Multiple Choice
1. Why bother putting a in the question? All four answers have a = 1 as a possibility. b and c can easily be found without finding a, just sub in x = i
2. Conjugate roots theorem, pretty straight forward.
3. Much easier if you go straight to
$latex e^2 = \frac {a^2 – b^2}{a^2}$
, especially with the possibility of silly mistakes playing with the fractions.
4. Did you use the idea of conjugate on the modulus squared for the reciprocal? Although in the case it would have been z on the modulus squared.
5. Simple rearrangement, mind you if you notioced that x = 1 is not in the domain, there is only one possibility.
6. Basic question, just plenty of opportunities to make silly mistakes.
7. I can’t help it, if I see 1 – sinx in the denominator, and nothing else, create the difference of two squares to get a single term. Mind you it does also help remembering the erivative of secx.
So Questions 1 to 7 I thought were basic bookwork, 8 to 10 require thought.
8. Really need to understand the idea of complex numbers being represented by vectors. Nice little question.
9. What I didn’t like is that once you had detrmined that A works, why do you need to look at the other options? I would of made the correct option D.
10. Great question, so many wrong paths you could go down. At the end of the day a simple substitution solves the problem. Nice!!!
Question 11
(a) Bookwork, only silly mistakes will stop you here.
(b) Integration by parts, but lets make it look harder. In hindsight I probably should have just made u = 3x – 1 instead of splitting into two integrals.
(c) Did you circle the origin? Technically the lines should be dotted outside the triangle, but will they penalise that??
(d) Addition of ordinates. Already in the form we would like to see, Q(x) = asymptote, etc.
(e) Strightforward cylindrical shells question, can’t see any traps here.
Question 12
(a) (i) I think the biggest mistake here would be to reflect below axis above, instead of required reflect right of axis to left.
(a) (ii) Did you circle x intercept at x = -1?
(b) (i) Wow, they really held your hand with this one. A standard type of question, did the “use De Moivre’s to show that cos 3x= etc” for you, so took the fun out of (i), then they told you what to substitute in!!! No wonder (i) was only worth 1 mark.
(b) (ii) Whilst it appears to be some more easy marks, you needed to remember , any three consecutive solutions of the trig equation will do, pi on 6 not pi on 3!!!, had to solve for x, not leave it as theta =. Lots of opportunities for silly mistakes.
(c) No twists here, just have to be careful with your implicit differentiation. 3 marks, I can see why it is 3, but it did seem generous.
(d) (i) Straightforward, inverse trig integral. Easy Ext 1 question, so it should be, thankyou for that mark.
(d) (ii) There is a reason that they wrote the question as I(n) + I(n -1) instead of I(n) =, it is meant to be a hint “just add the two integrals together and it will cancel nicely”. I hope you took their hint.
(d) (iii) Just had to be careful with negative signs, otherwise another 2 marks, thankyou.
Question 13
(a) Straightforward, t-result substitution althought the numbers are a bit fiddly in the substitution.
(b) If you recognise that the triangle on either side of the trapezium is your standard 30-60 triangle (sides are in ratio 1 – 2 – root 3), then the calculation of the trapezium’s height is quick. If you manage to find the area of the trapezium, then the integral is quite simple.
(c) (i) Straight substitution into equation, falls out nicely. Mind you, need to be careful with your algebra skills.
(c) (ii) Like part (i), the algebraic manipulation is the hardest part.
(c) (iii) As long as you remember the eccentricity definition, results falls out nicely.
(c) (iv) If you make t = e, just another substitution question. If you don’t realise this it could turn into a time consuming question.
Question 14
(a) (i) Multiple root theorem, couple of simple derivatives to find. Should be an easy 2 marks.
(a) (ii) Could play around with sum of the roots etc, simpler to just factorise and solve the quadratic.
(b) Now we get a bit interesting
(b) (i) I actually think the result in the HSC paper is wrong. Surely the result should contain the absolute value of sin theta cos theta. If P is in the second quadrant then sin theta cos theta would be negative, thus tan phi would be negative and thus will not find the acute angle phi. Are we meant to assume that P is located in the first quadrant, as in the diagram?
(b) (ii) You didn’t waste time using calculus because they used the word “maximum” did you? Your knowledge of trig functions should have been enough to determine the maximum.
(c) (i) Nice variation on using the terminal velocity. Certainly not a standard text book question. I always like that. Needed to use the terminal velocity in order to eliminate K.
(c) (ii) 4 marks!!! Thankyou. 4 marks for 4 lines of working makes for a happy day. Mind you, it did require you to remember the “non-standard integral”. i.e 1 over 2a log etc… If you didn’t, then some time spent doing partial fractions required. (I told you it would save you time)
Questions 11 to 14 – The unusual thing about this year’s paper is that we are yet to come across any Harder Extension 1 material. This probably means that we are going to have to maximise our marks in these Questions. The good news is that there really wasn’t anything out of the ordinary, so barring silly mistakes 60 marks + 10 marks multi choice, so far we haven’t lost too many marks. A good showing in the next two questions and Band E4 here we go.
