Unit 3: Mathematical induction, and further vectors, matrices and complex numbers

10 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

Topic 1: Proof by mathematical induction and further proof methods

Construct trigonometric proofs and apply direct proof, proof by contrapositive and proof by contradiction, choosing the appropriate method and writing each step with correct logical structure and justification

Proof methods beyond induction. Covers direct proof, proof by contrapositive, proof by contradiction and trigonometric proofs using identities, with a verified worked example and the logic errors QCAA markers penalise most.

8 min read →

Topic 1: Proof by mathematical induction

Understand and use the principle of mathematical induction to prove results involving sums of series, divisibility statements and inequalities for all positive integers, structuring the base step and the inductive step correctly

Proof by mathematical induction. Covers the base step and inductive step structure, summation proofs, divisibility arguments and inequality proofs, with the precise wording QCAA markers reward and the inductive-hypothesis logic that earns full method marks.

6 min read →

Topic 2: Vectors in two and three dimensions

Use vectors in three dimensions, compute scalar (dot) and vector (cross) products, find angles between vectors, scalar and vector projections, and apply these to geometric problems and the equations of lines and planes

Three-dimensional vectors. Covers component form, magnitude, the dot product and angle between vectors, scalar and vector projections, the cross product and its geometric meaning, and the vector and parametric equations of lines, with a verified worked example and the projection sign trap.

8 min read →

Topic 2: Vectors and matrices

Determine vector, parametric and Cartesian equations of a line in three dimensions, convert between these forms, and find the point of intersection of two lines or establish that they are parallel or skew

Lines in three dimensions. Covers the vector, parametric and Cartesian forms of a line, converting between them, and classifying two lines as intersecting, parallel or skew, with a verified worked example and the parameter-clash trap.

8 min read →

Use matrices beyond order two, including the determinant and inverse of a three by three matrix, to represent and solve systems of linear equations, and interpret unique, infinite and no-solution cases geometrically

Systems of linear equations. Covers representing systems in matrix form, the determinant and inverse of a three by three matrix, solving by the matrix inverse, and the geometric meaning of unique, infinitely many and no solutions, with a verified worked example and the determinant-zero trap.

8 min read →

Determine the vector equation and the Cartesian equation of a plane from a point and a normal or from three points, find the distance from a point to a plane, and find the line of intersection or angle between planes

Planes in three dimensions. Covers the normal vector, the vector equation r dot n equals a dot n, the Cartesian form, building a plane from three points, the distance from a point to a plane and the angle between planes, with a verified worked example and the normal-vector trap.

8 min read →

Topic 3: Complex numbers

Identify and sketch subsets of the complex plane determined by relations involving modulus, argument, distance and inequalities, including lines, circles, perpendicular bisectors, rays and regions

Subsets of the complex plane. Covers circles and discs from modulus relations, perpendicular bisectors from equal-distance relations, rays from argument conditions, and combining inequalities into regions, with a verified worked example and the boundary-inclusion trap.

8 min read →

Factorise polynomials with real coefficients over the complex field, apply the conjugate root theorem and the fundamental theorem of algebra, and find all roots of polynomial equations including those with complex coefficients

Factorising polynomials over the complex numbers. Covers the fundamental theorem of algebra, the conjugate root theorem for real coefficients, dividing out known factors and reconstructing real quadratic factors, with a verified worked example and the conjugate-pairs trap.

8 min read →

Topic 3: Further matrices

Use 2x2 matrices to represent linear transformations of the plane, compute determinants and inverses, interpret the determinant as a scaling factor, and combine transformations through matrix multiplication

Further matrices. Covers 2x2 matrix multiplication, determinants, inverses, the determinant as an area-scaling factor, standard transformation matrices for rotation, reflection, dilation and shear, and composition by matrix product, with a verified worked example and the order-of-multiplication trap.

8 min read →

Topic 4: Further complex numbers

Represent complex numbers in Cartesian, polar (modulus-argument) and exponential form, convert between forms, and apply de Moivre's theorem to compute powers and roots, locating results on the complex plane

Further complex numbers. Covers modulus and argument, polar and exponential form, multiplication and division as rotations, de Moivre's theorem for powers, and finding the n-th roots of a complex number with a fully worked example and the argument-range trap QCAA penalises.

6 min read →