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Topic Key:
Number
Algebra
Coord. Geom.
Geometry
Mensuration
Trigonometry
Matrices/Vectors
Statistics
Probability
A
Acute Angle
Geometry
An angle between 0° and 90°. An acute-angled triangle has all three interior angles less than 90°.
Alternate Angles
Geometry
When a transversal crosses two parallel lines, alternate angles are on opposite sides of the transversal between the parallel lines. They are equal. Also called Z-angles.
Ambiguous Case
Trigonometry
When using the sine rule to find an angle (SSA), there may be two valid solutions — one acute and one obtuse — because sin θ = sin(180°−θ). Always check both possibilities.
Arc
Mensuration
A portion of the circumference of a circle. Arc length = (θ/360) × 2πr where θ is the sector angle in degrees and r is the radius.
Area
Mensuration
The amount of two-dimensional space enclosed within a shape. Measured in square units (cm², m²). For similar shapes with linear scale factor k, area scales by k².
Arithmetic Sequence
Number
A sequence where each term is obtained by adding a fixed constant (the common difference d) to the previous term. The nth term = a + (n−1)d = dn + (a−d).
Asymptote
Algebra
A line that a curve approaches but never reaches. For y = 1/x, the x-axis and y-axis are both asymptotes. For y = tan x, vertical asymptotes occur at 90°, 270°, etc.
B
Bearing
Trigonometry
A direction measured clockwise from North, always written as a three-digit number (e.g. 045°, 270°). Back bearing: if A to B = θ°, then B to A = θ±180°.
Box-and-Whisker Plot
Statistics
A diagram showing the five-number summary of a dataset: minimum, lower quartile (Q1), median (Q2), upper quartile (Q3), and maximum. The box spans Q1 to Q3 (the IQR); whiskers extend to the extremes.
C
Class Interval
Statistics
A group used to organise continuous data in a frequency table. Defined by a lower and upper boundary. The midpoint of the class is used to estimate the mean of grouped data.
Co-interior Angles
Geometry
When a transversal crosses two parallel lines, co-interior angles are on the same side of the transversal between the parallel lines. They add up to 180°. Also called C-angles or allied angles.
Coefficient
Algebra
The numerical factor multiplying a variable in an algebraic expression. In 5x², the coefficient of x² is 5. In −3xy, the coefficient is −3.
Collinear Points
Matrices/Vectors
Three or more points lying on the same straight line. To prove A, B, C are collinear using vectors: show AB⃗ = k × AC⃗ for some scalar k (since they share point A).
Column Vector
Matrices/Vectors
A vector written vertically as (x over y), where x represents horizontal displacement and y represents vertical displacement. The magnitude is √(x²+y²).
Completing the Square
Algebra
Rewriting a quadratic expression ax²+bx+c in the form a(x+p)²+q. Used to solve quadratic equations, find the vertex/turning point of a parabola, and identify minimum or maximum values.
Composite Function
Algebra
A function formed by applying one function to the result of another. fg(x) = f(g(x)) — g is applied first, then f. Note: fg(x) ≠ gf(x) in general. The order of operations matters.
Compound Interest
Number
Interest calculated on both the original principal and the interest previously accumulated. Formula: A = P(1 + r/100)ⁿ where P = principal, r = rate %, n = number of periods.
Congruent
Geometry
Two shapes are congruent if they are identical in shape and size — all corresponding sides and angles are equal. Conditions for triangle congruence: SSS, SAS, ASA, RHS.
Continuous Data
Statistics
Numerical data that can take any value within a range — measured rather than counted. Examples: height, mass, time, temperature. Displayed using histograms (with frequency density on y-axis).
Correlation
Statistics
The relationship between two variables shown on a scatter diagram. Positive correlation: both increase together. Negative correlation: one increases as the other decreases. No correlation: no pattern. Strength described as strong or weak.
Cosine Rule
Trigonometry
Used in any triangle when two sides and the included angle (SAS) or all three sides (SSS) are known. Finding side: a² = b²+c²−2bc cosA. Finding angle: cosA = (b²+c²−a²)/2bc. No ambiguous case.
Cumulative Frequency
Statistics
A running total of frequencies up to and including each class. Plotted against the upper class boundary to produce a cumulative frequency curve (ogive). Used to find median, quartiles, and percentiles.
Cyclic Quadrilateral
Geometry
A quadrilateral whose four vertices all lie on a circle. Key theorem: opposite angles of a cyclic quadrilateral add up to 180°.
D
Denominator
Number
The bottom number of a fraction — the number of equal parts the whole is divided into. In ¾, the denominator is 4. To add fractions, find a common denominator first.
