Sri Lanka Advanced Level Combined Mathematics Syllabus

This syllabus integrates concepts from Pure Mathematics and Applied Mathematics. It aims to equip students with a strong mathematical foundation for higher studies in mathematics, engineering, physical sciences, and other related fields. The syllabus emphasizes problem-solving strategies, logical thinking, and practical application of mathematical knowledge. It is designed to be competency-based, focusing on the development of specific skills and abilities. The syllabus is structured to cater to students who have learned Mathematics at Grades 6-11 under the new curriculum reforms. It aims to bridge the gap between Ordinary Level Mathematics and Advanced Level Combined Mathematics.

Combined Mathematics I

Algebra

• Review of Basic Algebra:
• Expansion of algebraic expressions (including formulas like (a ± b)² , (a ± b)³ , and (a + b + c)²).
• Factorization of algebraic expressions (including formulas like a² - b² , a³ ± b³).
• Simplification of algebraic fractions (addition, subtraction, multiplication, and division).
• Solving equations (including equations with algebraic fractions, simultaneous equations up to three unknowns, and quadratic simultaneous equations).
• Simplification of expressions involving indices and logarithms.
• Properties and applications of proportions.
• Functions:
• Intuitive idea of a function, constants, variables, and expressions involving relationships between two variables.
• Functional notation, domain, codomain, and range.
• One-to-one and onto functions, inverse functions.
• Types of functions: constant, linear, piecewise, modulus (absolute value).
• Graph of a function, composite functions.
• Quadratic Functions and Equations:
• Definition of a quadratic function, completing the square, discriminant.
• Properties of a quadratic function: greatest value, least value, existence/non-existence of real zeros, graphs.
• Roots of a quadratic equation: sum and product of roots, nature of roots using discriminant, condition for common roots, transformation of equations.
• Polynomials:
• Polynomials of a single variable: terms, coefficients, degree, leading term, leading coefficient.
• Algebraic operations on polynomials: addition, subtraction, multiplication, division, and long division.
• Remainder theorem, factor theorem, and its converse.
• Solution of polynomial equations.
• Rational Functions:
• Definition of rational functions, proper and improper rational functions.
• Partial fractions decomposition (distinct linear factors, recurring linear factors, and quadratic factors).
• Index Laws and Logarithmic Laws:
• Index laws and logarithmic laws of base.
• Change of base.
• Inequalities:
• Basic properties of inequalities, including the trichotomy law.
• Numerical inequalities, representing inequalities on the real number line, introducing intervals.
• Inequalities involving simple algebraic functions (linear, quadratic, and rational), finding solutions algebraically and graphically.
• Inequalities involving modulus (absolute value) function.
• Angular Measurements:
• Angular measure, sign convention, degree and radian measures.
• Arc length and area of a circular sector.
• Circular Functions:
• Basic trigonometric functions: definitions, domain, and range.
• Values of circular functions at commonly used angles.
• Trigonometric relations of angles differing by multiples of π/2 and integer multiples of π.
• Graphs of basic trigonometric functions and their periodic properties.
• General solutions of trigonometric equations.
• Trigonometric Identities:
• Pythagorean identities.
• Sum and difference formulas.
• Product-sum and sum-product formulas.
• Double angle, triple angle, and half-angle formulas.
• Equations of the form a cos θ + b sin θ = c.
• Sine Rule and Cosine Rule:
• Sine rule and cosine rule for triangles.
• Applications of sine and cosine rules.
• Inverse Trigonometric Functions:
• Definition of inverse trigonometric functions, principal values.
• Graphs of inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹).
• Problems involving inverse trigonometric functions.
• Limits:
• Intuitive idea of limit, basic theorems on limits and their applications.
• Proof of lim (xⁿ - aⁿ)/(x - a) = naⁿ⁻¹ and its applications.
• Sandwich theorem.
• Proof of lim sin x/x = 1 and its applications.
• One-sided limits: right-hand limit and left-hand limit.
• Limits at infinity and horizontal asymptotes.
• Infinite limits and vertical asymptotes.
• Continuity at a point.
• Derivatives:
• Derivative as the slope of a tangent line, as a limit, and as a rate of change.
• Derivatives from first principles (xⁿ, basic trigonometric functions).
• Theorems on differentiation: constant multiple rule, sum rule, product rule, quotient rule, chain rule.
• Derivatives of inverse trigonometric functions.
• Natural exponential function and its derivative.
• Natural logarithmic function and its derivative.
• Differentiation of implicit and parametric functions (including parametric forms of parabola, ellipse, and hyperbola).
• Derivatives of higher order.
• Applications of Derivatives:
• Stationary points, increasing/decreasing functions, maximum/minimum points, points of inflection, first and second derivative tests.
• Concavity and points of inflection.
• Sketching curves (including horizontal and vertical asymptotes).
• Optimization problems.
• Integration:
• Integration as the reverse process of differentiation (anti-derivatives).
• Theorems on integration.
• Definite integral and its properties, fundamental theorem of calculus.
• Integration of rational functions using partial fractions and trigonometric identities.
• Integration by substitution and by parts.
• Applications of integration: area under a curve, area between two curves, volume of revolution.
• Rectangular Cartesian System and Straight Line:
• Rectangular Cartesian coordinates, distance between two points.
• Coordinates of a point dividing a line segment in a given ratio (internally and externally).
• Equation of a straight line: inclination, gradient, intercepts, various forms of the equation.
• Point of intersection of two lines.
• Equation of a line passing through the intersection of two given lines.
• Relative position of two points with respect to a given line.
• Angle between two lines, conditions for parallel and perpendicular lines.
• Parametric equation of a straight line, perpendicular distance from a point to a line.
• Equations of bisectors of angles between two lines.
• Principle of Mathematical Induction:
• Principle of mathematical induction and its applications (divisibility, summation, inequalities).
• Series:
• Finite series and their properties, sigma notation.
• Summation of series: arithmetic and geometric series, method of differences, method of partial fractions, principle of mathematical induction.
• Infinite series: sequences, partial sums, convergence and divergence, sum to infinity.
• Binomial Expansion:
• Binomial theorem for positive integral indices: binomial coefficients, general term, proof using mathematical induction.
• Relationships among binomial coefficients, specific terms.
• Complex Numbers:
• Imaginary unit, definition of a complex number, real and imaginary parts, purely imaginary numbers, equality of complex numbers.
• Algebraic operations on complex numbers.
• Complex conjugate and its properties.
• Modulus of a complex number and its properties.
• Argand diagram, geometrical representation of algebraic operations.
• Polar form of a complex number, argument and principal argument.
• De Moivre's theorem and its applications.
• Locus/region of a variable complex number.
• Permutations and Combinations:
• Factorial, fundamental principle of counting.
• Permutations: definition, notation, formulas.
• Combinations: definition, notation, formulas, distinction between permutations and combinations.
• Matrices:
• Definition and notation of matrices: elements, rows, columns, size, special types (row, column, square, null, identity, diagonal, symmetric, skew-symmetric, triangular).
• Equality of matrices, scalar multiplication, addition, subtraction, multiplication, transpose, inverse (for 2x2 matrices).
• Solution of simultaneous equations using matrices (graphically and algebraically).
• Circles:
• Equation of a circle with origin as the center and with a given center and radius.
• Conditions for intersection, tangency, and non-intersection of a line and a circle.
• Equation of tangent at a point on a circle and from an external point.
• Length of tangent from an external point, equation of chord of contact.
• Equation of a circle passing through the intersection of a line and a circle.
• Position and intersection of two circles, condition for orthogonal intersection.

