2014 ext2 sols

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

, 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.

First of all here are my solutions:Ext 2 HSC 2012 solutions

After sitting down and doing the paper, it actually was harder than it first appeared, I’d say it certainly is on the same level of difficulty as last year.

Obviously the new format has meant a change in style of the paper, so lets go through it.

Multiple Choice

Some very straight forward questions, and did you notice Question 3? Yes, it was identical to Question 3 in our trial, other than where they place z. (I’m claiming it, so that it is two years in a row I’ve picked a question, pause for some gloating!! )

Q8 & Q10 were probably the trickiest, Q8 needed some thought, and Q10 you really needed to know your definitions of odd / even funcvtions etc.

Question 11

A good chance to pick up some marks, nothing out of the ordinary (f) (ii) made you stop and think, hopefully you remembered the formal definition of |x|, if so should have got it.

Question 12

(a) basic integral

(b) as long as you are careful with your algebra, it shuld have come out nicely

(c) one of the easier reduction formulas, but then they didn’t ask you to use it!!

(d) ah vectors. (ii) I must admit when I got a fixed point for answer to the locus, I thought I had done something wrong, I was expecting some sort of curve. A tricky part.

Question 13

(a) resisted motion, but they gave you the acceleration formula. Disappointing, I would have liked them to make you show you understand resisted motion. Hey now I am looking at the paper again, I think I forgot to answer part (iii)!!! woops.

(b) a simple 2 unit geometry question, what is going on here?

(c) (i) using the definition of a conic makes this easier

(ii) ah that was why they asked b)

(iii) need to play with trig identities, but not too bad

Question 14

(a) partial fractions, hopefully you got this one.

(b) basically draw the reciprocal graph, again OK (by  the way the vertical asymptote turns into an x intercept, and it is included as they did not say draw y = 1/f(x), they actually wrote the new function)

(ii) just a polynomial division, the asymptote is the quotient

(c) Wow, that was tricky. You need to be on your game for this one and there was no lead you through it. I can imagine alot of the state missing out on this one. On a side note, how interesting to have a volume involving circles and there is no pi in the answer (or is that just me)

(d) I thougt this was one of the easier circle geometry questions there has been.

AND NOW LET THE FUN BEGIN!!!!!

Question 15

(a) (i) textbook question, should be a gift mark

(ii) move everything to the left, not too bad

(iii) OK, did you see the connection with the previous parts?? Well done if you did.

(iv) There won’t be too many getting this part out.

(b) (i)interesting, more than just real coefficients, therefore roots appear in conjugate pairs. you need to pint out conjugate of i is – i as well. I wonder how severe they will be, it is only one mark.

(ii) basic algebra, but I can see some overthinking or just leaving it out

(iii) did I hence show? I certainly used part (i) but I never used part (ii), I reckon it is OK

(iv) just used the same idea as part (iii), mayybe they were expecting something else in part (iii)

(v)did you link in the definition of alpha? should be a gift mark

(vi) WOW, that was tricky, how to get root 2 in to the question. There probably is another way of oing it, but I liked this way. I don’t think too many will get it out.

Question 16

(a) (i) there is the gift mark, hope you got it.

(ii) how to link in part (i), there must have been a reason for it. See my solution and the link becomes clear.

(b) (i) tan plus tan on one minus tan tan!!! Should get this one, however funny things happen at the end of the paper, people are getting tired, running out of time

(ii) induction, some heavy algebra, but use part (i) and it does fall together

(iii) if you didn’t get part (ii), you should still have been able to get this one.

(c) The first thing is trying to understand what the question is actually asking, I had to read it a few times. But a really tricky question, I would be surprised if there are any full marks for this part, especially part (iv)

Anyway, that was how I saw it. Now for Extension 1!!