Let the games begin……..
Question 15
(a) Inequalities. It just looks like you need to use AM > GM idea somewhere, but you don’t!!! A clever use of a < b < c , along with the expansion they recommend, and it all drops out nicely. Only the strong will get this out.
(b) (i) Nice variation on the “Show that z^n + 1/z ^n= 2cos ntheta” idea.
(b) (ii) Binomial Theorem, explaining your working is going to be the hard part here. There are some sophisticated ideas that need explaining, how do you know x^n is always going to equal one or did you just assume that?, that’s one example. Tough question to get full marks. Mind you, remember the golden rule, 3 or 4 marks, 1 mark should be easy to get. What will they end up giving 1 mark for, perhaps for just realising that you need to let x = i. We shall see.
(c) A nice resolving forces question, it seems like its been a while.
(c) (i) Standard divide vertical force equation by horizontal force equation to eliminate tension. But then a little twist, cos squared phi on the bottom of the fraction. Where did that come from??? Still, if you look at what you are trying to show, hopefully you noticed that r is missing, and needs eliminating, so another equation is required.
(c) (ii) Quadratic inequality, did you spot it. It really was a case of create a quadratic in terms of sin phi and plug into quadratic formula. Even if you couldn’t do part (i) these 2 marks are obtainable.
(c) (iii) Show that it is increasing, that’s 2 unit isn’t it?? Mind you the derivative might be a bit beyond the average 2 unit student!! Remember: to show a function is increasing, show the derivative is always positive.
(c) (iv) A good 1 mark to see who really understands what is going on here. Need to play with the idea of limits and what increasing function actually means. Nice little question.
Overall, I could be wrong, but I thought question 15 may have been tougher than question 16. I am looking forward to see the stats on this one.
Question 16
(a) Circle Geometry, expecting that, redrawing the diagram is going to be fun (I cheated in my solutions, I just photocopied, technically it is okay, question did say copy!!!)
(a) (i) Gift 2 marks: alternate segemant + alternate angles
(a) (ii) A hard 3 marks: I hope you did not show angles DPA and angle BPC are both 90 and therefore vertically opposite angles are equal. This, I think will be the most common mistake. cannot do it, as you do not know that APC is a straight line, that is what you are trying to prove.
(a) (iii) Now you can use vertically opposite angles, hopefully an easy 1 mark to complete the question.
(b) (i) If you recognised the geometric series youcan start the question. If you didn’t, then thanks for coming. I can imagine a high number of 0 marks here. If you do then 1 mark should be fine, but then the other 2!! Recognising you had to look at both the cases of even n and odd n, which traps the series between two expressions. Even then you had to use the idea that x^2 +1 > x^2, to get the required result. Tough question to get full marks. Only for the best.
(b) (ii) Gave us the hint of integrating, but, did you realise that it needed to be a definite integral between 0 and 1?
(b) (iii) The sandwich the inequality between the same values idea!!! Investigate the sum to infinity
(c) Wow this integral looks unusual, but if you try the add 1 and subtract 1 idea, magic happens. The first integral we do by parts, but don;t tough the second integral. Parts produces its negative, so no working is required. I told you it was magic. Last question, 3 hour paper, not many getting this one out.
Overall
I hate comparing papers betwen different years, they all have their gift marks, all have their different twists and turns. If I was forced into an opinion, I would say easier based on one thing alone;
Only 19 marks for Harder Extension 1. As you know, the hardest questions on an Extension 2 paper are the Extension 1 questions, 19% Harder Extension 1 is the lowest since 2011, the year prior to multi choice. For the record it had 22 marks in a 120 mark paper (18.3%). It is these “I’ve never seen a question like this before”, that makes the Extension 2 paper challenging, I just would like to see the usual 25 to 30%. The Syllabus states 30% of time should be devoted to Extension 1 work, so 19% of the exam is disappointing. Just my opinion, for what its worth. There are many that would disagree with me.
And from a personal point of view, as I mentioned before, really disappointed that a couple of old favourites where missing Perms & Combs and Induction. Oh, well, can’t have everything.
Having said that, the balance of marks for the Extension 2 topics was probably spot on with what was expected. Mind you, I didn’t have to do the paper under exam conditions, that alwasy makes a paper significantly harder than previous years. So if you feel it was harder under exam conditions, you are probably right. If you did find it hard, remember, if Extension 2 was easy, everybody would do it!!! Only the best 3000 to 4000 out of 70 000+ students are doing the paper, so well done, regardless of the final result.
So now we focus on Tuesday afternoon.