Mathematics Formulas

Factors and multiples

• Prime factorisation: write a number as a product of primes, e.g. $60 = 2^2 \times 3 \times 5$.
• $\text{GCD} \times \text{LCM} = \text{product of the two numbers}$ (for two numbers $a, b$: $\text{GCD}(a,b)\times\text{LCM}(a,b)=a\times b$).

Fractions

$$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \qquad \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \qquad \frac{a}{b} \div \frac{c}{d} = \frac{a}{b}\times\frac{d}{c}$$

Percentages

$$\text{Percentage} = \frac{\text{part}}{\text{whole}} \times 100\% \qquad \text{Percentage change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$$

• Increase by $r\%$: multiply by $\left(1 + \frac{r}{100}\right)$. Decrease by $r\%$: multiply by $\left(1 - \frac{r}{100}\right)$.
• Original amount $= \dfrac{\text{final amount}}{1 \pm \frac{r}{100}}$.

Squares, cubes and reciprocals

• $\sqrt{ab} = \sqrt{a}\sqrt{b}$, $\quad \sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}$
• Reciprocal of $x$ is $\dfrac{1}{x}$.

Ratios

• Ratio $a:b$ means $\dfrac{a}{b}$. Sharing $T$ in the ratio $a:b$: first share $= \dfrac{a}{a+b}\times T$, second share $= \dfrac{b}{a+b}\times T$.
• Increasing/decreasing in a ratio $a:b$: new value $= \dfrac{b}{a}\times \text{old value}$ (for $a:b$ read as old : new).

Rates

$$\text{Rate} = \frac{\text{amount}}{\text{time}} \qquad \text{Speed} = \frac{\text{distance}}{\text{time}}$$

Proportion

• Direct proportion: $\dfrac{y_1}{x_1} = \dfrac{y_2}{x_2}$.
• Inverse proportion: $x_1 y_1 = x_2 y_2$.

Rates of work

If one worker finishes a job in $a$ days and another in $b$ days, together they take $$\frac{ab}{a+b}\ \text{days}$$ (work done per day: $\dfrac{1}{a} + \dfrac{1}{b}$).

Mixtures

$$\text{Price of mixture per unit} = \frac{\text{total cost of ingredients}}{\text{total quantity}}$$ Selling price per unit $= \text{cost per unit} \times \left(1 + \dfrac{\text{profit \%}}{100}\right)$.

Laws of indices

$$a^m \times a^n = a^{m+n} \qquad \frac{a^m}{a^n} = a^{m-n} \qquad (a^m)^n = a^{mn}$$ $$a^0 = 1 \qquad a^{-n} = \frac{1}{a^n} \qquad a^{\frac{m}{n}} = \sqrt[n]{a^m} \qquad (ab)^n = a^n b^n$$

Laws of logarithms

If $a^x = N$ then $\log_a N = x$. $$\log_a (xy) = \log_a x + \log_a y \qquad \log_a \frac{x}{y} = \log_a x - \log_a y$$ $$\log_a x^n = n\log_a x \qquad \log_a a = 1 \qquad \log_a 1 = 0$$ $$\log_a x = \frac{\log_b x}{\log_b a} \quad\text{(change of base)}$$

$$\text{Absolute error} = |\text{approximate value} - \text{exact value}|$$ $$\text{Relative error} = \frac{\text{absolute error}}{\text{exact value}} \qquad \text{Percentage error} = \frac{\text{absolute error}}{\text{exact value}} \times 100\%$$

• A measurement given to the nearest unit $u$ lies in the range $\text{value} \pm \dfrac{u}{2}$.
• Maximum possible error $= \dfrac{1}{2}\times$ (smallest unit of measurement).
• Sum: upper bound $= \sum$ upper bounds; lower bound $= \sum$ lower bounds. Difference $a - b$: upper $= a_{\max} - b_{\min}$, lower $= a_{\min} - b_{\max}$.

Triangles

$$A = \tfrac{1}{2}\, b\, h \qquad A = \tfrac{1}{2}\, ab\sin C \qquad A = \sqrt{s(s-a)(s-b)(s-c)},\ \ s=\tfrac{a+b+c}{2}\ \text{(Heron)}$$

Quadrilaterals

Circle, sector and segment

$$C = 2\pi r = \pi d \qquad A = \pi r^2$$ $$\text{Arc length} = \frac{\theta}{360^\circ}\times 2\pi r \qquad \text{Sector area} = \frac{\theta}{360^\circ}\times \pi r^2$$ $$\text{Segment area} = \frac{\theta}{360^\circ}\pi r^2 - \tfrac{1}{2}r^2\sin\theta$$

Regular polygon with $n$ sides, side $s$, circumradius $r$

$$A = \tfrac{1}{2} n r^2 \sin\left(\frac{360^\circ}{n}\right)$$

Common region of two overlapping circles

Area $=$ sum of the two segments cut off by the common chord.