Determinant
Matrices/Vectors
A scalar value calculated from a square matrix. For 2×2 matrix [[a,b],[c,d]], the determinant = ad − bc. If the determinant is zero, the matrix is singular and has no inverse.
Direct Proportion
Algebra
Two quantities are in direct proportion if they increase or decrease at the same rate. Written y ∝ x, giving y = kx where k is the constant of proportionality. Graph is a straight line through the origin.
Discriminant
Algebra
The expression b² − 4ac in the quadratic formula. If Δ > 0: two distinct real roots. If Δ = 0: one repeated root. If Δ < 0: no real roots (the quadratic does not intersect the x-axis).
Discrete Data
Statistics
Numerical data that can only take specific separate values — usually whole numbers. Examples: number of children, shoe sizes, goals scored. Displayed using bar charts or frequency polygons.
Domain
Algebra
The set of allowable input values for a function. For f(x) = √x, the domain is x ≥ 0. For f(x) = 1/(x−2), the domain is x ≠ 2 (denominator cannot be zero).
E
Enlargement
Matrices/Vectors
A transformation that changes the size of a shape by a scale factor k from a fixed centre of enlargement. k > 1: enlargement. 0 < k < 1: reduction. k negative: image on opposite side of centre. Areas scale by k².
Equation of a Line
Coord. Geom.
The algebraic relationship between x and y for all points on a straight line. Standard form: y = mx + c (gradient m, y-intercept c). Point-slope form: y − y₁ = m(x − x₁).
Exterior Angle
Geometry
An angle formed outside a polygon by extending one side. For a triangle, the exterior angle equals the sum of the two non-adjacent interior angles. The sum of all exterior angles of any convex polygon = 360°.
F
Factor
Number
A whole number that divides exactly into another number with no remainder. The factors of 12 are 1, 2, 3, 4, 6, and 12. The Highest Common Factor (HCF) is the largest factor shared by two or more numbers.
Factorisation
Algebra
Expressing an algebraic expression as a product of its factors — the reverse of expanding. Methods: common factor, grouping, quadratic trinomial (ac method), difference of squares (a²−b²=(a+b)(a−b)).
Frequency Density
Statistics
The value plotted on the y-axis of a histogram: Frequency Density = Frequency ÷ Class Width. The area of each bar represents the frequency. Used when class widths are unequal to avoid misleading representations.
Function
Algebra
A rule that maps each input value (from the domain) to exactly one output value. Written as f(x). The range is the set of all possible output values. Every x in the domain maps to exactly one f(x).
G
Geometric Sequence
Number
A sequence where each term is multiplied by a fixed constant (the common ratio r). nth term = ar^(n−1). Sum of n terms = a(rⁿ−1)/(r−1) for r ≠ 1.
Gradient
Coord. Geom.
The steepness of a line — the ratio of vertical change to horizontal change (rise over run). Formula: m = (y₂−y₁)/(x₂−x₁). Positive: line rises left to right. Negative: line falls. Zero: horizontal line. Undefined: vertical line.
H
HCF (Highest Common Factor)
Number
The largest factor that divides exactly into two or more numbers. Found using prime factorisation — take common prime factors at their lowest power. HCF of 36 and 84: 36=2²×3², 84=2²×3×7 → HCF=2²×3=12.
Histogram
Statistics
A bar chart for continuous grouped data where the area of each bar (not height) represents frequency. The y-axis shows frequency density (= frequency ÷ class width). Bars have no gaps. Modal class = bar with greatest frequency density.
Hypotenuse
Trigonometry
The longest side of a right-angled triangle — always the side opposite the right angle. Used in Pythagoras' theorem: c² = a² + b². Also the denominator in SOH (sin = Opposite/Hypotenuse) and CAH (cos = Adjacent/Hypotenuse).
I
Identity Matrix
Matrices/Vectors
The 2×2 identity matrix is I = [[1,0],[0,1]]. Multiplying any 2×2 matrix M by I gives M. It acts like the number 1 in multiplication: MI = IM = M. Also: M × M⁻¹ = I.
Index / Indices
Algebra
The power to which a base is raised. In 2⁵, the index is 5. Key laws: aᵐ×aⁿ=aᵐ⁺ⁿ, aᵐ÷aⁿ=aᵐ⁻ⁿ, (aᵐ)ⁿ=aᵐⁿ, a⁰=1, a⁻ⁿ=1/aⁿ, a^(1/n)=ⁿ√a, a^(m/n)=(ⁿ√a)ᵐ.
