Question 1, HSC 2009 Extension 1 exam

2009 HSC, Extension 1 Mathematics Question 1 (12 marks)
(a) Factorise 8x^3 +27 (2 marks)

Whew! This question is lovely, because in terms of factorising a cubic, this is as easy as it gets. It's of the form (a^3+b^3), where a=2x and b=3. This is really easy if you can remember the formula:

(a^3+b^3)=(a+b)(a^2-ab+b^2)


If you don't remember the formula, don't fret! You can look at this problem and kind of just figure it out by inspection and a little trial and error.

We're dealing with cubics, and we've got 8, x^3, and 27.



So 2 is the cube root of 8, x is the cube root of x^3, and 3 is the cube root of 27. Even if you're just guessing after this, you've got a pretty good idea of where to start.

(2x+3)

The reason I would 'trial' with the (+) is simply because the original polynomial has a (+). If it doesn't work out, try again with a minus sign. As I mentioned before, this is just if you don't remember the formula and you're stuck but still want to spend the time to work it out. I wouldn't do trial and error in an exam unless I was pretty sure I knew I was heading in the right direction. It's not worth spending lots of time on something if you're going in totally blind.

Anyway, so you have (2x+3) as one potential factor. Now let's see what's left over when we pull that out.

(2x+3)(4x^2-6x+9)

8x^3 +12x-12x+18x-18x +27

YAAAAAY!!!




(b) Let f(x)=ln(x-3). What is the domain of f(x)? (1 mark)

Because this is one mark, it means you should be able to do this by inspection based on your knowledge of the domain of the natural log function, or it should take just one or two simple steps. This should not take you longer than a minute.

What do we know about the function 'ln'??

Well, ln0 is undefined. Basically the ln(x) function is saying 'e' to the power of what number equals x? So 'e' to the power of what number equals zero? In this case, that number is undefined. Equally, e to the power of what number equals a negative number? Also undefined.



So now we can very easily answer this question. What are the values for x that satisfy the condition that (x-3) must be greater than zero?

(x-3)>0

Add 3 to both sides to have x as the subject and you get...

Domain: x>3


(c) Find lim ((sin(2x))/x) (1 point)
x->0

That looks confusing trying to type it out, but you can see below. Once again, this is a 1 point question so should take you a minute or so. Just a couple of steps.

Okay, there are a couple of things to note here.

First of all, it's a good idea to just know this:

If you know this, then you should know how to do this question. It is based on l'Hopital's rule.

This working should be sufficient for a 1 point question in the HSC. You may want to just say something about "from l'Hopital's rule..." or whatever but really for 1 point I can't imagine they'd need a proof as long as you know it.

Here is an explanation of how to use l'Hopital's rule:

(d) Solve the inequality (x+3)/(2x) > 1 (3 points)

Okay, so since this is three points we probably need a few steps, but solving inequalities should just be a mechanical process and if you have to stress about this or you're spending too much time thinking about it rather than just going through the motions of solving it, I suggest you just do a whole heap of these in preparation for your exam because it should be easy marks. Anything mechanical like this you want to just be able to churn out with no stress or effort. There will be longer harder questions later that deserve that stress and effort, but not this one.

Okay, so we want to find the critical point. We know that x can't equal zero because the term '2x' is in the denominator and this would be undefined.

So change the inequality into an equality for the time being to find the critical point.


(e) Differentiate x(cos^2(x)) (2 marks)

Once again, a mechanical question. Worth 2 marks, so I guess 1 mark for knowing what the product rule is and another mark for actually getting the correct answer.

The product rule is what you use to differentiate a function that is two functions multiplied together.

In this case, one function is x and the other is cos^2(x).

The product rule is...

f(x) = uv, where u and v are functions

then,

f'(x)= uv'+vu'

Pretty straight forward. You just need to memorise it.

(f) I have no idea how to do integration symbols and whatnot keyboard, so here's part (f) written out. It's worth 3 marks.


So there you go, question 1 of the 2009 extension 1 mathematics exam.

There's nothing in there that should cause any problems if you practice. Don't waste time writing out pretty notes about stuff or doing flashcards or whatever. Just do actual past papers! Do question 1 of 2009, 2008, 2007, 2006, 2005... You will start to see the same types of questions over and over again and this is how you really truly internalise it. Don't sit around memorising what l'Hopital's rule looks like if you have no idea how or when to apply it.

Anyway, with Question 1 it's often all mechanical, going through the motions type maths.

If you have any questions, please let me know I'm happy to help!

xx

How to... win a car instead of a goat

My high school maths teacher (Dr V! I wonder what he's up to these days...) went through this probability question with us in a class one afternoon. I still love it, because it's totally logical but for some reason seems illogical when you first hear it. It's one of those things where it either makes you go "ugh, THIS is why I hate maths!" or, like me, you go, "ah, THIS is why I love maths!" So simple if you look at it one way, but totally confusing if you're thinking about it a different way.