For a cone, slant height $l = \sqrt{r^2 + h^2}$.

Frustum of a cone (radii $R$ and $r$, height $h$)

$$V = \tfrac{1}{3}\pi h\,(R^2 + Rr + r^2)$$ Find it as (big cone) $-$ (small cone removed). Curved surface area $= \pi l (R + r)$ where $l$ is the slant height of the frustum.

Capacity and mass

• $1\ \text{cm}^3 = 1\ \text{ml}$, $\quad 1000\ \text{cm}^3 = 1\ \text{litre}$, $\quad 1\ \text{m}^3 = 1000\ \text{litres}$. $$\text{Density} = \frac{\text{mass}}{\text{volume}} \qquad \text{Mass} = \text{density}\times\text{volume}$$

Pythagoras

$$c^2 = a^2 + b^2$$

Trigonometric ratios (right-angled triangle)

$$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}$$ $$\tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \sin^2\theta + \cos^2\theta = 1 \qquad \sin\theta = \cos(90^\circ - \theta)$$

Special angles

Any triangle (sides $a, b, c$ opposite angles $A, B, C$)

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \quad\text{(sine rule)}$$ $$a^2 = b^2 + c^2 - 2bc\cos A \quad\text{(cosine rule)} \qquad \cos A = \frac{b^2 + c^2 - a^2}{2bc}$$ $$\text{Area} = \tfrac{1}{2}ab\sin C$$

Double angle and graphs

$$\sin 2\theta = 2\sin\theta\cos\theta \qquad \cos 2\theta = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta = 2\cos^2\theta - 1$$ For $y = a\sin(bx + c)$ (or cosine): amplitude $= |a|$, period $= \dfrac{360^\circ}{b}$ (or $\dfrac{2\pi}{b}$). For $y = a\tan bx$: period $= \dfrac{180^\circ}{b}$.

Angles of elevation and depression

The angle of elevation equals the angle of depression (alternate angles); use $\tan\theta = \dfrac{\text{height}}{\text{horizontal distance}}$.

Identities

$$(a + b)^2 = a^2 + 2ab + b^2 \qquad (a - b)^2 = a^2 - 2ab + b^2 \qquad a^2 - b^2 = (a - b)(a + b)$$ $$a^3 + b^3 = (a + b)(a^2 - ab + b^2) \qquad a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$

Quadratic equation $ax^2 + bx + c = 0$

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

• Discriminant $\Delta = b^2 - 4ac$: $\Delta > 0$ two real roots, $\Delta = 0$ one repeated root, $\Delta < 0$ no real roots.
• Sum of roots $= -\dfrac{b}{a}$, product of roots $= \dfrac{c}{a}$.
• If the roots are $\alpha, \beta$: $x^2 - (\alpha + \beta)x + \alpha\beta = 0$.

Completing the square

$$x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 + c - \frac{b^2}{4}$$ Turning point of $y = ax^2 + bx + c$ is at $x = -\dfrac{b}{2a}$.

Factorising $ax^2 + bx + c$

Find two numbers with product $ac$ and sum $b$, split the middle term, then factorise by grouping.

Points $A(x_1, y_1)$ and $B(x_2, y_2)$: $$\text{Gradient } m = \frac{y_2 - y_1}{x_2 - x_1} = \tan\theta$$ $$\text{Distance } AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \qquad \text{Midpoint} = \left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)$$

Equation of a straight line

$$y = mx + c \qquad y - y_1 = m(x - x_1) \qquad \frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1} \qquad \frac{x}{a} + \frac{y}{b} = 1$$

• Horizontal line: $y = k$ (gradient $0$). Vertical line: $x = k$ (gradient undefined).
• For $ax + by + c = 0$: gradient $= -\dfrac{a}{b}$, $x$-intercept $= -\dfrac{c}{a}$, $y$-intercept $= -\dfrac{c}{b}$.

Parallel and perpendicular lines

• Parallel: $m_1 = m_2$.
• Perpendicular: $m_1 m_2 = -1$, i.e. $m_2 = -\dfrac{1}{m_1}$.