Inequality
Algebra
A mathematical statement that one expression is greater than or less than another. Symbols: > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to). When multiplying or dividing by a negative number, the inequality sign reverses.
Integer
Number
Any whole number — positive, negative, or zero. The set of integers is {..., −3, −2, −1, 0, 1, 2, 3, ...}. Integers do not include fractions or decimals. Note: 0 is an integer but is neither positive nor negative.
Interquartile Range (IQR)
Statistics
A measure of spread: IQR = Q3 − Q1. It represents the range of the middle 50% of the data. Less affected by outliers than the full range. A smaller IQR indicates more consistent data.
Inverse Function
Algebra
The function that reverses the effect of f. Written f⁻¹(x). To find: write y=f(x), swap x and y, solve for y. Key property: f(f⁻¹(x)) = x. The graph of f⁻¹ is the reflection of f in the line y=x.
Inverse Matrix
Matrices/Vectors
For M = [[a,b],[c,d]], the inverse is M⁻¹ = (1/det) × [[d,−b],[−c,a]] where det = ad−bc. Exists only when det ≠ 0. To find: swap main diagonal, negate off-diagonal, divide by determinant. Verify: M×M⁻¹ = I.
Inverse Proportion
Algebra
Two quantities are inversely proportional if one increases at the same rate as the other decreases. Written y ∝ 1/x, giving y = k/x or xy = k. Graph is a hyperbola (rectangular hyperbola).
Irrational Number
Number
A number that cannot be expressed as a fraction p/q where p and q are integers. Its decimal expansion is non-terminating and non-repeating. Examples: √2, √3, π, √5. Surds are irrational numbers in root form.
Isosceles Triangle
Geometry
A triangle with exactly two equal sides. The two angles opposite the equal sides (base angles) are also equal. Appears frequently in circle theorem problems when two radii form a triangle with a chord.
L
Law of Indices
Algebra
Seven rules governing operations with powers: (1) aᵐ×aⁿ=aᵐ⁺ⁿ (2) aᵐ÷aⁿ=aᵐ⁻ⁿ (3) (aᵐ)ⁿ=aᵐⁿ (4) a⁰=1 (5) a⁻ⁿ=1/aⁿ (6) a^(1/n)=ⁿ√a (7) a^(m/n)=(ⁿ√a)ᵐ. For fractional indices, always root first then power.
LCM (Lowest Common Multiple)
Number
The smallest positive integer that is a multiple of two or more given numbers. Found using prime factorisation — take all prime factors at their highest power. LCM of 36 and 84: 2²×3²×7=252.
Limits of Accuracy
Number
When a measurement is rounded, the true value lies within bounds: Lower Bound = x − ½u, Upper Bound = x + ½u, where u is the unit of rounding. The upper bound uses strict inequality (<, not ≤).
Line of Best Fit
Statistics
A straight line drawn through a scatter diagram to represent the trend, with approximately equal numbers of points on each side. Must pass through the mean point (x̄, ȳ). Used for interpolation (within data range) and extrapolation (outside range).
Locus (plural: Loci)
Geometry
The set of all points satisfying a given condition. Key loci: fixed distance from a point → circle; equidistant from two points → perpendicular bisector; equidistant from two lines → angle bisector; fixed distance from a line → pair of parallel lines.
M
Matrix
Matrices/Vectors
A rectangular array of numbers arranged in rows and columns. The order is m×n (m rows, n columns). Matrix multiplication AB requires columns of A = rows of B. AB ≠ BA in general — multiplication is non-commutative.
Mean
Statistics
The arithmetic average: sum of all values ÷ number of values. From a frequency table: x̄ = Σ(fx)/Σf. For grouped data, use class midpoints — the result is an estimate. Affected by extreme values (outliers).
Median
Statistics
The middle value when data is arranged in order. For n values, the median is at position (n+1)/2. From a cumulative frequency graph, read across from cf = n/2. Not affected by extreme values — preferred when data is skewed.
Midpoint
Coord. Geom.
The point exactly halfway between two points. Formula: M = ((x₁+x₂)/2, (y₁+y₂)/2). In vector form, the midpoint of AB has position vector OM⃗ = ½(a+b) where a and b are position vectors of A and B.
Mode
Statistics
The value that appears most frequently in a dataset. A dataset can have no mode, one mode, or multiple modes. For grouped data, the modal class is the class with the highest frequency (or frequency density in histograms).
Multiple
Number
The result of multiplying a number by any positive integer. Multiples of 6: 6, 12, 18, 24, 30, ... The Lowest Common Multiple (LCM) is the smallest multiple shared by two or more numbers.
