Category: Mathematics

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Maths Extension 2 tips from our top Maths tutors

Mathematics is one of the most rewarding subjects for your HSC, especially at the extension levels, where it’s one of the highest scaling subjects you can choose. Each year, the majority of students that score a 99+ ATAR did very well for Maths Extension 1 and 2. For maths extension 2 especially, each year the majority of the 21 or 22 students that score the perfect ATAR of 99.95 get 97+ for maths extension 2 as their HSC mark.

Even if you’re not aiming for such a high ATAR, doing well in any level of mathematics will get you closer to your goal in no small amount.

With every subject, there’s the content, then there’s technique. This fact is especially significant in maths extension 2 – things like knowing shortcuts / quick methods to verify the correctness of answers, how well you know your way around your Board-approved calculator, how well you understand the marking process and how to get partial marks, etc.

We all know the commonly told tips that your teachers at school have no doubt told you many times by now (e.g. writing partial answers for partial marks). Here’s a few lesser known tips our maths tuition classes give our students as part of our Extension 2 maths course:

  1. Volumes – easy way to check answer for volumes

Here’s a useful tip for the Volumes topic in Maths Extension 2 – Pappus’ Centroid Theorem. This theorem states that the volume of a solid of revolution generated by rotating a plane figure about an external axis is equal to the product of the cross-sectional area and the distance travelled by the cross section’s centroid.

Using this theorem, we can calculate all questions involving rotating a shape around an axis (e.g. circle around an axis to produce a torus) simply by finding the area of the cross section, then multiplying this with the distance travelled by the centroid of the cross section. It’s much quicker and reliable (because it’s simpler – less can go wrong) than using any of the prescribed methods, e.g. cylindrical shells or adding slices.

For example, what’s the volume of a circle, centre origin, radius 4, rotated about the line x=6? The answer is simply: the area of the circle – 16π multiplied by the distance travelled by the centre of the circle, which is 12π. The volume is therefore 192π^2 or 1894.96 cubic units. You can verify this with the cylindrical shells method.

Of course, in answering an exam question, you need to use the cylindrical shells method. The Pappus’ Centroid Theorem is just a useful tool to quickly check the correctness of your final answer. If there’s disagreement, you know you need to check your answer, and this would have easily saved you some lost marks!

  1. Volumes, integration and other applications – the area of an ellipse and the volume of an ellipsoid

Here’s another tip to make you quicker at checking your integration results. It’s also helpful for other topics (e.g. volumes).

It’s a good idea to memorise the area of an ellipse:

And the volume of an ellipsoid:

These are not too hard to remember because in the special case of the circle and sphere, r=a=b and r=a=b=c respectively, which causes both equations to reduce to the equations of the area and volume of a sphere respectively.

There are occasionally situations where you have a definite integral you need to evaluate, where the expression to be integrated is in the form of an ellipse. For volume questions, solids of rotation would often form ellipsoids and knowing the simple formula could give you a quick tool to check the correctness of your answer.

  1. Don’t be afraid to include explanations as part of your solution

Here’s a tip our extension 2 maths tutors love to give our students. In many situations in mathematics, especially in maths extension 2, some explanation saves you a lot of calculation.

For example, if you’re doing an integration and the form is that of a circle, instead of going through the x=sinα substitution, it’s easier to just write “This represents the area of a semi circle with radius r” – that’s all you need, then you can write the answer. Or if you need to evaluate a definite integral of an odd function with symmetrical limits, then the answer is always 0.

Sometimes you need to take cases – explain why some cases are impossible and this will save you time because you don’t need to cover them. Sometimes a graphical solution will show why one graph will never intersect another graph (hence no real roots to a related equation). The point I’m making is explanation goes a long way. The entire Mathematical Induction topic in 3 unit is based on explaining rather than using purely numbers, symbols and algebra.

