Combined Mathematics Revision

Quadratic functions

A quadratic function has the standard form f(x) = ax² + bx + c with a ≠ 0; its graph is a parabola opening upwards if a > 0 (minimum…

Polynomial functions

A polynomial function of degree n is f(x) = an*xⁿ + a(n-1)*xⁿ⁻¹ + ...

Partial fractions

A rational function is f(x) = P(x) / Q(x) where P(x) and Q(x) are polynomials with Q(x) ≠ 0; it is Proper if deg(P) < deg(Q), and Improper…

Indices and logarithms

Laws of Indices: am * aⁿ = am+n; am / aⁿ = am-n; (am)ⁿ = amn; (ab)ⁿ = aⁿ * bⁿ; a⁰ = 1 (a ≠ 0); a⁻ⁿ = 1 / aⁿ; ap/q = q√(ap).

Inequalities

Properties of inequalities: If a > b and c > 0, then ac > bc; if a > b and c < 0, the inequality sign REVERSES: ac < bc; if a > b, then a +…

Circular (trigonometric) functions

Radian measure: Angle subtended at circle center by arc of length equal to radius: π radians = 180°; arc length s = rθ and sector area A =…

Sine rule and cosine rule

Standard triangle notation: In any triangle ABC, angles are A, B, C and opposite side lengths are a, b, c; semi-perimeter s = (a + b + c) /…

Sine rule and cosine rule (Pure Maths)

Application of Sine and Cosine rules to heights, distances, and 3D surveying problems: angles of elevation (measured upward from…

Limits of a function

Intuitive and formal concept of limit: limx->a f(x) = L means f(x) can be made arbitrarily close to L by taking x sufficiently close to a…

Differentiation techniques

Derivative from first principles: f'(x) = limh->0 [f(x + h) - f(x)] / h; represents instantaneous rate of change of f with respect to x,…

The definite integral and the Fundamental Theorem

Fundamental Theorem of Calculus: If f(x) is continuous on [a, b] and F'(x) = f(x), then ∫ₐb f(x) dx = F(b) - F(a); connects differential…

Area bounded by curves

Area bounded by curve y = f(x), the x-axis, and vertical lines x = a and x = b: Area A = ∫ₐb |f(x)| dx; if f(x) ≥ 0, A = ∫ₐb f(x) dx; if…

Equation of a straight line

Gradient (Slope) of a straight line passing through points A(x1, y1) and B(x2, y2): m = (y2 - y1) / (x2 - x1) = tan θ, where θ is the…

Line through the intersection of two lines

Intersection of two lines: If L1: a1*x + b1*y + c1 = 0 and L2: a2*x + b2*y + c2 = 0 intersect at point P(x0, y0), the coordinates satisfy…

Position of two points relative to a line

Linear expression L(x, y) = ax + by + c defines three regions in the xy-plane: the straight line itself (L(x, y) = 0), and two open…

Cartesian coordinates and the straight line

Rectangular Cartesian coordinate system: Two mutually perpendicular directed real lines (x-axis and y-axis) intersecting at the Origin O(0,…

Perpendicular distance from a point to a line

Perpendicular distance d from point P(x1, y1) to straight line L: ax + by + c = 0 is d = |ax1 + by1 + c| / √(a² + b²).

Cartesian equation of a circle

Definition of a Circle: Locus of a point P(x, y) moving in a plane such that its distance from a fixed point (center C) remains constant…

Position of a straight line relative to a circle

Line-Circle geometric relationship: Let circle have center C and radius r, and let d be the perpendicular distance from center C to…

Tangents to a circle from an external point

From any external point P(x1, y1) lying outside a circle (S1 > 0), exactly TWO real tangents can be drawn to the circle.

Circle through the intersection of two circles

Pencil / Family of Circles through intersection of two circles: Let S1 = 0 and S2 = 0 be two non-concentric intersecting circles; the…

Position of two circles

Relative positions of two non-concentric circles: Let C1, C2 be centers, r1, r2 be radii (assume r1 ≥ r2), and d = C1C2 be the distance…

Condition for two circles to intersect orthogonally

Orthogonal Intersection Definition: Two circles intersect orthogonally if their tangents at either point of intersection are perpendicular…

Mathematical induction

Principle of Mathematical Induction (PMI): A deductive mathematical technique used to prove that a statement P(n) is true for all positive…

Finite series

Sequence vs Series: A sequence is an ordered progression of terms u1, u2, u3, ..., un; a Series is the sum of terms of a sequence: Sn =…

Sums of elementary series

Sum of first n natural numbers: ∑ᵣ₌₁ⁿ r = 1 + 2 + 3 + ...

Summation of series

Method of Differences (Telescoping Sums): Express the general term ur as the difference of two consecutive terms of another sequence: ur =…

Convergence and divergence of series

Sequence of Partial Sums: For an infinite series ∑ᵣ₌₁∞ ur, the n-th partial sum is Sn = ∑ᵣ₌₁ⁿ ur.

The complex number system

Imaginary unit i: Defined such that i² = -1; powers of i cycle every 4 steps: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, and i4k+r = ir.

Algebra of complex numbers

Addition and Subtraction of complex numbers: (x1 + iy1) ± (x2 + iy2) = (x1 ± x2) + i(y1 ± y2); add/subtract real and imaginary parts…

Complex conjugate

Definition of Complex Conjugate: For complex number z = x + iy, its complex conjugate is z̄ (or z*) = x - iy; reflection of z across the…

Modulus of a complex number

Definition of Modulus: For complex number z = x + iy, its modulus (absolute value) is |z| = √(x² + y²); geometrically represents Euclidean…

The Argand diagram

Argand Diagram: Geometric representation of complex numbers in a plane where the horizontal axis is the Real Axis (Re) and vertical axis is…

De Moivre's theorem

De Moivre's Theorem for Integer Exponents: For any real angle θ and any integer n ∈ ℤ: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ).