The Game Show Problem

So, you're a contestant on a game show and there are 3 closed doors.
Behind one door is a brand new car. The other two doors have a goat behind each one.

The game show host asks you to pick the door you think has the car. He then opens one of the other two doors, revealing a goat.



After opening the door, the host asks if you would like to switch your choice of door to the remaining closed door.



Should you switch your choice to the remaining closed door??

Well, instinct says it really doesn't matter because it's 50/50 at this point, right? There are two closed doors. One has a goat, one has a car.

But no.

You should actually change your choice to the other closed door. You will DOUBLE your chances of winning the car if you do! Weird, huh?

I remember sitting in class and we all looked at our teacher like "um... that's stoooopid, it's 50/50 chance at this point, so why change? duuur..." (because, you know, we as teenagers knew everything about everything... except probability, it turns out...)

But think about this -
Okay, so originally you have a 2/3 chance of choosing the goat and 1/3 chance of choosing the car. So, you have double the chance of choosing the goat as you do the car, right?

Then if one of the goat doors has been revealed and thus that option has been removed from the choices, that means that since your original choice was twice as likely to be a goat as a car then it's twice as likely that the other door is the car.

So switch!!!

If that doesn't make sense still, think about it for a while and it will come together. One of the best pieces of advice my maths lecturer at uni gave to me when I just could not understand a particular theory was that if I look at it long enough, even if I'm just kind of staring at it, it will start to come together eventually. This works.

Xxj

How to... find the size of an interior angle of a right triangle using sin/cos/tan

So yesterday's post was working with right triangles and trig functions when you know the interior angles and you have the length of one side of the triangle, but what about if you want to find the size of an interior angle? Well, we use the same trig functions - sin/cos/tan - as we do if we're trying to find the length of a side except we use the inverse.


By using the inverse functions we can find the size of the interior angle. It's important when you're doing this to make sure you still get your SOH CAH TOA correct and pay special attention to which side is the opposite (O) or adjacent (A) side in relation to whichever angle you're wanting to find. It's a common mistake to go "oh, I use sine" and then switch up the O, A or H. This is particularly common if there is a drawing and the right triangle is positioned with the hypotenuse facing the bottom of the page, like this...



Anyway, here's an example of using inverse trig functions -

Now, if you put this into your calculator it will probably return the answer 36.8698 etc.

Generally we don't talk about degrees with decimals, but rather as degrees and minutes. Just like with time, there are sixty minutes in one degree as there are sixty minutes in one hour. To change it on your calculator to read as degrees and minutes rather than with a decimal, find the button on your calculator that looks like it is a couple of apostrophes and degrees signs (the little bubble apostrophe thing :) ). It may be different on different calculators, so play around with it and you'll know when you've found it because it will change your answer to read as degrees and minutes. Don't wait for an exam situation to do this :)

Xx

How to... find information about a right triangle using sin/cos/tan

Before you use these rules, it is important to ensure the triangle you are working with is, in fact, a right triangle. Never assume a triangle is a right triangle! This is a rookie mistake. Unless the angle is marked as 90 degrees, assume it is not until you can prove it.

Okay, so if you really do have a right triangle, you can find relationships between the interior angles and the lengths of the sides of the triangle using the trigonometric functions sine, cosine and tangent (i.e. sin, cos and tan).

Now, PLEASE don't judge my handwriting. I just bought a Bamboo pad and pen. I love it, but it feels pretty wonky for now... I'm hoping I'll get better at it once I get more practice in so you won't have to read math tutorials that appear to be written by a 5 year old...
Ergh, I know it looks awful! But that's all there is to it. If you say "SOH CAH TOA" out loud a couple of times it will just be etched into your brain forever. I'm not sure why, but it's one of those things everyone seems to remember... That doesn't necessarily mean people remember what it means or how to use it, but the sounds are easy enough to remember.

Here's an example... Again, excuse my awful scrawl. Practice makes perfect :)




You might have noticed an interesting relationship between sine and cosine. Oh god, so nerdy. Interesting? Yeah, I know what you're thinking. But the more of this stuff you do the more you will realise how everything is related and you just start to see relationships all over the place. Before you know it you'll be going "Ngaaaa" and yelling out "PI IS EXACTLY THREE!" to shock people into being quiet and getting their attention. Yes, the Simpsons episode where Lisa discovers the bully antidote. Anyway, back to the interesting relationship! In the example above, sin40' turns out to be the same as cos50'.

In fact, sin(x)=cos(90-x) and cos(x)=sin(90-x).

Anyway, this post could continue for pages and pages as I go on and on about different trig functions. I'll post some more stuff in the coming days, but for now I hope this has been of some use to you. Please send through any questions you may have. I'm happy to answer or clarify anything.