CAPE Pure Mathematics Syllabus 2026

Students will learn to distinguish simple from compound propositions and identify logical connectives like disjunction, negation, conditional, and bi-conditional.

Students will construct truth tables for compound statements and formulate the converse, contrapositive, and inverse of conditional and bi-conditional statements.

Students will determine logical equivalence between statements, identify tautologies and contradictions, and apply propositional identities.

Students will apply fundamental arithmetic operations and understand their properties, including identity, closure, inverse, commutativity, associativity, and distributivity, along with related axioms.

Students will be able to perform and evaluate results of given binary operations.

Students will perform basic arithmetic operations including addition, subtraction, multiplication, and rationalization involving surds.

Students will interpret and use summation notation to expand and evaluate finite series, including natural number series.

Students will prove mathematical statements using various methods, including direct proof, proof by counter-example, and mathematical induction for addition and divisibility.

Students will use the Factor Theorem to find factors and evaluate unknown coefficients, and the Remainder Theorem to determine polynomial remainders.

Students will extract all factors of expressions in the form $a^n - b^n$ for positive integers $n ≤ 6$.

Students will identify and apply the concept of the order of polynomial expressions.

Students will define key function terms such as domain, range, injectivity, surjectivity, and bijectivity, and determine these properties for given functions.

Students will determine if an inverse function exists for a given function and, if so, how to find its expression.

Students will draw and interpret graphs illustrating the relationship between a function and its inverse, including transformations.

Students will simplify expressions by effectively using the product, quotient, and power laws of logarithms.

Students will define exponential functions and sketch their graphs, understanding the properties of $y = a^x$.

Students will define logarithmic functions and sketch their graphs, understanding the properties of $y = \log_a x$.

Students will define the exponential function $e^x$ and its inverse, the natural logarithm $\ln x$, and convert between logarithmic and exponential forms.

Students will solve exponential equations using logarithms and apply the change of base formula to solve logarithmic problems.

Students will define the modulus function, understand its basic properties like $|x| = \sqrt{x^2}$, and derive the triangle inequality $|x+y| \u2264 |x| + |y|$.

Students will draw graphs illustrating the transformation from a function $y = f(x)$ to its modulus function $y = |f(x)|$.

Students will solve equations and inequalities involving simple rational and modulus functions using algebraic or graphical methods.

Students will use the relationships among roots of polynomial equations, particularly for cubics, involving sums and products of roots.

Define and use the reciprocal trigonometric functions sec x, csc x, and cot x, and sketch their graphs based on their definitions.

Derive and apply compound angle formulae for sine, cosine, and tangent, including their direct application to double angle identities.

Derive and apply multiple-angle formulae for sin kA, cos kA, tan kA, and use factor formulae for sums and differences of sines and cosines.

Prove fundamental trigonometric identities including Pythagorean identities and those involving reciprocal functions.

Determine general solutions for basic trigonometric equations and apply them to find specific solutions within a given range.

Determine the limit of a function using graphical analysis and describe function behavior as a variable approaches a specific value.

Use standard limit notation and apply basic limit theorems to solve simple limit problems.

Understand and apply the special trigonometric limit involving sin(x)/x as x approaches zero, using a geometric approach.

Identify points of continuity and discontinuity in functions using graphs, algebraic functions, and left-hand or right-hand limits.

Define the derivative of a function at a point as a limit and differentiate simple functions from first principles.

Use standard differentiation notations and apply the sum, product, and quotient rules to differentiate functions.

Students will express complex numbers in the form $a+bi$ and perform fundamental arithmetic operations including addition, subtraction, multiplication, and division.

Students will compute complex roots of quadratic equations and understand that complex roots occur in conjugate pairs.

Students will calculate the square root of a given complex number.

Students will calculate the modulus and the principal value of the argument for a non-zero complex number.

Students will represent complex numbers on an Argand diagram and interpret their modulus and argument geometrically.