Mutually Exclusive Events
Probability
Events that cannot both occur at the same time — P(A and B) = 0. For mutually exclusive events: P(A or B) = P(A) + P(B). Example: rolling a 3 and rolling a 5 on one throw of a die.
N
nth Term
Number
A formula giving the value of any term in a sequence in terms of its position n. For arithmetic: dn + (a−d). To test if a value is in the sequence, set the nth term equal to that value and check if n is a positive integer.
O
Obtuse Angle
Geometry
An angle between 90° and 180°. The cosine of an obtuse angle is negative. When using the cosine rule, a negative cosine value indicates the angle opposite the longest side is obtuse.
Ogive
Statistics
Another name for the cumulative frequency curve — an S-shaped curve formed by plotting cumulative frequency against upper class boundaries and joining with a smooth curve. Used to read off median, quartiles, and percentiles.
P
Parabola
Algebra
The U-shaped (or ∩-shaped) curve formed by a quadratic function y = ax²+bx+c. If a > 0: U-shape with minimum turning point. If a < 0: ∩-shape with maximum turning point. The turning point is found by completing the square.
Parallel Lines
Coord. Geom.
Lines that never meet — they have equal gradients. If two lines have equations y=m₁x+c₁ and y=m₂x+c₂, they are parallel if and only if m₁ = m₂. Parallel lines have no point of intersection.
Percentile
Statistics
The value below which a given percentage of data falls. The 60th percentile = value at cumulative frequency 0.6n on the ogive. Q1 = 25th percentile; median = 50th percentile; Q3 = 75th percentile.
Perpendicular Bisector
Coord. Geom.
The line that is perpendicular to a line segment AB and passes through its midpoint. Every point on the perpendicular bisector is equidistant from A and B. Gradient = −1/m_AB. Used to find the centre of a circle through three points.
Perpendicular Lines
Coord. Geom.
Lines that meet at a right angle (90°). Their gradients multiply to −1: m₁ × m₂ = −1. So if one line has gradient m, the perpendicular line has gradient −1/m (the negative reciprocal).
Position Vector
Matrices/Vectors
The vector from the origin O to a point P, written OP⃗ or p. If A has position vector a and B has position vector b, then AB⃗ = b − a. The midpoint M of AB has position vector OM⃗ = ½(a+b).
Prime Number
Number
A whole number greater than 1 with exactly two factors: 1 and itself. First primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Note: 1 is NOT prime (only one factor). 2 is the only even prime number.
Probability
Probability
A measure of how likely an event is to occur, ranging from 0 (impossible) to 1 (certain). P(event) = favourable outcomes / total equally likely outcomes. P(A') = 1−P(A) is the complementary event.
Pythagoras' Theorem
Geometry
In a right-angled triangle: c² = a² + b² where c is the hypotenuse and a, b are the other two sides. Used to find any side given the other two, and to prove whether a triangle is right-angled. Extended to 3D using two applications.
Q
Quadratic Equation
Algebra
An equation of the form ax²+bx+c=0 (a≠0). Three solution methods: (1) Factorisation — quick when roots are rational. (2) Quadratic formula — always works. (3) Completing the square — gives vertex form. Has at most two solutions.
Quadratic Formula
Algebra
Formula for solving ax²+bx+c=0: x = [−b ± √(b²−4ac)] / 2a. The discriminant b²−4ac determines how many real roots exist. Use when asked for decimal answers or when factorisation is not obvious.
Quartile
Statistics
Values that divide an ordered dataset into four equal parts. Q1 (lower quartile) = 25th percentile = cf n/4. Q2 (median) = 50th percentile = cf n/2. Q3 (upper quartile) = 75th percentile = cf 3n/4. IQR = Q3−Q1.
R
Range
Statistics
The difference between the largest and smallest values in a dataset: Range = Maximum − Minimum. A simple measure of spread but sensitive to extreme values (outliers). The IQR is a more robust measure of spread.
Ratio
Number
A comparison of two or more quantities of the same kind, written as a:b. Always simplify by dividing by the HCF. To share an amount in ratio a:b: find total parts (a+b), value of one part = total÷(a+b), multiply each share.
Reflection
Matrices/Vectors
A transformation mapping every point to its mirror image in a mirror line. Size and shape are preserved (congruent image). Key reflections: x-axis: (x,y)→(x,−y); y-axis: (x,y)→(−x,y); y=x: (x,y)→(y,x); y=−x: (x,y)→(−y,−x).