Another situation would be proof questions (and we get many of these in Extension 2) that require you to prove LHS = RHS. Instead of starting somewhere random, you can start with what you’re trying to prove and write “If the above is true (what we’re required to prove), then:” and you can proceed to manipulate what you’re trying to prove. Just write ‘Then’ at the beginning of each line to signify you haven’t proven it yet, but if the original LHS = RHS, THEN so far these lines of working must all be true. Eventually, when you reach a situation that is in fact true (e.g. 1=1) then you’re finished, just write “And since the final line is true, the original LHS = RHS).

  1. Integration by parts – LIATE

We all should know how to do integration by parts by now – but sometimes choosing which one should be u and which should be v can get tricky. A reminder, here is the definition of integration by parts:

OR

Choosing which one should be u (or f(x) if you prefer the first line) is an important decision – choose the wrong one and you’ll waste precious exam time going down a path that may lead to an impossible integral.

The general rule of thumb is to remember LIATE. LIATE stands for:

  • Logarithm
  • Inverse trigonometric functions
  • Algebra (general polynomials)
  • Trigonometric functions
  • Exponential

You should give preference to the left-most function to be set as u. The reason for this is as you move from the left to the right in LIATE, functions become easier to integrate, so you should prefer to integrate the easy ones (e.g. trigonometric functions and exponentials are easy to integrate) and differentiate the hard ones (e.g. logarithms can’t easily be integrated).

  1. L’Hopital’s rule

In the Graphs topic, there are many compound graphs that give rise to situations where you have infinite multiplied by 0. When this limit occurs, 3 possibilities arise:

  • Graph turns to 0
  • Graph turns to infinite
  • Graph turns to a constant

To find out how, we suggest to students to learn how to use L’Hopital’s rule to discover the relative speed of curves. This rule only applies when the limit below turns to infinite or negative infinite, or 0.

For example, lets see what happens to the graph at y=xlnx close to x=0. We set f(x) = lnx and g(x) = 1/x so that f(x)/g(x) = xlnx. When we evaluate the limit for f’(x)/g’(x) we find it equals (1/x)/(-1/x^2) which equals –x. This would turn to 0 as x approaches 0 so this shows the graph y=xlnx turns to 0 instead of negative infinite at x near 0. Incidentally, this is also one of our HR manager’s favourite questions to ask potential candidates for maths tutors, even if they tutor lower levels of maths (you’d be surprised how many people apply to be a mathematics tutor but can’t even graph y=xlnx).

  1. Use your calculator memory effectively

Become proficient in the use of your Board-approved calculator that you can take with you to the exam room. If you’re swift with using your calculator’s memory slots, this makes rechecking over your answers so much faster and accurate (if you happen to finish your exam with 20 minutes to spare, you can literally go through the entire exam once or even twice! If you’re good with your calculator, that is). And this helps even more in subjects like HSC Physics and HSC Chemistry where there’s lots of calculator work that requires a final definite numerical answer.

Another reason is exact values. Your calculator’s memory actually stores something to the order of 100 digits (much more than what’s only shown on screen when you press ‘=’). Sometimes when you’re supposed to get an exact value, if you put in your written down answer (that’s only rounded to 3-4 digits at most) you will get some answer like 1.99837734 but if you subbed in your memory-stored calculator, you will get an exact value of 2. In complex questions, even simple clues like knowing something is an exact value could be the difference between doing the question and skipping it altogether.

  1. Multipart questions (attempt part 2)

We should all know to be able to attempt the next subpart of an exam question by now. E.g. if you can’t do part a, which required you to prove a result to be used for part b, you should use the result to do part b so that you can still score partial marks for the question.

Sometimes even within a single part worth multiple marks, there are two parts to the question (e.g. “Show that LHS = RHS and hence derive an expression for acceleration of the particle.”) In these examples, if you can’t do the question as they intended, just use the result you’re given and finish the second part of the question for partial marks. If the question was worth 3 marks, you should at least get a mark for your efforts.

  1. Maths tutoring helps

Sometimes if you’re stuck with a teacher at school that’s not very knowledgeable, it would be wise to seek outside help. This situation is particularly common in disadvantaged schools – not all schools even offer extension 2, and for schools that do, not all their teachers assigned to the course are actually capable of teaching all parts of the course effectively. Getting outside help also gives you an important advantage over your peers when it comes to internal assessments (where you’re up against your peers for the top assessment ranks).

