Formulae
What to learn by heart, and what the exam gives you. Year 13 students need both years.
Learn by heart
In the formulae booklet
Further Maths
Maths
Physics
Year 12
Year 12 and 13
Not in the Further Maths formulae booklet, so learn them by heart. You also need everything on the Maths list.
Year 12 (AS level)
Complex numbers Loci ∣ z − a ∣ = r is a circle |z - a| = r \text{ is a circle} arg ( z − a ) = θ is a half-line \arg(z - a) = \theta \text{ is a half-line} Modulus, argument and conjugate pairs ∣ z ∣ = x 2 + y 2 |z| = \sqrt{x^2 + y^2} tan arg z = y x \tan\arg z = \tfrac{y}{x} z z ∗ = ∣ z ∣ 2 z z^* = |z|^2 Matrices Determinant and inverse of a 2 × 2 matrix ( a b c d ) − 1 = 1 a d − b c ( d − b − c a ) \begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \dfrac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix} Order of transformations and inverses A B : apply B first \mathbf{AB}: \text{apply } \mathbf{B} \text{ first} ( A B ) − 1 = B − 1 A − 1 (\mathbf{AB})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1} Area scale factor area of image = ∣ det M ∣ × area of object \text{area of image} = |\det\mathbf{M}| \times \text{area of object} Series Sum of the first n integers ∑ r = 1 n r = 1 2 n ( n + 1 ) \sum_{r=1}^{n} r = \tfrac{1}{2}n(n+1) Roots of polynomials Roots of quadratics and cubics α + β = − b a \alpha + \beta = -\tfrac{b}{a} α β = c a \alpha\beta = \tfrac{c}{a} ∑ α = − b a \sum\alpha = -\tfrac{b}{a} ∑ α β = c a \sum\alpha\beta = \tfrac{c}{a} α β γ = − d a \alpha\beta\gamma = -\tfrac{d}{a} Roots of a quartic ax⁴ + bx³ + cx² + dx + e = 0 ∑ α = − b a \sum\alpha = -\tfrac{b}{a} ∑ α β = c a \sum\alpha\beta = \tfrac{c}{a} ∑ α β γ = − d a \sum\alpha\beta\gamma = -\tfrac{d}{a} α β γ δ = e a \alpha\beta\gamma\delta = \tfrac{e}{a} Sum of squares of roots ∑ α 2 = ( ∑ α ) 2 − 2 ∑ α β \sum\alpha^2 = \left(\sum\alpha\right)^2 - 2\sum\alpha\beta Calculus Volumes of revolution V x = π ∫ y 2 d x V_x = \pi\int y^2\,\mathrm{d}x V y = π ∫ x 2 d y V_y = \pi\int x^2\,\mathrm{d}y Vectors Scalar product, lines and planes a ⋅ b = ∣ a ∣ ∣ b ∣ cos θ \mathbf{a}\cdot\mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta r = a + t b \mathbf{r} = \mathbf{a} + t\mathbf{b} ( r − a ) ⋅ n = 0 (\mathbf{r} - \mathbf{a})\cdot\mathbf{n} = 0 Mechanics Momentum, impulse and energy momentum = m v \text{momentum} = mv impulse = m v − m u \text{impulse} = mv - mu KE = 1 2 m v 2 \text{KE} = \tfrac{1}{2}mv^2 PE = m g h \text{PE} = mgh
Year 13 (A level only)
