Combined Mathematics Theory Notes

Quadratic functions

Imaginary Unit and Complex Numbers: A complex number is a number of the form z = a + ib, where a and b are real numbers, and i is the…

Polynomial functions

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Partial fractions

A rational function is a function that can be written as the ratio of two polynomials, P(x) and Q(x), where Q(x) is not identically zero.

Indices and logarithms

Indices (Exponents): An index (or exponent) indicates how many times a base number is multiplied by itself.

Inequalities

An inequality is a mathematical statement that compares two expressions using an inequality symbol (>, <, ≥, ≤, ≠).

Circular (trigonometric) functions

Trigonometric Equations: A trigonometric equation is an equation that involves one or more trigonometric functions of a variable angle.

Sine rule and cosine rule

Definition and Cartesian Equations of a Circle: A circle is defined as the locus of a point that moves in a plane such that its distance…

Sine rule and cosine rule (Pure Maths)

The Sine Rule states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant for all three…

Limits of a function

The Concept of a Limit: The limit of a function f(x) as x approaches a specific value 'a' is the value that f(x) gets arbitrarily close to,…

Differentiation techniques

Linear momentum is a vector quantity defined as the product of an object's mass and its velocity.

The definite integral and the Fundamental Theorem

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Area bounded by curves

Scalars vs.

Equation of a straight line

Evolution of the Number System: The historical development of numerical concepts and symbols, starting from simple counting to complex…

Line through the intersection of two lines

Understanding Functions: A function is a special type of relation where each element in the domain (input set) is mapped to exactly one…

Position of two points relative to a line

Angular Measure (Degrees and Radians): Angular measure quantifies the amount of rotation or the extent of an angle, using units such as…

Cartesian coordinates and the straight line

Angle Between Two Straight Lines: When two non-parallel straight lines intersect, they form two pairs of vertically opposite angles.

Perpendicular distance from a point to a line

Trigonometric Ratios in the Cartesian Plane: Trigonometric ratios (sine, cosine, tangent) are defined for any angle using a point P(x,y) on…

Cartesian equation of a circle

A circle is the locus of a point P(x, y) moving in a plane at a constant distance r (radius) from a fixed point C(a, b) (centre).

Position of a straight line relative to a circle

The relative geometric position of a straight line ax + by + k = 0 with respect to a circle (x - a)² + (y - b)² = r² is determined either…

Tangents to a circle from an external point

From an external point P(x₁, y₁) lying outside a circle S ≡ x² + y² + 2gx + 2fy + c = 0, exactly two real tangents can be drawn to the…

Circle through the intersection of two circles

The equation of any circle passing through the points of intersection of a given circle S ≡ x² + y² + 2gx + 2fy + c = 0 and a straight line…

Position of two circles

The relative geometric position of two circles with centres C₁, C₂ and radii r₁, r₂ is governed by comparing the distance between their…

Condition for two circles to intersect orthogonally

Two circles are said to intersect orthogonally if the tangents at either point of intersection are mutually perpendicular.

Mathematical induction

The Principle of Mathematical Induction is a rigorous deductive proof method for statements P(n) asserted for all positive integers n ∈ ℤ⁺.

Finite series

A finite series is the sum of terms of a sequence, denoted using sigma notation as Sₙ = ∑ᵣ₌₁ⁿ uᵣ = u₁ + u₂ + ...

Sums of elementary series

Evaluating polynomial series requires standard summation identities for powers of natural numbers: (1) Sum of the first n natural numbers:…

Summation of series

The Method of Differences (telescoping series) is a powerful algebraic technique used to sum series where standard power formulas do not…

Convergence and divergence of series

An infinite series ∑ᵣ₌₁∞ uᵣ is investigated by examining the sequence of its partial sums {Sₙ}, where Sₙ = ∑ᵣ₌₁ⁿ uᵣ.

The complex number system

The complex number system extends real numbers to provide solutions to polynomial equations with negative discriminants, introducing the…

Algebra of complex numbers

Fundamental algebraic operations on complex numbers follow standard field axioms with i² = -1: (1) Addition: (x₁ + iy₁) + (x₂ + iy₂) = (x₁…

Complex conjugate

The complex conjugate of a complex number z = x + iy is defined as z̄ = x - iy, formed by reflecting z across the real axis on the Argand…

Modulus of a complex number

The modulus of a complex number z = x + iy is defined as the non-negative real distance from the origin to z on the Argand plane: |z| =…

The Argand diagram

The Argand diagram represents complex numbers geometrically, mapping z = x + iy to the point (x, y) or the position vector from origin O to…

De Moivre's theorem

De Moivre's Theorem states that for any real number θ and any integer n ∈ ℤ: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ).

