometric progression — sum
S_n = \dfrac{a(r^n - 1)}{r - 1}, \quad r \ne 1
Sum of n terms of a GP.
Infinite GP sum
S_\infty = \dfrac{a}{1 - r}, \quad |r| < 1
Sum of an infinite GP that converges.
Logarithms
Logarithm laws
\log_b(xy) = \log_b x + \log_b y, \quad \log_b\!\dfrac{x}{y} = \log_b x - \log_b y
Product and quotient rules for logs.
Power rule & change of base
\log_b(x^n) = n\log_b x, \quad \log_b a = \dfrac{\ln a}{\ln b}
Bring down powers; change the base.
Binomial Theorem
Binomial theorem
(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{\,n-k} b^{\,k}
Expansion of (a + b) to a positive integer power.
Trigonometry
Pythagorean identity
\sin^2\theta + \cos^2\theta = 1
The fundamental trigonometric identity.
Secant & cosecant identities
1 + \tan^2\theta = \sec^2\theta, \quad 1 + \cot^2\theta = \csc^2\theta
Derived Pythagorean identities.
Sine of a sum
\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B
Compound-angle formula for sine.
Cosine of a sum
\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B
Compound-angle formula for cosine.
Double-angle formulae
\sin 2\theta = 2\sin\theta\cos\theta, \quad \cos 2\theta = 2\cos^2\theta - 1
Sine and cosine of twice an angle.
Law of sines
\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} = 2R
Relates sides to opposite angles; R is the circumradius.
Law of cosines
c^2 = a^2 + b^2 - 2ab\cos C
Generalised Pythagoras for any triangle.
Coordinate Geometry
Distance between two points
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
Length of the segment joining two points.
Midpoint formula
M = \left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)
Midpoint of a line segment.
Section formula
P = \left(\dfrac{m x_2 + n x_1}{m + n}, \dfrac{m y_2 + n y_1}{m + n}\right)
Point dividing a segment in ratio m : n.
Slope of a line
m = \dfrac{y_2 - y_1}{x_2 - x_1}
Gradient through two points.
Distance from a point to a line
d = \dfrac{|a x_0 + b y_0 + c|}{\sqrt{a^2 + b^2}}
Perpendicular distance to ax + by + c = 0.
Equation of a circle
(x - h)^2 + (y - k)^2 = r^2
Centre (h, k), radius r.
Area of a triangle (coordinates)
A = \tfrac{1}{2}\,|\,x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\,|
Area from three vertices.
Mensuration
Circle — area & circumference
A = \pi r^2, \quad C = 2\pi r
Area and perimeter of a circle.
Heron's formula
A = \sqrt{s(s-a)(s-b)(s-c)}, \quad s = \dfrac{a+b+c}{2}
Area of a triangle from its three sides.
Sphere — surface area & volume
S = 4\pi r^2, \quad V = \tfrac{4}{3}\pi r^3
Surface area and volume of a sphere.
Cylinder — surface area & volume
\text{TSA} = 2\pi r(r + h), \quad V = \pi r^2 h
Total surface area and volume of a cylinder.
Cone — surface area & volume
\text{TSA} = \pi r(r + l), \quad V = \tfrac{1}{3}\pi r^2 h, \quad l = \sqrt{r^2 + h^2}
Curved/total surface area, volume and slant height.
Cuboid — surface area & volume
\text{TSA} = 2(lb + bh + hl), \quad V = lbh
Surface area and volume of a cuboid.
Calculus — Derivatives
Power rule
\dfrac{d}{dx}\,x^n = n\,x^{\,n-1}
Derivative of a power of x.
Derivatives of sine & cosine
\dfrac{d}{dx}\sin x = \cos x, \quad \dfrac{d}{dx}\cos x = -\sin x
Standard trigonometric derivatives.
Derivatives of eˣ and ln x
\dfrac{d}{dx}e^x = e^x, \quad \dfrac{d}{dx}\ln x = \dfrac{1}{x}
Exponential and natural-log derivatives.
Product rule
(uv)' = u'v + uv'
Derivative of a product.
Quotient rule
\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}
Derivative of a quotient.
Chain rule
\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}
Derivative of a composite function.
Calculus — Integrals
Power rule for integration
\int x^n\,dx = \dfrac{x^{\,n+1}}{n+1} + C, \quad n \ne -1
Indefinite integral of a power.
Integral of 1/x
\int \dfrac{1}{x}\,dx = \ln|x| + C
Integral giving a natural logarithm.
Integrals of sine & cosine
\int \sin x\,dx = -\cos x + C, \quad \int \cos x\,dx = \sin x + C
Standard trigonometric integrals.
Permutations & Combinations
Permutations
{}^nP_r = \dfrac{n!}{(n-r)!}
Ordered arrangements of r from n.