Complex numbers Euler's relation e i θ = cos θ + i sin θ \mathrm{e}^{\mathrm{i}\theta} = \cos\theta + \mathrm{i}\sin\theta Multiplying and dividing in exponential form r 1 e i θ 1 × r 2 e i θ 2 = r 1 r 2 e i ( θ 1 + θ 2 ) r_1\mathrm{e}^{\mathrm{i}\theta_1} \times r_2\mathrm{e}^{\mathrm{i}\theta_2} = r_1r_2\mathrm{e}^{\mathrm{i}(\theta_1 + \theta_2)} r 1 e i θ 1 r 2 e i θ 2 = r 1 r 2 e i ( θ 1 − θ 2 ) \dfrac{r_1\mathrm{e}^{\mathrm{i}\theta_1}}{r_2\mathrm{e}^{\mathrm{i}\theta_2}} = \dfrac{r_1}{r_2}\mathrm{e}^{\mathrm{i}(\theta_1 - \theta_2)} Hyperbolic functions Definitions cosh x = e x + e − x 2 \cosh x = \dfrac{\mathrm{e}^x + \mathrm{e}^{-x}}{2} sinh x = e x − e − x 2 \sinh x = \dfrac{\mathrm{e}^x - \mathrm{e}^{-x}}{2} tanh x = sinh x cosh x \tanh x = \dfrac{\sinh x}{\cosh x} Calculus d d x sinh k x = k cosh k x \dfrac{\mathrm{d}}{\mathrm{d}x}\sinh kx = k\cosh kx d d x cosh k x = k sinh k x \dfrac{\mathrm{d}}{\mathrm{d}x}\cosh kx = k\sinh kx The basic identity cosh 2 x − sinh 2 x ≡ 1 \cosh^2 x - \sinh^2 x \equiv 1 Differential equations Simple harmonic motion x ¨ = − ω 2 x \ddot{x} = -\omega^2 x Integrating factor d y d x + P ( x ) y = Q ( x ) : \frac{\mathrm{d}y}{\mathrm{d}x} + P(x)y = Q(x): IF = e ∫ P ( x ) d x \text{IF} = \mathrm{e}^{\int P(x)\,\mathrm{d}x} Auxiliary equation solutions real roots: A e m 1 x + B e m 2 x \text{real roots: } A\mathrm{e}^{m_1x} + B\mathrm{e}^{m_2x} repeated: ( A + B x ) e m x \text{repeated: } (A + Bx)\mathrm{e}^{mx} p ± q i : e p x ( A cos q x + B sin q x ) p \pm q\mathrm{i}: \mathrm{e}^{px}(A\cos qx + B\sin qx) Mechanics Elastic strings and springs T = λ x l T = \dfrac{\lambda x}{l} EPE = λ x 2 2 l \text{EPE} = \dfrac{\lambda x^2}{2l} SHM results x = a cos ω t x = a\cos\omega t v 2 = ω 2 ( a 2 − x 2 ) v^2 = \omega^2(a^2 - x^2) T = 2 π ω T = \dfrac{2\pi}{\omega} Polar coordinates Converting polar and Cartesian x = r cos θ x = r\cos\theta y = r sin θ y = r\sin\theta r 2 = x 2 + y 2 r^2 = x^2 + y^2 Calculus Mean value of a function y ˉ = 1 b − a ∫ a b f ( x ) d x \bar{y} = \dfrac{1}{b - a}\int_a^b f(x)\,\mathrm{d}x These are printed in the formulae booklet you get in the exam, so you don't need to memorise them, but you do need to know how to use them. Each exam board sets its booklet out slightly differently.
Year 12 (AS level)
Series Sums of squares and cubes ∑ r = 1 n r 2 = 1 6 n ( n + 1 ) ( 2 n + 1 ) \sum_{r=1}^{n} r^2 = \tfrac{1}{6}n(n+1)(2n+1) ∑ r = 1 n r 3 = 1 4 n 2 ( n + 1 ) 2 \sum_{r=1}^{n} r^3 = \tfrac{1}{4}n^2(n+1)^2 Matrices Rotation through θ anticlockwise ( cos θ − sin θ sin θ cos θ ) \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} Reflection in y = (tan θ)x ( cos 2 θ sin 2 θ sin 2 θ − cos 2 θ ) \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix} Vectors Distance from a point to a plane ∣ a x 1 + b y 1 + c z 1 − d ∣ a 2 + b 2 + c 2 \dfrac{|ax_1 + by_1 + cz_1 - d|}{\sqrt{a^2 + b^2 + c^2}}
Year 13 (A level only)
