Syllabus Year 12, Mathematics

The goals of mathematics education will be achieved at different levels in the K-12 area according to the developmental stages of the students at those levels. Awareness of the place of mathematics in solving everyday problems and in contributing to the development of our society. Students should be encouraged to become aware of the relevance of mathematics to their lives.

Opportunities to use mathematical processes and to formulate and solve problems should be provided in all areas of mathematics. Mathematics learning should be appropriate for each student's current stage of development and should build on previous experiences and achievements.

Introduction

Aims and objectives

Syllabus Structure

Assessment

Evaluation

Basic Curves
Drawing graphs by addition and subtraction of ordinates The student is able to
Sketching functions by division of ordinates The student is able to
Drawing graphs of the form [/(x)] n The student is able to
General approach to curve sketching The student is able to
GRAPHS
GRAPHS Applications, Implications and Considerations

The first emphasis in this topic is on working with graphs of these basic functions to produce a graph of a more complex function (for example, the graph of y = x logex will be developed by the properties of the graphs of y = x and y to consider = logex). After roughening the shape of the curve by adding ordinates, the position of stationary points and inflection points, if any, can be obtained. A good first idea of the behavior of functions of the form f g can be obtained by looking at the graphs of f and g independently.

The exact positions of the stationary point and the inflection point can be determined by calculus. Areas in the number plane in which the graph exists can then be shaded, discontinuities determined, intersections with coordinate axes marked, and the behavior of the function for x —• ±«>.

The number of roots of the equation can be investigated graphically (e.g. find the number of solutions. Consider a seat of point mass M kg suspended from a weightless rod h meters below a pivot placed R meters from the axis of rotation.

COMPLEX NUMBERS

Arithmetic of complex numbers and solving quadratic equations The student is able to
Geometric representation of a complex number as a point The student is able to
Geometrical representations of a complex number as a vector The student is able to
Powers and roots of complex numbers The student is able to

The arithmetic and algebra of complex numbers would then be developed and eventually it could be shown that 2 complex roots exist for a complex number. This then leads to the discovery that a quadratic equation with complex coefficients will have 2 complex roots. Examining graphs of these curves for different values of a and b leads to the conclusion that two roots will always exist for a complex number.

The geometric meaning of the modulus and argument for z = x + iy must be given and the following definitions used:. O with a factor c, should be used in simple geometric exercises. a) Let OABC be a square in an Argand diagram where. Familiarity with the vector representation of a complex number is extremely useful when dealing with curves and loci.

Students must use the expression | can interpret z — (a + ib) | as the magnitude of a vector connecting (a,b) to the point representing z. Students should recognize that the expression arg(z — z{) refers to the angle, which a vector joining the point representing z\ to the point representing z makes with the positive direction of the real axis. cos 9 + i sin 6) = cos n9 + i sin n9 for negative integers n follows algebraically from the previous result.

CONICS

The main properties of an ellipse must be proved both for a general ellipse with center O and for ellipses with given values of a and b. The parametric representation x = a cos 0, y = b sin 9 is useful in graphically displaying an ellipse from an auxiliary circle. The definition of the focus directrix leads to a simple proof that the sum of the focal lengths is constant.

The chord of contact is useful as a tool in proving many properties of the ellipse. The reflecting property of an ellipse can be approximated by the result that the angle bisector of a triangle divides the opposite side into two intervals whose lengths are in the same ratio as the lengths of the other two sides.

CONICS Applications, Implications and Considerations

The most important properties of the hyperbola must be proved both for the general hyperbola with center O and for hyperbolas with given values of a and b. The same geometry theorem, as used in the case of the ellipse, is useful in proving the reflection property of the hyperbola. Geometric properties of the rectangular hyperbola must be proved for the rectangular hyperbola xy = \ a2 for varying values.

Students are expected to be able to proceed from some parametric equations to obtain a locus that can be expressed by a linear equation (perhaps with constraints on x or y).

Continued)

General descriptive properties of conies The student is able to

Applications, Implications and Considerations

INTEGRATION

Integration

Some of the results listed in the standard integral table will need to be established as an appropriate method is developed. Some integrals can be changed to a form that can be integrated using some simple algebra, e.g. Includes squares of all trigonometric functions, as well as those that can be found by simple substitution - e.g.

The work on integration by division must contain the integrands sin- 1 x, e3* cos bx, m x, xn in x (n an integer). Integration by division must be extended to certain types of recurrence relations, e.g. Jxnexdx, j 1^2 cosnx dx. Recurrence relations such as xm( l — x )n dx, involving more than one integer parameter, are excluded.).

