← Specialist Mathematics syllabus
Unit 3: Mathematical induction, and further vectors, matrices and complex numbers
10 dot points across 7 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.
Topic 4: Further complex numbers
Topic 2: Vectors and matrices
- 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
A focused answer to the QCE Specialist Mathematics Unit 3 dot point on 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.
6 min answer → - 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
A focused answer to the QCE Specialist Mathematics Unit 3 dot point on 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.
6 min answer → - 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
A focused answer to the QCE Specialist Mathematics Unit 3 dot point on 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.
6 min answer →
Topic 3: Complex numbers
- 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
A focused answer to the QCE Specialist Mathematics Unit 3 dot point on 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.
6 min answer → - 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
A focused answer to the QCE Specialist Mathematics Unit 3 dot point on 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.
6 min answer →