Simultaneous equations

Solve by elimination or substitution; graphically the solution is the intersection point of the two lines.

Section formula

Point dividing $AB$ in the ratio $m:n$ (from $A$): $$\left(\frac{n x_1 + m x_2}{m + n},\ \frac{n y_1 + m y_2}{m + n}\right)$$

• Adding or subtracting the same number on both sides keeps the inequality sign.
• Multiplying or dividing by a negative number reverses the sign.
• Compound inequality: $a < x \le b$ means both $x > a$ and $x \le b$.

Graphing $y \ge mx + c$ (or similar)

1. Draw the boundary line $y = mx + c$: solid for $\le$ or $\ge$, dashed for $<$ or $>$.
2. Test a point (e.g. the origin) to decide which side to shade (or leave unshaded, as the question asks).

Linear programming

1. Define the variables ($x$, $y$) and write the constraints as inequalities.
2. Write the objective function, e.g. $P = ax + by$ (profit to maximise or cost to minimise).
3. Graph the region that satisfies all constraints.
4. Evaluate the objective function at each vertex of the region, or slide a search line; the best vertex gives the optimum.
5. Use whole numbers when the variables are items (people, machines...).

$$\sqrt{a}\times\sqrt{b} = \sqrt{ab} \qquad \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \qquad \sqrt{a^2 b} = a\sqrt{b}$$ $$(\sqrt{a})^2 = a \qquad (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b$$

Rationalising the denominator

$$\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a} \qquad \frac{1}{a + \sqrt{b}} = \frac{a - \sqrt{b}}{a^2 - b} \qquad \frac{1}{\sqrt{a} - \sqrt{b}} = \frac{\sqrt{a} + \sqrt{b}}{a - b}$$ Multiply top and bottom by the conjugate of the denominator.

Arithmetic progression (A.P.) - first term $a$, common difference $d$

$$T_n = a + (n - 1)d \qquad S_n = \frac{n}{2}\left[2a + (n - 1)d\right] = \frac{n}{2}(a + l)$$ where $l$ is the last term.

Geometric progression (G.P.) - first term $a$, common ratio $r$

$$T_n = ar^{\,n-1} \qquad S_n = \frac{a(r^n - 1)}{r - 1} = \frac{a(1 - r^n)}{1 - r}\ (r \neq 1)$$ $$S_\infty = \frac{a}{1 - r}\quad\text{for } |r| < 1$$

Useful

• Common difference $d = T_{n+1} - T_n$; common ratio $r = \dfrac{T_{n+1}}{T_n}$.
• $T_n = S_n - S_{n-1}$.

Pascal's triangle (coefficients of $(a + b)^n$)

n=0:            1
n=1:          1   1
n=2:        1   2   1
n=3:      1   3   3   1
n=4:    1   4   6   4   1
n=5:  1   5  10  10   5   1

Each number is the sum of the two above it.

Expansion

$$(a + b)^n = a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \dots + b^n$$ $$\binom{n}{r} = \frac{n!}{r!\,(n - r)!}$$

• The number of terms is $n + 1$.
• General term: $T_{r+1} = \binom{n}{r}a^{n-r}b^r$.
• $(1 + x)^n \approx 1 + nx + \dfrac{n(n-1)}{2}x^2 + \dots$ (use for approximations with small $x$).

• Direct variation: $y \propto x \Rightarrow y = kx$ (also $\dfrac{y_1}{x_1} = \dfrac{y_2}{x_2}$).
• Inverse variation: $y \propto \dfrac{1}{x} \Rightarrow y = \dfrac{k}{x}$ (also $x_1y_1 = x_2y_2$).
• Joint variation: $y \propto xz \Rightarrow y = kxz$.
• Partial variation: $y = a + kx$ (part constant, part varying).

Steps

1. Write the relationship with the constant $k$.
2. Use given values to find $k$.
3. Write the full equation, then substitute to find the required value.

Changing the subject

Undo the operations in reverse order, doing the same to both sides. Collect the terms containing the new subject together and factorise if it appears more than once.

Simple interest

$$I = \frac{PRT}{100} \qquad A = P + I = P\left(1 + \frac{RT}{100}\right)$$ $P$ = principal, $R$ = rate per year (%), $T$ = time in years, $A$ = amount.

