| Term | Definition | |------|------------| | Laws of indices | These are rules that govern how exponents are manipulated, such as $a^m \times a^n = a^{m+n}$ and $(a^m)^n = a^{mn}$. They apply to rational exponents, including fractions and negative numbers. | | Surds | Surds are expressions involving roots, typically square roots, that cannot be simplified to a rational number. Examples include $\sqrt{2}$ and $\sqrt{3}$. Manipulating surds involves simplifying expressions and rationalizing denominators. | | Discriminant of a quadratic function | The discriminant is a part of the quadratic formula that indicates the nature of the roots of a quadratic equation. For a quadratic equation $ax^2 + bx + c = 0$, the discriminant is $\Delta = b^2 - 4ac$. If $\Delta > 0$, there are two distinct real roots; if $\Delta = 0$, there is one repeated real root; if $\Delta < 0$, there are no real roots. | | Factor Theorem | The Factor Theorem states that a polynomial $f(x)$ has a factor $(x-a)$ if and only if $f(a)=0$. This is a direct consequence of the Remainder Theorem. | | Remainder Theorem | The Remainder Theorem states that when a polynomial $f(x)$ is divided by $(x-a)$, the remainder is $f(a)$. This theorem helps in finding remainders without performing polynomial long division. | | Recurrence relation | A recurrence relation defines a sequence where each term is defined as a function of preceding terms. For example, $x_{n+1} = f(x_n)$ or $x_{n+1} = ax_n + b$. | | Geometric series | A geometric series is a series where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The sum of a finite geometric series is given by $S_n = \frac{a(1-r^n)}{1-r}$ where $a$ is the first term and $r$ is the common ratio. | | Binomial expansion | The binomial expansion is a formula for expanding powers of a binomial. For a positive integer $n$, $(1+x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k$. The notation $\binom{n}{k}$ represents the binomial coefficient. | | Coordinate geometry of the circle | This involves using algebraic equations to describe properties of circles. The standard forms of a circle's equation are $(x-a)^2 + (y-b)^2 = r^2$ for a circle with centre $(a,b)$ and radius $r$, and $x^2 + y^2 + cx + dy + e = 0$ for a general circle. | | Radian measure | Radians are a unit of angle measurement, where one radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. $2\pi$ radians is equivalent to 360 degrees. | | Trigonometric functions | These are functions that relate angles of a right-angled triangle to the ratios of its sides. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). Their graphs exhibit periodicity and symmetry. | | Logarithms | Logarithms are the inverse operations to exponentiation. The equation $a^b = c$ is equivalent to $b = \log_a c$. Laws of logarithms allow for the manipulation of logarithmic expressions, such as $\log_a x + \log_a y = \log_a (xy)$. | | Differentiation | Differentiation is the process of finding the rate of change of a function. The derivative of a function $f(x)$ at a point represents the gradient of the tangent to the graph of $y=f(x)$ at that point. Notation includes $f'(x)$ and $\frac{dy}{dx}$. | | Stationary points | Stationary points on a graph are points where the derivative of the function is zero, indicating a horizontal tangent. These can be local maxima, local minima, or points of inflection. | | Integration | Integration is the process of finding the area under a curve or the antiderivative of a function. A definite integral $\int_a^b f(x) dx$ represents the area between the curve $y=f(x)$ and the x-axis from $x=a$ to $x=b$. The Fundamental Theorem of Calculus links differentiation and integration. | | Trapezium rule | The trapezium rule is a numerical method for approximating the definite integral (area under a curve). It divides the area into trapezoids and sums their areas. | | Transformations of graphs | These are operations that alter the position, size, or shape of a graph. Common transformations include translations (shifting), scaling (stretching or compressing), and reflections. These can be represented as $y=af(x)$, $y=f(x)+a$, $y=f(x+a)$, and $y=f(ax)$. | | Algebraic division | This refers to the process of dividing polynomials using algebraic methods, similar to long division for numbers. It can be used to factorize polynomials or to simplify rational expressions. | | Direct proportion | Two quantities are directly proportional if they increase or decrease at the same rate. This can be expressed as $y = kx$, where $k$ is the constant of proportionality. Their graphs are straight lines passing through the origin. | | Inverse proportion | Two quantities are inversely proportional if as one increases, the other decreases proportionally. This can be expressed as $y = \frac{k}{x}$ or $xy = k$, where $k$ is the constant of proportionality. Their graphs are hyperbolas. | | Congruence criteria | These are conditions used to determine if two triangles are congruent (identical in shape and size). The standard criteria are SSS (side-side-side), SAS (side-angle-side), ASA (angle-side-angle), and RHS (right angle-hypotenuse-side). | | Similarity | Two shapes are similar if they have the same shape but potentially different sizes. Corresponding angles are equal, and corresponding sides are in proportion. Scale factors relate the lengths of corresponding sides. | | Pythagoras’ theorem | A theorem stating that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, $a^2 + b^2 = c^2$. | | Circle theorems | These are a set of geometric theorems relating angles, radii, tangents, and chords in a circle. Examples include the angle at the centre being double the angle at the circumference, and the angle in a semicircle being 90 degrees. | | Vectors | Vectors are mathematical objects that have both magnitude and direction. They can be represented by arrows or in column notation, e.g., $\begin{pmatrix} x \\ y \end{pmatrix}$. Vector operations include addition, subtraction, and scalar multiplication. | | Histograms | Histograms are graphical representations of the distribution of numerical data. They use bars to represent the frequency of data within specified intervals (classes). For unequal class intervals, frequency density is used on the vertical axis. | | Cumulative frequency graph | A graph that plots the cumulative frequency of data against the upper class boundaries. It is used to estimate median, quartiles, and percentiles. | | Correlation | Correlation describes the statistical relationship between two variables. It can be positive (variables tend to increase together), negative (one variable tends to increase as the other decreases), or zero (no linear relationship). Correlation does not imply causation. | | Conditional probability | The probability of an event occurring given that another event has already occurred. It is denoted as $P(A|B)$ and calculated as $P(A|B) = \frac{P(A \cap B)}{P(B)}$. | | Tree diagrams | Tree diagrams are used to represent the outcomes of a sequence of events, particularly in probability. They are useful for calculating probabilities of combined events, especially when probabilities are dependent on previous outcomes. | | Mathematical logic | The study of reasoning and argumentation. Key concepts include truth values (true/false), logical connectives (and, or, not), conditional statements (if...then), and their converses and contrapositives. | | Proof by contradiction | A method of proof where one assumes the statement to be proven is false and then derives a contradiction, thereby showing the original statement must be true. | | Disproof by counterexample | To disprove a statement, it is sufficient to find a single instance (a counterexample) where the statement does not hold true. | | Argumentative reasoning | The process of constructing and evaluating logical arguments. This involves understanding premises, conclusions, implications, and the validity of deductive and inductive reasoning. | | Necessary and sufficient conditions | A condition that is necessary for an event to occur must be met. A condition that is sufficient guarantees the event will occur. If a condition is both necessary and sufficient, it is equivalent to the event. |