Complex numbers De Moivre's theorem ( r ( cos θ + i sin θ ) ) n = r n ( cos n θ + i sin n θ ) \bigl(r(\cos\theta + \mathrm{i}\sin\theta)\bigr)^n = r^n(\cos n\theta + \mathrm{i}\sin n\theta) Roots of unity z n = 1 : z^n = 1: z = e 2 π k i / n z = \mathrm{e}^{2\pi k\mathrm{i}/n} k = 0 , 1 , … , n − 1 k = 0, 1, \dots, n - 1 Series Maclaurin series f ( x ) = f ( 0 ) + x f ′ ( 0 ) + x 2 2 ! f ′ ′ ( 0 ) + … f(x) = f(0) + xf'(0) + \dfrac{x^2}{2!}f''(0) + \dots Standard Maclaurin series e x = 1 + x + x 2 2 ! + … \mathrm{e}^x = 1 + x + \dfrac{x^2}{2!} + \dots ln ( 1 + x ) = x − x 2 2 + x 3 3 − … \ln(1 + x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dots sin x = x − x 3 3 ! + … \sin x = x - \dfrac{x^3}{3!} + \dots cos x = 1 − x 2 2 ! + … \cos x = 1 - \dfrac{x^2}{2!} + \dots Hyperbolic functions Inverse hyperbolic functions arsinh x = ln ( x + x 2 + 1 ) \operatorname{arsinh} x = \ln\bigl(x + \sqrt{x^2 + 1}\bigr) arcosh x = ln ( x + x 2 − 1 ) \operatorname{arcosh} x = \ln\bigl(x + \sqrt{x^2 - 1}\bigr) artanh x = 1 2 ln 1 + x 1 − x \operatorname{artanh} x = \tfrac{1}{2}\ln\dfrac{1 + x}{1 - x} Calculus Derivatives of inverse functions arcsin x → 1 1 − x 2 \arcsin x \to \dfrac{1}{\sqrt{1 - x^2}} arctan x → 1 1 + x 2 \arctan x \to \dfrac{1}{1 + x^2} arsinh x → 1 1 + x 2 \operatorname{arsinh} x \to \dfrac{1}{\sqrt{1 + x^2}} arcosh x → 1 x 2 − 1 \operatorname{arcosh} x \to \dfrac{1}{\sqrt{x^2 - 1}} Standard integrals ∫ d x a 2 − x 2 = arcsin x a \int \dfrac{\mathrm{d}x}{\sqrt{a^2 - x^2}} = \arcsin\dfrac{x}{a} ∫ d x a 2 + x 2 = 1 a arctan x a \int \dfrac{\mathrm{d}x}{a^2 + x^2} = \dfrac{1}{a}\arctan\dfrac{x}{a} ∫ d x x 2 + a 2 = arsinh x a \int \dfrac{\mathrm{d}x}{\sqrt{x^2 + a^2}} = \operatorname{arsinh}\dfrac{x}{a} Polar coordinates Area of a sector A = 1 2 ∫ α β r 2 d θ A = \tfrac{1}{2}\int_\alpha^\beta r^2\,\mathrm{d}\theta
Not in the formulae booklet, so learn them by heart.
Year 12 (AS level)
Algebra Quadratic formula x = − b ± b 2 − 4 a c 2 a x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} Laws of indices a m a n = a m + n a^m a^n = a^{m+n} a m ÷ a n = a m − n a^m \div a^n = a^{m-n} ( a m ) n = a m n (a^m)^n = a^{mn} Laws of logarithms log a x + log a y = log a x y \log_a x + \log_a y = \log_a xy log a x − log a y = log a x y \log_a x - \log_a y = \log_a \tfrac{x}{y} k log a x = log a x k k\log_a x = \log_a x^k Definition of a logarithm log a x = n ⟺ x = a n \log_a x = n \iff x = a^n Coordinate geometry Straight line y − y 1 = m ( x − x 1 ) y - y_1 = m(x - x_1) Perpendicular gradients m 1 m 2 = − 1 m_1 m_2 = -1 Trigonometry Sine rule a sin A = b sin B = c sin C \dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} Cosine rule a 2 = b 2 + c 2 − 2 b c cos A a^2 = b^2 + c^2 - 2bc\cos A Area of a triangle Area = 1 2 a b sin C \text{Area} = \tfrac{1}{2}ab\sin C Basic identities sin 2 θ + cos 2 θ ≡ 1 \sin^2\theta + \cos^2\theta \equiv 1 tan θ ≡ sin θ cos θ \tan\theta \equiv \dfrac{\sin\theta}{\cos\theta} Mensuration Circle C = 2 π r C = 2\pi r A = π r 2 A = \pi r^2 Trapezium and prism A = 1 2 ( a + b ) h A = \tfrac{1}{2}(a+b)h V = cross-section × length