Compound interest

$$A = P\left(1 + \frac{R}{100}\right)^n \qquad \text{Compound interest} = A - P$$ $n$ = number of compounding periods. If the interest is compounded $m$ times a year at annual rate $R\%$ for $t$ years: $$A = P\left(1 + \frac{R}{100m}\right)^{mt}$$

Appreciation and depreciation

$$\text{Appreciation: } V = V_0\left(1 + \frac{r}{100}\right)^n \qquad \text{Depreciation: } V = V_0\left(1 - \frac{r}{100}\right)^n$$

Hire purchase

$$\text{Hire purchase price} = \text{deposit} + (\text{instalment} \times \text{number of instalments})$$ $$\text{Extra paid} = \text{hire purchase price} - \text{cash price}$$

Profit and loss

$$\text{Profit} = \text{selling price} - \text{cost price} \qquad \text{Profit \%} = \frac{\text{profit}}{\text{cost price}}\times 100\%$$ $$\text{Discount} = \text{marked price} - \text{selling price} \qquad \text{Discount \%} = \frac{\text{discount}}{\text{marked price}}\times 100\%$$

Tax

$$\text{VAT} = \text{rate} \times \text{price before tax} \qquad \text{Commission} = \text{rate} \times \text{sales}$$ $$\text{Taxable income} = \text{gross income} - \text{allowable deductions} \qquad \text{Net pay} = \text{gross pay} - \text{deductions}$$

Angle facts

• Angles on a straight line add up to $180^\circ$; angles at a point add up to $360^\circ$.
• Vertically opposite angles are equal.
• Parallel lines: corresponding angles are equal, alternate angles are equal, co-interior angles add up to $180^\circ$.
• Angles of a triangle add up to $180^\circ$; exterior angle $=$ sum of the two opposite interior angles.

Polygons ($n$ sides)

$$\text{Sum of interior angles} = (n - 2)\times 180^\circ \qquad \text{Sum of exterior angles} = 360^\circ$$ $$\text{Each interior angle (regular)} = \frac{(n - 2)\times 180^\circ}{n} \qquad \text{Each exterior angle (regular)} = \frac{360^\circ}{n}$$ $$\text{Number of diagonals} = \frac{n(n - 3)}{2}$$

Circle theorems

1. The angle at the centre is twice the angle at the circumference on the same arc.
2. The angle in a semicircle is $90^\circ$.
3. Angles in the same segment are equal.
4. Opposite angles of a cyclic quadrilateral add up to $180^\circ$; an exterior angle equals the interior opposite angle.
5. A tangent is perpendicular to the radius at the point of contact.
6. Tangents drawn from an external point are equal in length.
7. Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
8. Intersecting chords: $AP \times PB = CP \times PD$. Tangent and secant: $PT^2 = PA \times PB$.

For similar figures with linear scale factor $k$: $$\text{Length ratio} = k \qquad \text{Area ratio} = k^2 \qquad \text{Volume ratio} = k^3$$ So $k = \dfrac{\text{new length}}{\text{old length}}$, and $k = \sqrt{\dfrac{A_2}{A_1}} = \sqrt[3]{\dfrac{V_2}{V_1}}$.

Enlargement (centre $O$, scale factor $k$)

$$\text{Image distance from centre} = k \times \text{object distance from centre}$$

• $k > 1$: image larger. $0 < k < 1$: image smaller. $k < 0$: image on the opposite side of the centre, inverted.
• To find the centre, join corresponding points; the lines meet at the centre.
• Area scale factor of an enlargement $= k^2$.

Congruence tests

SSS, SAS, ASA (or AAS), RHS.

Ungrouped data

$$\text{Mean } \bar{x} = \frac{\sum x}{n} \qquad \text{Range} = \text{largest} - \text{smallest}$$ Median: middle value when ordered (average of the two middle values if $n$ is even); position $= \dfrac{n + 1}{2}$.

Frequency table

$$\bar{x} = \frac{\sum fx}{\sum f}$$ Using an assumed mean $A$: $\ \bar{x} = A + \dfrac{\sum fd}{\sum f}$ where $d = x - A$.

Grouped data (use class mid-points $x$)

$$\bar{x} = \frac{\sum fx}{\sum f}$$ $$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f}\right)\times c$$ $L$ = lower boundary of the median class, $N = \sum f$, $F$ = cumulative frequency before the median class, $f$ = frequency of the median class, $c$ = class width. $$\text{Mode} = L + \left(\frac{\Delta_1}{\Delta_1 + \Delta_2}\right) c$$ where $\Delta_1 = f_m - f_{\text{before}}$ and $\Delta_2 = f_m - f_{\text{after}}$ (frequency of the modal class minus the frequency of the class before and after it), $L$ = lower boundary of the modal class, $c$ = class width.