Only rational functions whose denominators can be resolved into a product of distinct linear factors, or of a distinct quadratic factor and a linear factor, or of two distinct quadratic factors, are to be considered, e.g.

VOLUMES

The purpose of this topic is to provide practical examples of using a definite integral to represent a quantity (in this case, a volume) whose value can be considered the limit of a suitable approximate sum. Emphasis should be placed on understanding the various approximation methods given, deriving the corresponding approximate expression for the corresponding volume element and proceeding from this to the expression of the volume as a definite integral. Evaluations of infinite series by a definite integral, or of integrals by summation of series, are not covered in this topic.

Volumes of revolution could range from questions involving rotation about a coordinate axis to rotation about a line parallel to the coordinate axis—for example, find the volume of a solid that results when the region bounded by y = lyjx, the x-axis, and x = 4 revolves around the line x = 4. Students should be encouraged to draw a sketch of the shape of the volume to be found and a cross-sectional sketch. They must then derive an expression for the volume of a cross-sectional slice in a form that leads directly to an expression for the total volume as an integral. Express the areas of both shaped sections as a function of y, the distance of the plane from the origin.

The process of writing the limiting sum as an integral must be extended to cases where cross sections are other than circular. These cases should only involve problems in which the geometric shape can be visualized - eg. prove that the volume of a pyramid of height h on a square base of side a I a2h. Two typical strips of width 8r whose centerlines are distance r from the y-axis are shown. a) Show that the indicated strips generate shells of approximate volume 2nf(-t)(s - t)ot, 2nf(t)(s + t)&t respectively.

A donut shape is formed by rotating a circular disk of radius r about an axis in its own plane at a distance s (s> r) from the center of the disk.

MECHANICS

Mathematical Representation of a motion described in physical terms

Resisted Motion along a horizontal line
Motion of a particle moving upwards in a resisting medium and under the influence of gravity
Motion of a particle falling downwards in a resisting medium and under the influence of gravity
Motion of a particle around a circle The student is able to
Motion of a particle moving with uniform angular velocity around a circle

The classical statement of Newton's First and Second Laws of Motion should be given as an illustration of the application of calculus to the physical world. Find when the deck was level with the wharf, if the motion of the tide were simply harmonic. So express the horizontal distance from P to the foot of the wall in terms of h, a.

The initial direction of the beam varies continuously between angles of 15° and 60° from the horizontal. A typical example that requires an understanding of the mathematical description of a motion is question 10(ii), 3 Unit H.S.C. ii) The motion of a pendulum can be approximated by the equation. where x is the displacement at time t. The constants g and L represent the value of gravity at the Earth's surface and the length of the pendulum, respectively.

The pendulum swings in accordance with the initial conditions. using the approximate equation of motion above, find the first value of x where v = 0. c) The motion of the pendulum can be more accurately represented by. Typical cases to consider include those in which the resistance is proportional to the speed and the square of the speed. Analysis of the motion of a particle must include consideration of the behavior of the particle as t increases.

Cases, except where the resistance is proportional to the first or second power of the speed, need not be investigated. If the motion of a particle is considered both upward and then downward, the position of the origin must be changed once the particle reaches its maximum height. Problems should include a study of the complete motion of a particle, projected vertically upwards, which then returns to its starting point.

Resolving these forces in the direction of the tangent and the normal leads to = —mRco2 and. Find the slope of the track to the horizontal, if there will be no tendency for the car to slide sideways.

POLYNOMIALS

Multiple Roots The student is able to
Fundamental Theorem of Algebra The student is able to
Factoring Polynomials The student is able to
Roots and Coefficients of a Polynomial Equation The student is able to

All possible integer roots of such a polynomial therefore lie among the positive and negative integer divisors of its constant term. The "Fundamental Theorem of Algebra" asserts that every polynomial P(x) of degree n over the complex numbers has at least one root. Using this result, the factor theorem must now be used to prove (by induction on the degree) that a polynomial of degree n > 0 with real (or complex) coefficients has exactly n complex roots (each .. counted according to jcTit's muUiplicity) iand can be expressed.as a product - — — /of “exactly n complex linear factors.

POLYNOMIALS Applications, Implications and Considerations

UNIT TOPICS

HARDER 3 UNIT

Geometry of the Circle The student is able to
Induction

Students should have experience in solving more difficult geometry problems than those indicated in the 3 Unit syllabus. The following examples on concyclic points illustrate the depth of treatment Unit 4 students must encounter with all results in the course on deductive geometry. Circles are drawn through the vertices of the four train angles ABH, HDE, FBD and FAE.

In the 4-unit course, students must see proofs by mathematical induction on a variety of topics. Using induction, show that for every positive integer n there are unique positive integers pn and qn such that.