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ScotlandMaths

Geometry, Proof and Systems of Equations

4 dot points across 4 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do you work with complex numbers in Cartesian and polar form, and use de Moivre's theorem for powers and roots?

Perform arithmetic with complex numbers in Cartesian form, represent them on an Argand diagram, convert to polar (modulus-argument) form, and use de Moivre's theorem to find powers and the nth roots of a complex number.
A focused answer to the SQA Advanced Higher Mathematics complex numbers content, covering arithmetic in Cartesian form, the complex conjugate, the Argand diagram, modulus and argument, polar form, and de Moivre's theorem for finding powers and the nth roots of a complex number.
12 min answer →

How do you compute with matrices and use them to solve a system of linear equations?

Add, subtract and multiply matrices, find the determinant and inverse of 2x2 and 3x3 matrices, and solve systems of linear equations using the inverse matrix and Gaussian elimination, identifying unique, no, and infinitely many solutions.
A focused answer to the SQA Advanced Higher Mathematics matrices and systems of equations content, covering matrix addition, subtraction and multiplication, the determinant and inverse of 2x2 and 3x3 matrices, solving systems by the inverse matrix and by Gaussian elimination, and recognising unique, no, and infinitely many solutions.
12 min answer →

How do you prove a mathematical statement directly, by contradiction or by contrapositive, and how do you use the Euclidean algorithm?

Construct proofs using direct proof, proof by contradiction and proof by contrapositive, disprove a conjecture by counterexample, and use the Euclidean algorithm to find the highest common factor and express it as a linear combination.
A focused answer to the SQA Advanced Higher Mathematics number theory and methods of proof content, covering direct proof, proof by contradiction, proof by contrapositive, disproof by counterexample, the fundamental theorem of arithmetic, and the Euclidean algorithm for the highest common factor and its linear combination.
12 min answer →

How do you describe lines and planes in three dimensions and use the scalar and vector products?

Use the scalar and vector products of vectors in three dimensions, find the equation of a line in three dimensions and the equation of a plane in vector, parametric and Cartesian form, and find angles and intersections between lines and planes.
A focused answer to the SQA Advanced Higher Mathematics vectors content, covering the scalar and vector products in three dimensions, the equation of a line in symmetric and parametric form, the equation of a plane in vector and Cartesian form, and finding angles and intersections between lines and planes.
12 min answer →