Combined Mathematics II

Vectors

• Introduction to Vectors:
• Scalar and vector quantities, magnitude and direction, vector notation (algebraic and geometric), null vector, modulus, equality of vectors, triangle law of vector addition, scalar multiplication, difference of vectors, unit vectors, parallel vectors.
• Algebraic system for vectors, position vectors, applications of position vectors.
• Scalar product: definition, properties, condition for perpendicularity.
• Vector product: definition and properties.
• Systems of Coplanar Forces:
• Concept of a particle and a force, representation of forces, dimension and unit of force, types of forces, resultant force.
• Resultant of two forces: parallelogram law, equilibrium, resolution of a force.
• Resultant of a system of coplanar forces: method of resolution, graphical method, conditions for equilibrium.
• Equilibrium of a particle under three forces: triangle law, Lami's theorem.
• Resultant of coplanar forces acting on a rigid body: principle of transmission of forces, moment of a force, couple, equivalent couples, equilibrium.
• Reduction of a system of coplanar forces: single force, couple, conditions for equilibrium.
• Equilibrium of three coplanar forces acting on a rigid body.
• Friction:
• Smooth and rough surfaces, frictional force and its nature, laws of friction, coefficient of friction, angle of friction.
• Jointed Rods and Frameworks:
• Types of simple joints, movable and rigid joints, forces acting at a smooth joint, applications involving jointed rods.
• Frameworks with light rods, conditions for equilibrium at each joint, Bow's notation and stress diagram.
• Center of Mass:
• Definition of center of mass, center of mass of symmetrical bodies, composite bodies, and remaining bodies.
• Center of gravity, stability of equilibrium.
• Motion in a Straight Line:
• Distance, speed, displacement, velocity, acceleration, kinematic equations, graphs (displacement-time, velocity-time).
• Relative motion in a straight line: relative displacement, velocity, and acceleration.
• Motion in a Plane:
• Position vector, velocity, and acceleration as functions of time.
• Relative motion in a plane: relative displacement, velocity, and acceleration.
• Shortest distance between two particles, collision of two bodies.
• Projectiles:
• Motion of a projectile, velocity and displacement components, equation of path, maximum height, time of flight, horizontal range.
• Newton's Laws of Motion:
• Newton's laws of motion, mass, linear momentum, inertial frame of reference, absolute and gravitational units of force, weight and mass, action and reaction.
• Work, Power, Energy:
• Work, energy, kinetic energy, potential energy, work-energy equations, dissipative and conservative forces, principle of conservation of mechanical energy.
• Power, tractive force.
• Impulse and Collisions:
• Impulse, impulsive action, loss of kinetic energy due to impulse.
• Newton's law of restitution, coefficient of restitution, direct impact.
• Principle of conservation of linear momentum.
• Circular Motion:
• Angular velocity and acceleration, velocity and acceleration for circular motion.
• Motion in a horizontal circle, conical pendulum.
• Motion in a vertical circle, conservation of energy.
• Simple Harmonic Motion:
• Definition and characteristics of simple harmonic motion, differential equation and solutions, velocity and displacement as functions of time, amplitude, period, representation using uniform circular motion.
• Simple harmonic motion on a horizontal line and a vertical line, Hooke's law, combination with free motion under gravity.
• Probability:
• Random experiments, sample space, sample points, events, probability, conditional probability, independence, Bayes' theorem.
• Statistics:
• Descriptive statistics, measures of central tendency (mean, mode, median), measures of relative position (quartiles, percentiles, box plots), measures of dispersion (range, interquartile range, mean deviation, variance, standard deviation), measures of skewness.

This detailed syllabus provides a comprehensive outline of the topics covered in the Sri Lanka Advanced Level Combined Mathematics examination. It is designed to be a challenging and rewarding course of study, preparing students for a wide range of future academic and professional pursuits.