General tips for maths exams

We will start with the obvious things that you may have heard before. These tips may sound obvious, but they’re among the more important / commonly applicable ones, so be sure to remember them!

  • Cross out incorrect answers with a single line

HSC Markers read everything that can be read, even if you’ve crossed out an answer. If you’ve written an answer but change your mind afterwards and write another answer, cross your old answer out with a single diagonal line using your pen. Do not use liquid paper. This ensures that even if your final answer is wrong, there’s more chance you’ll receive partial marks for the question (as long as the marker can see you did SOME things correctly).

  • Show ALL working out

Some students prefer to write things out step by step – that’s generally the better / safer approach, as showing working out ensures you will get at least partial marks, even if your final answer is incorrect.

In the past, one of our top students (who later went on to achieve a state rank) preferred to do entire questions just by using his calculator’s memory, storing everything into the A, B, C … to M memory slots! We always had to remind him to remember to write out his ‘working out’ after he wrote his final answer – it was also a great way to check his answer.

  • Look for clues from previous parts of a question

All HSC maths exams (from General maths, 2 unit to Extension 2) structure their questions in terms of part a, b, c, etc. Use the answer from the previous parts as a clue to your current part (even if it’s not a ‘hence’ or ‘hence or otherwise’ question).

  • Use your calculator’s memory!

For questions / parts that require you to use a numerical result from a previous question / part, you’re better off using the stored number in your calculator rather than your rounded written answer. This applies especially true in subjects like HSC Physics and HSC Chemistry where you’ll be doing much more numerical calculations.

For Mathematics Extension 1 & 2 students

  • Work a proof question from BOTH sides

For questions that require you to show LHS = RHS (e.g. typical induction questions like “Show that f(x) = g(x) is true for all x > 0″), realise that you don’t need to work strictly from LHS to RHS.

Instead, start with the LHS, see if you can simplify it / progress it as usual. Then when you’re stuck, check the RHS and try progressing with that. Usually you will find this approach makes equating LHS and RHS much easier.

Think of these types of questions as requiring you to make LHS and RHS meet, but there’s a valley in the middle. Instead of pushing LHS all the way through the valley (down the valley, then up the valley), push LHS all the way down, then push RHS all the way down, so they meet at the bottom.

  • Don’t be afraid to use graphs as part of your answer

Sometimes, graphs are appropriate as part of a mathematical proof. For example, if you’re required to prove some inequality, you can use a graph (and some calculus of course) to show that a line is tangential to a curve, in order to support your inequality.

  • REMEMBER the definition of the log integral:
  • int dx/x = ln |x| + C

Remember that when you integrate 1/x you get the log of the ABSOLUTE VALUE of x, not just x by itself. Although you won’t lose a mark for not including the absolute value signs, some questions with definite integrals (e.g. requiring you to find the area under a curve) will result in logs of negative numbers and hence impossible to evaluate unless you remember to include the absolute value signs. Don’t get tricked!

  • Strategies for ‘hence or otherwise’ questions

In multipart questions, the last part is usually either a ‘hence’ or ‘hence or otherwise’ question. When you have ‘hence’, you have no choice but to use the previous result(s) to do the question. When you have ‘hence or otherwise’ you have an option either to use your previous result(s), or take a wholly new route to the answer.

Here’s the tip: if you can see that the question reduces to anything you recognise, its often actually FASTER to use your ‘otherwise’ option. For example, in tricky Extension 2 question 8 type questions, you are often required to show LHS = RHS, or LHS > RHS, or LHS < RHS. If you can re-formulate the equation into something you recognise, then it’s just a matter of writing out your proof for that thing you recognise, then reshuffling it back into the required form.

The reason why this is a better approach is because for harder questions, the amount of time you could sit there potentially thinking (on how to do it using your previous result(s)) is highly variable (could take a very long time), and risky (you may not even see the answer after spending plenty of exam time). If you can reduce it to a recognised form and write out a memorised proof for it, even if it’s not the most elegant / efficient proof, you will score full marks, and the time you take is only dependant on how much you need to write out.

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