V = \text{cross-section} \times \text{length} Calculus Differentiating powers d d x x n = n x n − 1 \dfrac{\mathrm{d}}{\mathrm{d}x}x^n = nx^{n-1} Differentiating exponentials d d x e k x = k e k x \dfrac{\mathrm{d}}{\mathrm{d}x}\mathrm{e}^{kx} = k\mathrm{e}^{kx} Integrating powers ∫ x n d x = x n + 1 n + 1 + c \int x^n\,\mathrm{d}x = \dfrac{x^{n+1}}{n+1} + c n ≠ − 1 n \ne -1 Area under a curve Area = ∫ a b y d x \text{Area} = \int_a^b y\,\mathrm{d}x Vectors Magnitude in two dimensions ∣ x i + y j ∣ = x 2 + y 2 |x\mathbf{i} + y\mathbf{j}| = \sqrt{x^2 + y^2} Statistics Mean x ˉ = ∑ x n = ∑ f x ∑ f \bar{x} = \dfrac{\sum x}{n} = \dfrac{\sum fx}{\sum f} Mechanics Weight and Newton's second law W = m g W = mg F = m a F = ma Calculus in kinematics v = d r d t v = \dfrac{\mathrm{d}r}{\mathrm{d}t} a = d v d t a = \dfrac{\mathrm{d}v}{\mathrm{d}t} r = ∫ v d t r = \int v\,\mathrm{d}t v = ∫ a d t v = \int a\,\mathrm{d}t
Year 13 (A level only)
Sequences nth term of an arithmetic sequence u n = a + ( n − 1 ) d u_n = a + (n-1)d nth term of a geometric sequence u n = a r n − 1 u_n = ar^{n-1} Trigonometry Reciprocal identities sec 2 θ ≡ 1 + tan 2 θ \sec^2\theta \equiv 1 + \tan^2\theta cosec 2 θ ≡ 1 + cot 2 θ \operatorname{cosec}^2\theta \equiv 1 + \cot^2\theta Double angle formulae sin 2 A ≡ 2 sin A cos A \sin 2A \equiv 2\sin A\cos A cos 2 A ≡ cos 2 A − sin 2 A \cos 2A \equiv \cos^2 A - \sin^2 A tan 2 A ≡ 2 tan A 1 − tan 2 A \tan 2A \equiv \dfrac{2\tan A}{1 - \tan^2 A} Arc length and sector area (radians) s = r θ s = r\theta A = 1 2 r 2 θ A = \tfrac{1}{2}r^2\theta Calculus Standard derivatives d d x sin k x = k cos k x \dfrac{\mathrm{d}}{\mathrm{d}x}\sin kx = k\cos kx d d x cos k x = − k sin k x \dfrac{\mathrm{d}}{\mathrm{d}x}\cos kx = -k\sin kx d d x ln x = 1 x \dfrac{\mathrm{d}}{\mathrm{d}x}\ln x = \dfrac{1}{x} Product and chain rules d d x ( u v ) = u d v d x + v d u d x \dfrac{\mathrm{d}}{\mathrm{d}x}(uv) = u\dfrac{\mathrm{d}v}{\mathrm{d}x} + v\dfrac{\mathrm{d}u}{\mathrm{d}x} d y d x = d y d u ⋅ d u d x \dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{\mathrm{d}y}{\mathrm{d}u}\cdot\dfrac{\mathrm{d}u}{\mathrm{d}x} Standard integrals ∫ cos k x d x = 1 k sin k x + c \int \cos kx\,\mathrm{d}x = \tfrac{1}{k}\sin kx + c ∫ e k x d x = 1 k e k x + c \int \mathrm{e}^{kx}\,\mathrm{d}x = \tfrac{1}{k}\mathrm{e}^{kx} + c ∫ 1 x d x = ln ∣ x ∣ + c \int \tfrac{1}{x}\,\mathrm{d}x = \ln|x| + c Vectors Magnitude in three dimensions ∣ x i + y j + z k ∣ = x 2 + y 2 + z 2 |x\mathbf{i} + y\mathbf{j} + z\mathbf{k}| = \sqrt{x^2 + y^2 + z^2} Statistics Standardising a normal variable Z = X − μ σ Z = \dfrac{X - \mu}{\sigma} These are printed in the formulae booklet you get in the exam, so you don't need to memorise them, but you do need to know how to use them. Each exam board sets its booklet out slightly differently.