Combinations
{}^nC_r = \dfrac{n!}{r!\,(n-r)!}
Unordered selections of r from n.
Probability & Statistics
Probability of an event
P(A) = \dfrac{\text{favourable outcomes}}{\text{total outcomes}}
Classical definition of probability.
Addition rule
P(A \cup B) = P(A) + P(B) - P(A \cap B)
Probability of A or B.
Conditional probability
P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}
Probability of A given that B occurred.
Mean
\bar{x} = \dfrac{\sum f_i x_i}{\sum f_i}
Arithmetic mean of grouped data.
Variance & standard deviation
\sigma^2 = \dfrac{\sum (x_i - \bar{x})^2}{n}, \quad \sigma = \sqrt{\sigma^2}
Spread of data about the mean.
Physics
Kinematics
First equation of motion
v = u + at
Velocity after time t under constant acceleration.
Second equation of motion
s = ut + \tfrac{1}{2}at^2
Displacement under constant acceleration.
Third equation of motion
v^2 = u^2 + 2as
Velocity–displacement relation.
Laws of Motion
Newton's second law
F = ma
Force equals mass times acceleration.
Momentum & impulse
p = mv, \quad J = F\,\Delta t = \Delta p
Linear momentum and impulse–momentum theorem.
Work, Energy & Power
Work done
W = F\,d\cos\theta
Work by a constant force at angle θ.
Kinetic energy
KE = \tfrac{1}{2}mv^2
Energy of a moving body.
Potential energy (gravitational)
PE = mgh
Energy due to height in a gravitational field.
Power
P = \dfrac{W}{t} = Fv
Rate of doing work.
Gravitation
Newton's law of gravitation
F = \dfrac{G m_1 m_2}{r^2}
Force between two masses.
Acceleration due to gravity
g = \dfrac{GM}{R^2}
g at the surface of a planet of mass M, radius R.
Oscillations
Time period of a spring
T = 2\pi\sqrt{\dfrac{m}{k}}
SHM of a mass on a spring of stiffness k.
Time period of a pendulum
T = 2\pi\sqrt{\dfrac{L}{g}}
Simple pendulum of length L.
Waves
Wave speed
v = f\lambda
Speed equals frequency times wavelength.
Current Electricity
Ohm's law
V = IR
Voltage equals current times resistance.
Electric power
P = VI = I^2 R = \dfrac{V^2}{R}
Power dissipated in a resistor.
Optics
Lens formula
\dfrac{1}{f} = \dfrac{1}{v} - \dfrac{1}{u}
Thin-lens equation (sign convention applies).
Mirror formula
\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}
Spherical-mirror equation (sign convention applies).
Magnification (lens)
m = \dfrac{v}{u} = \dfrac{h'}{h}
Ratio of image to object size.
Mechanics of Fluids
Pressure
P = \dfrac{F}{A}
Force per unit area.
Density
\rho = \dfrac{m}{V}
Mass per unit volume.
Thermal Physics
Heat energy
Q = mc\,\Delta T
Heat to change temperature by ΔT; c is specific heat.
Modern Physics
Mass–energy equivalence
E = mc^2
Energy equivalent of mass.
Photon energy
E = hf = \dfrac{hc}{\lambda}
Energy of a quantum of light.
Chemistry
Mole Concept
Number of moles
n = \dfrac{m}{M} = \dfrac{N}{N_A}
From mass/molar mass, or particles/Avogadro's number.
Moles of a gas at STP
n = \dfrac{V}{22.4\ \text{L}}
Molar volume of an ideal gas at STP.
Solutions
Molarity
M = \dfrac{\text{moles of solute}}{\text{volume of solution (L)}}
Concentration in mol per litre.
Molality
m = \dfrac{\text{moles of solute}}{\text{mass of solvent (kg)}}
Concentration in mol per kg of solvent.
States of Matter
Ideal gas equation
PV = nRT
Relates pressure, volume, moles and temperature.
Boyle's law
P_1 V_1 = P_2 V_2
At constant temperature and amount of gas.
Charles's law
\dfrac{V_1}{T_1} = \dfrac{V_2}{T_2}
At constant pressure and amount of gas (T in kelvin).
Acids, Bases & Salts
pH and pOH
\text{pH} = -\log[\text{H}^+], \quad \text{pH} + \text{pOH} = 14
Acidity from hydrogen-ion concentration (at 25 °C).
Thermodynamics
Gibbs free energy
\Delta G = \Delta H - T\,\Delta S
ΔG < 0 means a spontaneous process.
Equilibrium
Equilibrium constant
K_c = \dfrac{[\text{C}]^c[\text{D}]^d}{[\text{A}]^a[\text{B}]^b}
For aA + bB ⇌ cC + dD, in terms of concentrations.
Electrochemistry
Cell potential
E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}}
Standard EMF of an electrochemical cell.
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