Loci and regions of a complex number

Loci and regions on the Argand plane describe the geometric paths and areas traced by a variable complex number z = x + iy satisfying…

Binomial expansion

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

Binomial theorem

Practical applications of the Binomial Theorem encompass finding specific terms, determining the term independent of x, identifying…

Factorials

The factorial of a non-negative integer n, denoted by n!, represents the product of all positive integers less than or equal to n: n!

Fundamental principle of counting

The Fundamental Principle of Counting encompasses two core logical rules that govern combinatorial mathematics: (1) The Multiplication…

Permutations

A permutation is an arrangement of objects in a specific order.

Combinations

A combination is an unordered selection of r distinct items chosen from a set of n distinct objects, denoted by ⁿCᵣ or (n \atop r) = n!

Properties of matrices

A matrix is a rectangular array of numbers arranged in m rows and n columns, said to have order m × n.

Special square matrices

A square matrix has an equal number of rows and columns (n × n).

Transpose and inverse of a matrix

The transpose of an m × n matrix A, denoted by AT, is the n × m matrix formed by interchanging the rows and columns of A.

Simultaneous equations by matrices

A system of two linear simultaneous equations in two variables: a₁x + b₁y = d₁ and a₂x + b₂y = d₂ can be structured in matrix form as AX =…

Systems of coplanar forces

In mechanics, a particle is an idealized body that possesses mass but has negligible dimensions, meaning its size and shape are ignored.

Systems of coplanar forces (statics)

A rigid body is a body in which the distance between any two points remains unchanged when it is subjected to external forces of any…

Stresses in the rods of a framework

The Concept of an Inverse Function: An inverse function 'undoes' the action of the original function, mapping the output of the original…

Centre of mass of uniform bodies

The centre of mass (centre of gravity in a uniform gravitational field) of a rigid body is the unique point at which the entire mass of the…

Centre of mass by integration

For continuous bodies, the centre of mass is obtained by integrating element mass moments over the body: x̄ = (∫ x dm) / (∫ dm).

Centre of mass of composite bodies

The centre of mass of a composite body assembled from multiple sub-bodies is determined using the Principle of Moments: Mtotal XG = ∑ mᵢ…

Centre of gravity

The Centre of Gravity (CG) of a body is the single point where the entire weight of the body appears to act.

Stability of bodies in equilibrium

The equilibrium of a body subjected to conservative forces is classified into three states based on its behavior following an infinitesimal…

Angle of inclination of suspended bodies

When a rigid body is suspended freely from a fixed pivot point O, it attains static equilibrium under two forces: the tension/reaction at O…

Newtonian model of motion

Describing Motion: Distance is the total path length covered by an object, a scalar quantity.

Velocity and acceleration in circular motion

The Natural Exponential Function (ex): The natural exponential function, denoted as ex, is a mathematical function where 'e' is Euler's…

Motion in a horizontal circle

Natural Logarithmic Function (ln x): The natural logarithmic function, denoted as ln x, is the inverse function of the natural exponential…

Motion in a vertical circle

Motion of a particle of mass m in a vertical circle of radius a involves non-uniform speed due to gravity.

Simple harmonic motion

Simple Harmonic Motion (SHM) is rectilinear oscillatory motion where the acceleration ẍ of a particle is directly proportional to its…

Simple harmonic motion on a horizontal line

Horizontal Simple Harmonic Motion arises when a particle of mass m moves on a smooth horizontal table attached to the ends of elastic…

Simple harmonic motion on a vertical line

Vertical Simple Harmonic Motion (SHM): Vertical Simple Harmonic Motion occurs when a particle oscillates along a vertical line under the…

Events of a random experiment

A random experiment is an observational procedure whose outcome cannot be predicted with certainty, but whose set of all possible…

Probability models for random events

Classical probability models define the likelihood of an event A in an equiprobable sample space S as P(A) = n(A) / n(S), satisfying…

Conditional probability

Conditional probability measures the probability of an event A occurring given that another event B has already occurred with P(B) > 0,…

Bayes' theorem

Bayes' Theorem provides a mathematical method for computing posterior probabilities (reversing cause and effect) given observed evidence.

The nature of statistics

Statistics is the mathematical discipline concerned with collecting, organizing, presenting, analyzing, and interpreting numerical data to…

Measures of central tendency

Measures of central tendency summarize a data distribution by identifying a single representative central location: (1) Mean (Arithmetic…

Measures of relative position

Measures of relative position (partition values) divide a sorted frequency distribution into specified fractional proportions: (1)…

Measures of dispersion

Measures of dispersion quantify the extent of spread, scatter, or variability of data values around their central mean: (1) Range:…