Year 12 (AS level)
Algebra Binomial expansion (positive integer n) ( a + b ) n = a n + ( n 1 ) a n − 1 b + ( n 2 ) a n − 2 b 2 + ⋯ + b n (a + b)^n = a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \dots + b^n Logarithms and exponentials log a x = log b x log b a \log_a x = \dfrac{\log_b x}{\log_b a} e x ln a = a x \mathrm{e}^{x\ln a} = a^x Mensuration Sphere and cone sphere surface = 4 π r 2 \text{sphere surface} = 4\pi r^2 cone curved surface = π r l \text{cone curved surface} = \pi r l Statistics Probability P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) \mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cap B) P ( A ∩ B ) = P ( A ) P ( B ∣ A ) \mathrm{P}(A \cap B) = \mathrm{P}(A)\mathrm{P}(B \mid A) Binomial distribution P ( X = x ) = ( n x ) p x ( 1 − p ) n − x \mathrm{P}(X = x) = \binom{n}{x}p^x(1 - p)^{n - x} mean n p \text{mean } np variance n p ( 1 − p ) \text{variance } np(1 - p) Variance and standard deviation σ 2 = ∑ x 2 n − x ˉ 2 = S x x n \sigma^2 = \dfrac{\sum x^2}{n} - \bar{x}^2 = \dfrac{S_{xx}}{n} Mechanics Constant acceleration v = u + a t v = u + at s = u t + 1 2 a t 2 s = ut + \tfrac{1}{2}at^2 v 2 = u 2 + 2 a s v^2 = u^2 + 2as s = 1 2 ( u + v ) t s = \tfrac{1}{2}(u + v)t
Year 13 (A level only)
Sequences Arithmetic series S n = 1 2 n ( a + l ) = 1 2 n ( 2 a + ( n − 1 ) d ) S_n = \tfrac{1}{2}n(a + l) = \tfrac{1}{2}n\bigl(2a + (n - 1)d\bigr) Geometric series S n = a ( 1 − r n ) 1 − r S_n = \dfrac{a(1 - r^n)}{1 - r} S ∞ = a 1 − r , ∣ r ∣ < 1 S_\infty = \dfrac{a}{1 - r},\ |r| < 1 Algebra Binomial series (any n) ( 1 + x ) n = 1 + n x + n ( n − 1 ) 2 ! x 2 + … (1 + x)^n = 1 + nx + \dfrac{n(n-1)}{2!}x^2 + \dots ∣ x ∣ < 1 |x| < 1 Trigonometry Addition formulae sin ( A ± B ) ≡ sin A cos B ± cos A sin B \sin(A \pm B) \equiv \sin A\cos B \pm \cos A\sin B cos ( A ± B ) ≡ cos A cos B ∓ sin A sin B \cos(A \pm B) \equiv \cos A\cos B \mp \sin A\sin B Tan addition formula tan ( A ± B ) ≡ tan A ± tan B 1 ∓ tan A tan B \tan(A \pm B) \equiv \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B} Small angle approximations sin θ ≈ θ \sin\theta \approx \theta cos θ ≈ 1 − θ 2 2 \cos\theta \approx 1 - \tfrac{\theta^2}{2} tan θ ≈ θ \tan\theta \approx \theta Calculus Further derivatives tan k x → k sec 2 k x \tan kx \to k\sec^2 kx sec x → sec x tan x \sec x \to \sec x\tan x cot x → − cosec 2 x \cot x \to -\operatorname{cosec}^2 x cosec x → − cosec x cot x \operatorname{cosec} x \to -\operatorname{cosec} x\cot x Quotient rule d d x ( u v ) = v u ′ − u v ′ v 2 \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{u}{v}\right) = \dfrac{v u' - u v'}{v^2} Further integrals ∫ sec 2 k x d x = 1 k tan k x + c \int \sec^2 kx\,\mathrm{d}x = \tfrac{1}{k}\tan kx + c ∫ tan k x d x = 1 k ln ∣ sec k x ∣ + c \int \tan kx\,\mathrm{d}x = \tfrac{1}{k}\ln|\sec kx| + c Integration by parts ∫ u d v d x d x = u v − ∫ v d u d x d x \int u\dfrac{\mathrm{d}v}{\mathrm{d}x}\,\mathrm{d}x = uv - \int v\dfrac{\mathrm{d}u}{\mathrm{d}x}\,\mathrm{d}x Numerical methods Trapezium rule ∫ a b y d x ≈ 1 2 h { ( y 0 + y n ) + 2 ( y 1 + ⋯ + y n − 1 ) } \int_a^b y\,\mathrm{d}x \approx \tfrac{1}{2}h\bigl\{(y_0 + y_n) + 2(y_1 + \dots + y_{n-1})\bigr\} h = b − a n h = \dfrac{b - a}{n} Newton–Raphson x n + 1 = x n − f ( x n ) f ′ ( x n ) x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)} Statistics Conditional probability P ( A ∣ B ) = P ( A ∩ B ) P ( B ) \mathrm{P}(A \mid B) = \dfrac{\mathrm{P}(A \cap B)}{\mathrm{P}(B)} Distribution of the sample mean X ˉ ∼ N ( μ , σ 2 n ) \bar{X} \sim \mathrm{N}\left(\mu, \dfrac{\sigma^2}{n}\right)
Both boards give you a data and formulae sheet, so you rarely need to recall an equation. You do need to know what every symbol means, its unit, and when to use each equation.
