Sydney Morning Herald Mathematics Survival

In the frantic preparation for exams and the desire to maximise your marks, it is easy to lose sight of the relevance of mathematics to life beyond the Higher School Certificate.

Studying mathematics is not an end in itself. The ways of thinking and solving problems developed through mathematics are applicable in both the professional and personal areas of your life.

The process and strategies used in mathematics become the important long-term learning goals.

Doing as well as possible in the examination is, however, an important short-term goal and it is important to learn from the past.

After each HSC exam, the examiners produce a report for each subject. This is to help students in the following year to improve their performance. Each year many of the comments are similar. The following comments from the 1999 HSC Examination Report are typical.

"Candidates should be aware of the fact that it is their responsibility to indicate to the examiners the process by which they have obtained their answer.

"In marking, each individual mark is allocated to a step or process, which is essential to a correct solution of that question.

"Those who provide sufficient evidence that the appropriate step, or its equivalent, has been completed are awarded the mark, which then cannot be lost for a subsequent error.

"Candidates who give only a single word or figure as their response forgo any possibility of earning any marks unless their answer is completely correct.

"Sometimes, in cases where examiners believe that the correct answer is easily guessed without doing the work required to establish the result, a mere correct answer without any supporting justification may not earn all of the available marks."

Use the reading time at the start of the examination to get a feel for the papers so that you are aware of the parts that you feel that you can do easily.

Work through the paper quickly but carefully, and if you come to a part that you feel you cannot work out, leave it and move on to later parts.

Mark on your question paper those parts that you leave, so that you can come back to them at the end. It is often better to start the question again when you come back to it. Most of us find it difficult to locate our errors by re-reading our work.

Do not cross-out your first attempt until you have replaced it with a better attempt. Should you need to cross-out work, simply put a line through it. Do not obliterate the work or use white-out, as this failed attempt may gain you marks if it shows evidence of correct processes.

2-Unit Mathematics
http://hsc.csu.edu.au/maths/courses/#course1

Remember to use the table of Standard Integrals given on the back of the question paper. The information provided in the table can help you to check your differentiation.

Half of the 1999 examination paper involved calculus and its applications. This is similar to previous years. When finding indefinite integrals or primitive functions, do not forget to include the arbitrary constant, C.

It is also important to understand the difference between finding the value of a definite integral and the area under a curve. These ideas are related but they are not identical.

Question 5(c) on the 1999 examination paper asked students to evaluate a definite integral, using areas shown on a diagram. The value of the definite integral was determined by adding the areas above the x-axis and subtracting the area below the axis. Many candidates did not attempt this part at all, even though the calculation was straightforward.

When finding areas and volumes it is a good idea to have a sketch of the function with the appropriate region shown.

The expressions differentiate, find the equation of the tangent or normal, find the stationary points and determine their nature, local maximum or minimum, minimise, velocity and acceleration, will generally require you to differentiate a given function at least once. You may then have to substitute a value into this derivative or solve an equation formed by putting the derivative equal to zero.

You can expect to be asked to differentiate functions in which you will have to use the "function of a function" rule, product rule, quotient rule or combinations of these rules. To help you remember these rules you should always write down the rule in all practice examples.

When you have to differentiate or integrate trigonometric functions, such as sin x, cos x, tan x or sec²x, remember that x must be in radians. If the question is in degrees, remember to convert, using the result that
radians

NSW HSC On-Line http://hsc.csu.edu.au/maths/courses/2unit/trigfunc/ has a range of useful tutorials on trigonometric functions.

Geometry questions require you to give reasons for your answers. It is important that your reasons contain enough information to make your reasoning clear. When dealing with angles from parallel lines, you should name the parallel lines that you are using. Your reason should say "Alternate angles equal, AB // CD" in preference to "Alternate angles". Always copy the diagram into your answer booklet, as any information that you mark on the diagram is then available for the marker when assessing your work. It might compensate for a poorly worded reason and get you the mark.

Questions on arithmetic and geometric series often assess practical applications of series work. You may need to identify the required mathematical information in the question and write it as a series before you can answer the question.

3-Unit Mathematics
http://hsc.csu.edu.au/maths/courses/#course2

The depth of mathematical understanding required in 3-Unit is greater than that of 2-Unit and this is reflected in the difficulty of the paper. It is important to remember that the 3-Unit course includes the 2-Unit course and might include some questions that are based on harder 2-Unit work. The word hence is frequently used, telling you that you should use the result of the previous part in what you are now doing.

The first question usually consists of unrelated parts drawn from the whole syllabus and tests your skills. In the following questions, the first part is often easier than the last part of the preceding question. Use the reading time to pick out the parts of the paper that you can do easily and do them first.

You need to know when to use methods for counting combinations or permutations based on the words used in the question. When selecting committees, order is not important, but when arranging objects, order is important.

Writing proofs for the topic of circle geometry requires information to be given in a logical order, with appropriate reasons stated. You should copy any given diagrams into your answer booklet. When a question asks you to show or prove several results in order, use the result given in an earlier part, even if you haven't been able to prove it. Many questions are intentionally designed to build up to a result by guiding you through the intermediate steps.

The trigonometric results for the sum and difference of angles and double angles must be known. They are used in solving trigonometric equations, finding integrals such as

and evaluating some expressions involving inverse trigonometric functions.

Parametric equations and the parabola terminology from 2-Unit are often used on the 3-Unit paper. Finding the locus of a point requires you to find the parametric co-ordinates of the point and then eliminate the parameter to determine the Cartesian equation of the locus.

When proving results using mathematical induction, make sure that your steps follow in logical order and that you finish your proof with statements outlining the process of mathematical induction.

On the mathematics node of NSW HSC On-Line in the mathematical induction section http://hsc.csu.edu.au/maths/courses/3unit/math_ind/ there are a variety of induction questions for you to practise.

Questions on projectile motion usually ask you to show the results needed, so simply stating the formulae will not gain any marks. You need to understand and be able to work through the process and then use the results. Simple Harmonic Motion (SHM) questions often require you to be able to rearrange a given equation in the standard form to show that the motion is SHM and then state the period and amplitude of the motion.

Binomial theorem questions have often appeared in the final question in the paper. You should practise manipulating binomial expansions in various forms.

The table of standard integrals may be required to evaluate integrals that are of the form given in that table. For example, "Show that

requires you to substitute into
and then simplify the result.

4-Unit Mathematics
http://hsc.csu.edu.au/maths/courses/#course3

This course includes the whole of the 2-Unit and 3-Unit courses. In fact, the hardest parts of the examination are usually based on the harder 3-Unit work. Many students do not complete this paper in the three hours allocated.

Being able to use time-efficient methods of solving the problems helps to relieve the time pressure.

It is important that you work through this paper, doing those parts that seem the easiest to you, without getting tied up in difficult algebra. Then come back and complete the parts that you have left. Use the reading time to plan the order in which you will complete the questions.

The first question often involves different integrals and tests the various methods of integration studied in the course, such as integration by parts and the use of substitution and partial fractions. Students who have had plenty of practice do well in this question.

The question assessing complex numbers is usually well done by many students who have prepared adequately for the examination. The other early questions tend to cover the 4-Unit course. The topics: conics, graphs, polynomials, volumes and mechanics all have additional examples on NSW HSC On-Line (http://hsc.csu.edu.au/maths/courses/

Work through these questions quickly but carefully. It is not unusual for Questions 7 and 8 to be very challenging, even for the best-prepared students.