Following on from my last article, we’ll now have a look at some of the more difficult and prominent topics in the year 12 mathematics extension 1 syllabus, with some hot tips and my personal recommendations on how to master them. So without further ado, let’s dive right in!
Topic 1: Proofs
What this topic involves:
You will be asked to prove statements using mathematical induction – this is an inductive method following a series of steps to prove a statement is true
These questions can be to prove sum results or divisibility results
You can also be asked to find the mistake in a line of proof/identify where it doesn’t work
How to approach these questions:
The key to proof questions is really to set out your working in the best way possible, making sure the marker is able to see your reasoning along the way I like to think of proof questions as equal parts logic and equal parts math – include sentences/conclusions that show you have understood the process and what this really means
Set out your proof using the following 3 steps:
• Step 1. Prove the statement is true for the first possible case
• Step 2. Assume the statement is true for some n = k, where k is any positive integer
• Step 3. Prove the statement is true for the next term n = k + 1 using the assumption
The key to these questions is also the assumption that you make – ensure you make the assumption for the correct amount of terms and similarly prove step 1 for one or more terms as needed
Don’t forget the final conclusive statement in order to complete the proof!!
Note: even if you are struggling with a proofs question, I highly recommend you do as many steps as you can as you can get one mark just for showing step 1, and so on. So attempt these questions even if you can’t see the answer right away.
Topic 2: Trigonometric Equations
What
does this topic involve:
Solving trig equations using auxiliary angles
Solving quadratic trigonometric equations
Using the compound angle/double angle formulae and t formulae to solve trig equations
How to approach these questions:
Firstly, familiarise yourself with the trigonometry section of the formula sheet:
o This contains each of the formulas you will be using
When using the t formulae, remember to sub π back into your equation as tan(90) is undefined
When solving quadratic equations involving trigonometry, be aware of the specified domain in which to list your answers, and also sub each one back in to ensure it truly satisfies the equation
Rookie mistake: but ensure calculator is in radian mode, not degrees!
Topic 3: Binomial Distribution
What does this topic involve:
Bernoulli trials
The binomial distribution and finding the mean and variance
The normal approximation for the sample proportion and finding the mean and variance
How to approach these questions:
Firstly, understand that a Bernoulli trial is simply an experiment with only two possible outcomes, either a success or a failure
The success is given a probability of 1 and the failure a probability of 0
A binomial distribution makes use of the Bernoulli trial except it models the probability distribution if this experiment was conducted n number of times
When calculating probabilities, first convert to z-scores and if you aren’t given a standardised probability chart, then think about the normal curve as shown on the formula sheet and use that to estimate probabilities within a certain number of standard deviations from the mean:
And that’s all from me folks! Once again, I hope these tips have been helpful and that you’re cruising steadily through your revision and preparation in the lead up to the exam. In the next article, I’ll be covering some tips on exactly that!
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