Loci and regions of a complex number

Locus of a variable complex number z = x + iy: The path or region traced out in the Argand plane by z satisfying a given algebraic or…

Binomial expansion

Binomial Theorem for positive integer index n: (a + b)ⁿ = ∑ᵣ₌₀ⁿ nCr * an-r * br = nC0 * aⁿ + nC1 * aⁿ⁻¹b + ...

Binomial theorem

Binomial Theorem for any rational index n: If n is rational and |x| < 1, then (1 + x)ⁿ = 1 + nx + [n(n-1)/2!]x² + [n(n-1)(n-2)/3!]x³ + ...

Factorials

Definition of Factorial: For a non-negative integer n, n factorial (denoted n!) is the product of all positive integers less than or equal…

Fundamental principle of counting

Fundamental Counting Principle (Multiplication Rule): If an operation can be performed in m ways, and following this a second independent…

Permutations

Definition of Permutation: An arrangement of objects in a definite order (order of selection MATTERS).

Combinations

Definition of Combination: A selection of objects without regard to order (order of selection DOES NOT matter).

Properties of matrices

Definition of a Matrix: A rectangular array of real numbers arranged in m rows and n columns, having order (dimension) m × n.

Special square matrices

Square Matrix: A matrix with equal numbers of rows and columns (m = n, order n × n).

Transpose and inverse of a matrix

Transpose of a Matrix: The matrix AT obtained by interchanging rows and columns of A; if A is m × n, then AT is n × m.

Simultaneous equations by matrices

Matrix formulation of linear system: System a1*x + b1*y = d1 and a2*x + b2*y = d2 is written in matrix form as A * X = B, where coefficient…

Systems of coplanar forces

Coplanar Forces: Forces whose lines of action all lie in the same geometric plane.

Systems of coplanar forces (statics)

Reduction of a general coplanar force system: Any system of coplanar forces can be reduced to a single resultant force R acting at an…

Stresses in the rods of a framework

Pin-Jointed Truss / Framework: A rigid structure composed of light rods smoothly pinned (hinged) together at their ends, carrying loads…

Centre of mass of uniform bodies

Center of Mass (C.M.) Definition: The unique point G at which the total mass M of a body or system of particles may be considered to be…

Centre of mass by integration

Continuous Mass Center via Calculus: xG = (1/M) ∫ x dm, where M = ∫ dm is total mass and dm is an infinitesimal mass element.

Centre of mass of composite bodies

Composite Body Principle: A composite body composed of parts with masses m1, m2, ..., mn and respective centers of mass (x1, y1), (x2, y2),…

Centre of gravity

Center of Gravity (C.G.) Definition: The single point G through which the resultant gravitational force (total weight W = Mg) acts in all…

Stability of bodies in equilibrium

Three States of Equilibrium for a body resting under gravity: Stable, Unstable, and Neutral equilibrium.

Angle of inclination of suspended bodies

Freely suspended body equilibrium: When a body is freely suspended from a fixed point O, it hangs in equilibrium with its Center of Gravity…

Newtonian model of motion

Newton's Laws of Motion: First Law (law of inertia), Second Law (resultant force equals time rate of change of linear momentum: F = dp/dt =…

Velocity and acceleration in circular motion

Circular Motion Kinematics: Angular displacement θ (radians), angular velocity ω = dθ/dt (rad/s), and angular acceleration α = dω/dt =…

Motion in a horizontal circle

Horizontal Circular Motion: Particle moves along a circle in a horizontal plane with radius r at constant angular velocity ω and constant…

Motion in a vertical circle

Vertical Circular Motion: Non-uniform circular motion in a vertical plane under gravity where speed v and tension/normal reaction T vary…

Simple harmonic motion

Definition of Simple Harmonic Motion (SHM): Rectilinear motion of a particle where acceleration is directly proportional to its…

Simple harmonic motion on a horizontal line

Hooke's Law for elastic strings and springs: Tension T is directly proportional to extension x beyond natural length l: T = λ * x / l,…

Simple harmonic motion on a vertical line

Vertical Elastic String/Spring system: Particle of mass m hangs in static equilibrium from fixed support by an elastic string of natural…

Events of a random experiment

Random Experiment: An experiment whose outcome cannot be predicted with certainty in advance, but whose set of all possible outcomes is…

Probability models for random events

Classical (Equally Likely) Probability Definition: If sample space S has n equally likely outcomes, and event A contains m outcomes, then…

Conditional probability

Conditional Probability Definition: The probability of event A occurring given that event B has ALREADY occurred is P(A | B) = P(A ∩ B) /…

Bayes' theorem

Bayes' Theorem: Calculates the posterior (revised / inverse) probability of a cause Bk given that effect A has occurred.

The nature of statistics

Statistics: The scientific discipline concerned with collecting, organizing, summarizing, presenting, analyzing, and interpreting numerical…

Measures of central tendency

Measures of Central Tendency: Statistical summary metrics identifying the central or typical value around which data cluster: Mean, Median,…

Measures of relative position

Measures of Relative Position (Quantiles): Values that divide an ordered distribution into equal proportioned partitions: Quartiles (4…

Measures of dispersion

Measures of Dispersion (Variation): Statistical metrics quantifying the degree of spread, scatter, or variability of data values around…