Year 12 (AS level)
Units SI base units kg , m , s , A , K , mol \text{kg},\ \text{m},\ \text{s},\ \text{A},\ \text{K},\ \text{mol} Prefixes T 1 0 12 , G 1 0 9 , M 1 0 6 , k 1 0 3 , c 1 0 − 2 , m 1 0 − 3 , μ 1 0 − 6 , n 1 0 − 9 , p 1 0 − 12 , f 1 0 − 15 \text{T}\ 10^{12},\ \text{G}\ 10^{9},\ \text{M}\ 10^{6},\ \text{k}\ 10^{3},\ \text{c}\ 10^{-2},\ \text{m}\ 10^{-3},\ \mu\ 10^{-6},\ \text{n}\ 10^{-9},\ \text{p}\ 10^{-12},\ \text{f}\ 10^{-15} Electronvolt 1 eV = 1.60 × 1 0 − 19 J 1\ \text{eV} = 1.60 \times 10^{-19}\ \text{J} Maths skills Areas and volumes circle π r 2 \text{circle } \pi r^2 sphere 4 3 π r 3 \text{sphere } \tfrac{4}{3}\pi r^3 cylinder π r 2 h \text{cylinder } \pi r^2 h Uncertainties % unc. of A B or A B = % A + % B \%\text{ unc. of } AB \text{ or } \tfrac{A}{B} = \%A + \%B % unc. of A n = n × % A \%\text{ unc. of } A^n = n \times \%A
Year 13 (A level only)
Maths skills Radians and small angles π rad = 18 0 ∘ \pi\ \text{rad} = 180^\circ sin θ ≈ tan θ ≈ θ \sin\theta \approx \tan\theta \approx \theta cos θ ≈ 1 \cos\theta \approx 1 Straight-line graphs from exponentials ln N = ln N 0 − λ t \ln N = \ln N_0 - \lambda t The key equations. Almost all of them are printed on your exam's data and formulae sheet, so you don't need to memorise them, but you do need to know how to use them.
Year 12 (AS level)
Mechanics Constant acceleration (suvat) v = u + a t v = u + at s = u t + 1 2 a t 2 s = ut + \tfrac{1}{2}at^2 v 2 = u 2 + 2 a s v^2 = u^2 + 2as s = 1 2 ( u + v ) t s = \tfrac{1}{2}(u+v)t Force, weight and momentum F = m a F = ma W = m g W = mg p = m v p = mv F = Δ p Δ t F = \dfrac{\Delta p}{\Delta t} Energy and power E k = 1 2 m v 2 E_k = \tfrac{1}{2}mv^2 Δ E p = m g Δ h \Delta E_p = mg\Delta h W = F s cos θ W = Fs\cos\theta P = W t = F v P = \dfrac{W}{t} = Fv Moments and efficiency moment = F d \text{moment} = Fd efficiency = useful output total input \text{efficiency} = \dfrac{\text{useful output}}{\text{total input}} Materials Density and Hooke's law ρ = m V \rho = \dfrac{m}{V} F = k Δ x F = k\Delta x E e l = 1 2 F Δ x E_{el} = \tfrac{1}{2}F\Delta x Stress, strain and the Young modulus σ = F A \sigma = \dfrac{F}{A} ε = Δ x x \varepsilon = \dfrac{\Delta x}{x} E = σ ε E = \dfrac{\sigma}{\varepsilon} Stokes' law Edexcel only F = 6 π η r v F = 6\pi\eta r v Electricity Current, p.d. and resistance I = Δ Q Δ t I = \dfrac{\Delta Q}{\Delta t} V = W Q V = \dfrac{W}{Q} R = V I R = \dfrac{V}{I} Power and resistivity P = V I = I 2 R = V 2 R P = VI = I^2R = \dfrac{V^2}{R} R = ρ L A R = \dfrac{\rho L}{A} Resistors and e.m.f. R T = R 1 + R 2 R_T = R_1 + R_2 1 R T = 1 R 1 + 1 R 2 \dfrac{1}{R_T} = \dfrac{1}{R_1} + \dfrac{1}{R_2} ε = I ( R + r ) \varepsilon = I(R + r) Drift velocity Edexcel only I = n q v A I = nqvA Waves Wave equation and refraction v = f λ v = f\lambda n = c v n = \dfrac{c}{v} n 1 sin θ 1 = n 2 sin θ 2 n_1\sin\theta_1 = n_2\sin\theta_2 sin C = 1 n \sin C = \dfrac{1}{n} Diffraction grating d sin θ = n λ d\sin\theta = n\lambda Young's double slits AQA only w = λ D s w = \dfrac{\lambda D}{s} Quantum Photons and the photoelectric effect E = h f = h c λ E = hf = \dfrac{hc}{\lambda} h f = ϕ + E k ( max ) hf = \phi + E_{k(\max)} de Broglie wavelength λ = h p = h m v \lambda = \dfrac{h}{p} = \dfrac{h}{mv}
Year 13 (A level only)
Circular motion Angular speed and centripetal force ω = 2 π T = v r \omega = \dfrac{2\pi}{T} = \dfrac{v}{r} a = v 2 r = ω 2 r a = \dfrac{v^2}{r} = \omega^2 r F = m v 2 r F = \dfrac{mv^2}{r} Oscillations Simple harmonic motion a = − ω 2 x a = -\omega^2 x x = A cos ω t x = A\cos\omega t T = 2 π m k T = 2\pi\sqrt{\dfrac{m}{k}} T = 2 π l g T = 2\pi\sqrt{\dfrac{l}{g}} Thermal physics Specific heat and latent heat Q = m c Δ θ Q = mc\Delta\theta Q = m L Q = mL Ideal gases p V = n R T = N k T pV = nRT = NkT p V = 1 3 N m c 2 ‾ pV = \tfrac{1}{3}Nm\overline{c^2} 1 2 m c 2 ‾ = 3 2 k T \tfrac{1}{2}m\overline{c^2} = \tfrac{3}{2}kT Fields Gravitational fields F = G m 1 m 2 r 2 F = \dfrac{Gm_1m_2}{r^2} g = G M r 2 g = \dfrac{GM}{r^2} V = − G M r V = -\dfrac{GM}{r} Electric fields F = Q 1 Q 2 4 π ε 0 r 2 F = \dfrac{Q_1Q_2}{4\pi\varepsilon_0 r^2} E = F Q = V d E = \dfrac{F}{Q} = \dfrac{V}{d} V = Q 4 π ε 0 r V = \dfrac{Q}{4\pi\varepsilon_0 r} Capacitors Capacitance and discharge C = Q V C = \dfrac{Q}{V} W = 1 2 Q V = 1 2 C V 2 W = \tfrac{1}{2}QV = \tfrac{1}{2}CV^2 Q = Q 0 e − t / R C Q = Q_0\mathrm{e}^{-t/RC} Magnetic fields Forces and induction F = B I l F = BIl F = B Q v F = BQv Φ = B A \Phi = BA ε = − N Δ Φ Δ t \varepsilon = -N\dfrac{\Delta\Phi}{\Delta t} Nuclear physics Radioactive decay A = λ N A = \lambda N N = N 0 e − λ t N = N_0\mathrm{e}^{-\lambda t} t 1 / 2 = ln 2 λ t_{1/2} = \dfrac{\ln 2}{\lambda} Mass–energy Δ E = c 2 Δ m \Delta E = c^2\Delta m Nuclear radius AQA only R = R 0 A 1 / 3 R = R_0A^{1/3} Space Stars and the universe Edexcel only L = σ A T 4 L = \sigma AT^4 λ max T = 2.898 × 1 0 − 3 m K \lambda_{\max}T = 2.898 \times 10^{-3}\ \text{m K} I = L 4 π d 2 I = \dfrac{L}{4\pi d^2} v = H 0 d v = H_0 d