Reflex Angle
Geometry
An angle greater than 180° and less than 360°. In circle theorem problems, the angle at the centre (reflex) is twice the angle at the circumference subtended by the same arc, even when the reflex angle is involved.
Rotation
Matrices/Vectors
A transformation turning every point through a given angle about a fixed centre of rotation. Size and shape preserved. Key matrices about origin: 90° CCW: [[0,−1],[1,0]]; 90° CW: [[0,1],[−1,0]]; 180°: [[−1,0],[0,−1]].
S
Scale Factor
Geometry
The ratio by which lengths are multiplied in an enlargement or between similar shapes. If scale factor = k, then areas scale by k² and volumes scale by k³. If k > 1: enlargement. If 0 < k < 1: reduction.
Scatter Diagram
Statistics
A graph plotting two variables against each other as individual points to investigate correlation. A line of best fit is drawn through the mean point (x̄, ȳ) to represent the trend. Correlation does not imply causation.
Sector
Mensuration
A "pizza slice" region of a circle bounded by two radii and an arc. Area = (θ/360)×πr². Perimeter = arc length + 2r = (θ/360)×2πr + 2r. The sector angle θ is measured in degrees at the centre.
Significant Figures
Number
A method of expressing the precision of a number. Count from the first non-zero digit. Leading zeros are NOT significant (0.0045 has 2 s.f.). Zeros between non-zero digits ARE significant (3.04 has 3 s.f.). Trailing zeros after decimal ARE significant (2.50 has 3 s.f.).
Similar
Geometry
Two shapes are similar if they have the same shape (all corresponding angles equal) but different sizes. Corresponding sides are in the same ratio (scale factor k). Area ratio = k², Volume ratio = k³. Triangles are similar if AA (two equal angles).
Simultaneous Equations
Algebra
Two or more equations that must be satisfied by the same values of the unknowns simultaneously. Methods: elimination (make coefficients equal, add/subtract), substitution (express one variable in terms of the other), or matrix method (X = M⁻¹B).
Sine Rule
Trigonometry
Used in any triangle when two angles and one side (AAS/ASA) or two sides and a non-included angle (SSA) are known. Formula: a/sinA = b/sinB = c/sinC. Beware the ambiguous case when finding an angle.
Standard Form
Number
A way of writing very large or small numbers as A × 10ⁿ where 1 ≤ A < 10 and n is an integer. Large numbers: positive n. Small numbers: negative n. After multiplying/dividing, always check A is still between 1 and 10.
Surd
Number
An irrational number left in root form because it cannot be simplified to a rational number. Examples: √2, √3, 3√5. Key rules: √a×√b=√(ab), √a÷√b=√(a/b). To rationalise 1/√a, multiply top and bottom by √a.
T
Tangent (to a Circle)
Geometry
A line that touches a circle at exactly one point (the point of tangency). Key theorems: (1) Tangent ⊥ radius at point of contact. (2) Two tangents from an external point are equal in length. (3) Alternate segment theorem.
Transformation
Matrices/Vectors
A mapping of every point of a shape (object) to a new position (image). The four transformations are: reflection (mirror line), rotation (centre + angle + direction), translation (vector), and enlargement (centre + scale factor). Must be described fully.
Translation
Matrices/Vectors
A transformation moving every point the same distance in the same direction, described by a column vector (a over b). Effect: (x,y)→(x+a, y+b). Shape, size, and orientation are all preserved. No invariant points (unless vector is zero).
Tree Diagram
Probability
A branching diagram showing all possible outcomes of two or more successive events. Each branch shows an outcome and its probability. Rules: multiply along branches (AND), add across branches (OR). All branches from one point must sum to 1.
V
Variable
Algebra
A letter representing an unknown quantity or a quantity that can change. In 3x²+2y−5, x and y are variables. A constant has a fixed value. In f(x), x is the independent variable and f(x) is the dependent variable (output).
Vector
Matrices/Vectors
A quantity with both magnitude (size) and direction. Represented as a column vector, bold letter, or arrow. Operations: addition (add components), scalar multiplication (multiply each component), magnitude: |v| = √(x²+y²).
Volume
Mensuration
The amount of three-dimensional space enclosed by a solid. Measured in cm³, m³, or litres (1 litre = 1000 cm³). Key formulae: cylinder πr²h, cone ⅓πr²h, sphere (4/3)πr³, prism = cross-section area × length. For similar solids: volume ratio = k³.
Y
y-intercept
Coord. Geom.
The point where a line or curve crosses the y-axis (where x = 0). In the equation y = mx + c, the y-intercept is c — found by substituting x = 0. In real-world graphs, it often represents a fixed cost or starting value.