Quartiles

$$Q_1 \text{ at } \frac{N}{4}, \quad Q_2 \text{ at } \frac{N}{2}, \quad Q_3 \text{ at } \frac{3N}{4} \text{ (from the cumulative frequency curve)}$$ $$\text{Interquartile range} = Q_3 - Q_1 \qquad \text{Quartile deviation} = \frac{Q_3 - Q_1}{2}$$

Variance and standard deviation

$$\text{Variance } \sigma^2 = \frac{\sum (x - \bar{x})^2}{n} = \frac{\sum x^2}{n} - \bar{x}^2 \qquad \sigma = \sqrt{\text{variance}}$$ Grouped: $\ \sigma^2 = \dfrac{\sum f x^2}{\sum f} - \left(\dfrac{\sum fx}{\sum f}\right)^2$.

Data representation

• Pie chart: angle $= \dfrac{\text{frequency}}{\text{total}}\times 360^\circ$.
• Histogram: area of a bar $\propto$ frequency; frequency density $= \dfrac{\text{frequency}}{\text{class width}}$.

$$P(A) = \frac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}} \qquad 0 \le P(A) \le 1$$ $$P(\text{not } A) = 1 - P(A) \qquad \text{sum of all probabilities} = 1$$

Combined events

• Independent events (AND): $P(A \text{ and } B) = P(A)\times P(B)$.
• Mutually exclusive events (OR): $P(A \text{ or } B) = P(A) + P(B)$.
• General OR: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
• Conditional (without replacement): $P(A \text{ then } B) = P(A)\times P(B \mid A)$.

Tree diagrams

Multiply along the branches, add the results of the different paths that give the required outcome.

Expected number

$$\text{Expected number of occurrences} = P(\text{event})\times\text{number of trials}$$

Column vector and magnitude

$$\overrightarrow{AB} = \begin{pmatrix} x_2 - x_1 \\ y_2 - y_1 \end{pmatrix} = \mathbf{b} - \mathbf{a} \qquad \left|\overrightarrow{AB}\right| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ For $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$: $\ |\mathbf{v}| = \sqrt{x^2 + y^2}$.

Operations

$$\begin{pmatrix} a \\ b \end{pmatrix} \pm \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a \pm c \\ b \pm d \end{pmatrix} \qquad k\begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} ka \\ kb \end{pmatrix}$$

Position vectors

• Midpoint of $AB$: $\ \dfrac{\mathbf{a} + \mathbf{b}}{2}$.
• Point $P$ dividing $AB$ in the ratio $m:n$: $\ \mathbf{p} = \dfrac{n\mathbf{a} + m\mathbf{b}}{m + n}$.

Parallel and collinear

• $\mathbf{a}$ and $\mathbf{b}$ are parallel if $\mathbf{a} = k\mathbf{b}$.
• Points $A$, $B$, $C$ are collinear if $\overrightarrow{AB} = k\,\overrightarrow{BC}$ (or $\overrightarrow{AB}$ is parallel to $\overrightarrow{AC}$).

Matrices

$$\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{pmatrix}$$

• Matrix multiplication is generally not commutative: $AB \ne BA$.
• Identity matrix $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$, and $AI = IA = A$.

Determinant and inverse of $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$

$$\det A = ad - bc \qquad A^{-1} = \frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$ If $\det A = 0$ the matrix is singular and has no inverse.

Solving simultaneous equations

$\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} p \\ q \end{pmatrix}$ gives $\begin{pmatrix} x \\ y \end{pmatrix} = A^{-1}\begin{pmatrix} p \\ q \end{pmatrix}$.

Transformation matrices (about the origin)

Areas and combined transformations

$$\text{Area of image} = |\det M| \times \text{area of object}$$ For the transformation $A$ followed by $B$, the single matrix is $BA$ (apply the first transformation on the right). To find the image of $(x, y)$: $\ \begin{pmatrix} x' \\ y' \end{pmatrix} = M\begin{pmatrix} x \\ y \end{pmatrix}$.

Translation

Translation by vector $\begin{pmatrix} a \\ b \end{pmatrix}$: $(x, y) \to (x + a, y + b)$.

