Course Content

CAPE Applied Mathematics SBA

MODULE 1: DISCRETE MATHEMATICS (Linear Programming) On completion of this Module, students should: 1. understand the concept of linear programming to formulate models in a realworld context; 2. understand the terms, concepts and methods used in graph theory; 3. understand basic network concepts; 4. understand basic concepts and applications of Boolean Algebra; 5. have the ability to employ truth table techniques to establish the validity of statements; 6. appreciate the application of discrete methods in efficiently addressing realworld situations.
 Derive and graph linear inequalities in two variables;
 Determine whether a selected trial point satisfies a given inequality;
 Determine the solution set that satisfies a set of linear inequalities in two variables;
 Determine the feasible region of a linear programming problem;
 Identify the objective function and constraints of a linear programming problem;
 Determine a unique optimal solution (where it exists) of a linear programming problem;
 Formulate linear programming models in two variables from realworld data.

MODULE 1: DISCRETE MATHEMATICS (Assignment Models)
 Model a weighted assignment (or allocation) problem as an m x n matrix (where m is the number of rows and n is the number of columns);
 Convert nonsquare matrix models to square matrix models with the addition of rows or columns of dummy entries that take the maximum (minimum) value of all entries in the minimisation (maximisation) assignment problem;
 Convert a maximisation assignment problem into a minimisation problem (by changing the sign of each entry);
 Solve a minimisation assignment problem (of complexity 5 x 5 or less) by the Hungarian algorithm. (The convention of reducing rows before columns will be followed.

MODULE 1: DISCRETE MATHEMATICS (Graph Theory and Critical Path Analysis)
 Identify the vertices and sequence of edges that make up a path;
 Determine the degree of a vertex;
 Use networks as models of realworld situations;
 Use the activity network algorithm in drawing a network diagram to model a realworld problem (activities will be represented by vertices and the duration of activities by edges);
 Calculate the earliest start time, latest start time, earliest finish time, latest finish time and float time;
 Identify the critical path in an activity network;
 Use the activity network in decision making
 Use networks as models of realworld situations;
 Use the activity network algorithm in drawing a network diagram to model a realworld problem (activities will be represented by vertices and the duration of activities by edges);
 Calculate the earliest start time, latest start time and float time;
 Identify the critical path in an activity network;
 Use the critical path in decision making.

MODULE 1: DISCRETE MATHEMATICS (Logic and Boolean Algebra)
 Formulate (in symbols or in words):
 Establish the truth value of:
 State the converse, inverse and contrapositive of implications of propositions;
 Use truth tables;
 Use the laws of Boolean algebra (idempotent, complement, identity, commutative, associative, distributive, absorption, de Morgan’s Law) to simplify Boolean expressions;
 Derive a Boolean expression from a given switching or logic circuit;
 Represent a Boolean expression by a switching or logic circuit;
 Use switching and logic circuits to model realworld situations.

MODULE 2: PROBABILITY AND DISTRIBUTIONS (Probability) On completion of this Module, students should: 1. apply counting techniques and calculus in probability; 2. appreciate that probability models can be used to describe realworld situations; 3. apply appropriate distributional approximations to data; 4. assess the appropriateness of distributions to data.
 Calculate the number of selections of n distinct objects taken r at a time, with or without restrictions;
 Calculate the number of ordered arrangements of n objects taken r at a time, with or without restrictions;
 Calculate probabilities of events (which may be combined by unions or intersections) using appropriate counting techniques;
 Calculate and use probabilities associated with conditional, independent or mutually exclusive events

MODULE 2: PROBABILITY AND DISTRIBUTIONS (Discrete Random Variables)
 Apply the properties:
 Formulate and use the probability function f(x) = P(X=x) where f is a simple polynomial or rational function;
 Calculate and use the expected values and variance of linear combinations of independent random variables;
 Model practical situations in which the discrete uniform, binomial, geometric or Poisson distributions are suitable;
 Apply the formulae:
 Use the formulae for E(X) and Var(X) where X follows a discrete uniform, binomial, geometric or Poisson distribution;
 Use the Poisson distribution as an approximation to the binomial distribution;

MODULE 2: PROBABILITY AND DISTRIBUTIONS (Continuous Random Variables)
 Apply the properties of the probability density function f of a continuous random variable X;
 Use the cumulative distribution function;
 Use the result
 Calculate expected value, variance, median and other quartiles;
 Solve problems involving probabilities of the normal distribution using Z – scores;
 Use the normal distribution, with a continuity correction, to approximate the Poisson distribution, as appropriate ( λ > 15)

MODULE 2: PROBABILITY AND DISTRIBUTIONS (χ2 test)

MODULE 3: PARTICLE MECHANICS (Coplanar Forces and Equilibrium) On completion of this Module, students should: 1. understand forces and their applications; 2. understand the concepts of work, energy and power; 3. appreciate the application of mathematical models to the motion of a particle.
 Identify forces (including gravitational forces) acting on a body in a given situation;
 Use vector notation to represent forces;
 Represent the contact force between two surfaces in terms of its normal and frictional component;
 Calculate the resultant of two or more coplanar forces;
 Resolve forces, on particles, in mutually perpendicular directions (including those on inclined planes);
 Use the principle that, for a particle in equilibrium, the vector sum of its forces is zero, (or equivalently the sum of its components in any direction is zero);
 Use the appropriate relationship F = µR or F ≤ µR for two bodies in limiting equilibrium;
 Solve problems involving concurrent forces in equilibrium, (which may involve the use of Lami’s Theorem).

MODULE 3: PARTICLE MECHANICS (Kinematics and Dynamics)
 Distinguish between distance and displacement, and speed and velocity;
 Draw and use displacementtime and velocitytime graphs;
 Calculate and use displacement, velocity, acceleration and time in simple equations representing the motion of a particle in a straight line;
 Apply Newton’s laws of motion;
 Apply where appropriate the following rates of change;
 Formulate and solve first order differential equations as models of the linear motion of a particle when the applied force is proportional to its displacement or its velocity (only differential equations where the variables are separable will be required);
 Apply the principle of conservation of linear momentum to the direct impact of two inelastic particles moving in the same straight line. (Knowledge of impulse is required. Problems may involve twodimensional vectors).

MODULE 3: PARTICLE MECHANICS (Projectiles)

MODULE 3: PARTICLE MECHANICS (Work, Energy and Power)
 Calculate the work done by a constant force;
 Calculate the work done by a variable force in onedimension;
 Solve problems involving kinetic energy and gravitational potential energy;
 Apply the principle of conservation of energy;
 Solve problems involving power;
 Apply the workenergy principle in solving problems.