Differentiation

$$\frac{d}{dx}(ax^n) = anx^{n-1} \qquad \frac{d}{dx}(c) = 0 \qquad \frac{d}{dx}(u \pm v) = \frac{du}{dx} \pm \frac{dv}{dx}$$

• Gradient of a curve at a point $=\dfrac{dy}{dx}$ at that point. Equation of the tangent: $y - y_1 = m(x - x_1)$ with $m = \dfrac{dy}{dx}$.
• Turning points: $\dfrac{dy}{dx} = 0$. Maximum if $\dfrac{d^2y}{dx^2} < 0$; minimum if $\dfrac{d^2y}{dx^2} > 0$.
• Rate of change of $y$ with respect to $x$ is $\dfrac{dy}{dx}$. If $s$ is displacement: $v = \dfrac{ds}{dt}$, $a = \dfrac{dv}{dt}$.

Integration

$$\int ax^n\,dx = \frac{ax^{n+1}}{n+1} + c \quad (n \ne -1) \qquad \int a\,dx = ax + c$$ $$\int_a^b f(x)\,dx = F(b) - F(a)$$

• Area under a curve between $x = a$ and $x = b$: $\ \displaystyle\int_a^b y\,dx$. Areas below the $x$-axis come out negative - use the positive value.
• Displacement $= \displaystyle\int v\,dt$, velocity $= \displaystyle\int a\,dt$.

Approximating areas

$$\text{Trapezium rule: } A \approx \frac{h}{2}\left[y_0 + y_n + 2(y_1 + y_2 + \dots + y_{n-1})\right]$$ $$\text{Mid-ordinate rule: } A \approx h\,(y_1 + y_2 + \dots + y_n) \text{ using mid-ordinates}$$ where $h$ is the width of each strip.

• $1$ nautical mile (nm) $=$ the distance along a great circle subtending $1'$ (one minute of arc) at the centre of the Earth. $\ 1\ \text{nm} = 1.853\ \text{km}$ (about $1.85$ km).
• Radius of the Earth $R \approx 6370\ \text{km}$ (use the value given in the question).

Distance along a meridian or the Equator

$$\text{Distance (nm)} = 60\times\theta^{\circ}\ \ (\theta = \text{angle between the two points in degrees})$$ $$\text{Distance (km)} = \frac{\theta}{360}\times 2\pi R$$

Distance along a parallel of latitude $\alpha$

$$\text{Distance (nm)} = 60\times\Delta\lambda\times\cos\alpha \qquad \text{Distance (km)} = \frac{\Delta\lambda}{360}\times 2\pi R\cos\alpha$$ where $\Delta\lambda$ is the difference in longitude. The radius of the circle of latitude $\alpha$ is $R\cos\alpha$.

Time difference

$$\text{Time difference} = \Delta\lambda \times 4\ \text{minutes} \qquad (15^\circ \text{ of longitude} = 1\text{ hour})$$ Places to the east are ahead (add the difference); places to the west are behind (subtract).

Speed

$$\text{Speed (knots)} = \frac{\text{distance in nautical miles}}{\text{time in hours}}$$ ($1$ knot $=$ $1$ nm per hour.)

$$\text{Speed} = \frac{\text{distance}}{\text{time}} \qquad \text{Distance} = \text{speed}\times\text{time} \qquad \text{Time} = \frac{\text{distance}}{\text{speed}}$$ $$\text{Average speed} = \frac{\text{total distance}}{\text{total time}}$$

• Convert km/h to m/s: multiply by $\dfrac{5}{18}$. Convert m/s to km/h: multiply by $\dfrac{18}{5}$.

Relative speed

• Objects moving in opposite directions (towards each other or apart): add the speeds.
• Objects moving in the same direction: subtract the speeds. $$\text{Time to meet} = \frac{\text{distance apart}}{\text{relative speed}}$$

Acceleration

$$a = \frac{v - u}{t} \qquad \text{Deceleration (retardation)} = -a$$

Velocity-time graph

• Gradient $=$ acceleration.
• Area under the graph $=$ distance travelled.
• Average velocity $= \dfrac{\text{total displacement}}{\text{total time}}$.

Standard loci

• Points at a fixed distance from a point: a circle.
• Points equidistant from two points: the perpendicular bisector of the line joining them.
• Points equidistant from two intersecting lines: the angle bisector.
• Points at a fixed distance from a line: two parallel lines (with semicircular ends).

Bearings

• A bearing is measured clockwise from North, with three figures (e.g. $045^\circ$, $270^\circ$).
• Back bearing $=$ bearing $\pm 180^\circ$.

Scale

$$\text{Linear scale} = \frac{\text{map length}}{\text{actual length}} \qquad \text{Area scale} = (\text{linear scale})^2$$ For a scale $1 : n$, $1\ \text{cm}$ on the map represents $n\ \text{cm}$ on the ground.