HSC 2016 MX1 Marathon (archive) (1 Viewer)

Arucerious · Nov 21, 2015

Re: HSC 2016 3U Marathon

A polynomials question
How to do c) and d)?

leehuan · Nov 21, 2015

Re: HSC 2016 3U Marathon

$\\ a{ \left( x-1 \right) }^{ 2 }+b\left( x-1 \right) +c\\ =a{ x }^{ 2 }-2ax+a+bx-b+c\\ =a{ x }^{ 2 }+\left( -2a+b \right) x+(a-b+c) \\ $Hence $a{ x }^{ 2 }+\left( -2a+b \right) x+(a-b+c)\equiv { x }^{ 2 }$

From there, we can equate coefficients to deduce directly that a=1,
-2a+b=0 -> b=2
a-b+c=0 -> 1-2+c=0 -> c=1

d) is done similarly but is more tedious because of the simultaneous equations that will follow.

leehuan · Nov 22, 2015

Re: HSC 2016 3U Marathon

Basically all I meant was
sin(x)/cos(x) = (O/H)/(A/H) = O/A = tan(x)
-----------------------------------
NEXT QUESTION
$\\ $a) Explain the geometrical significance of the result $\frac { dy }{ dx } \frac { dx }{ dy } =1 \\ $b) Show that $\frac { d }{ dx } \arcsin { x } =\frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } }$

davidgoes4wce · Nov 25, 2015

Re: HSC 2016 3U Marathon

There was a mistake in Grove (2nd edition) Exercise 1.6 Q 3e. The solution showed the wrong letter.

davidgoes4wce · Nov 25, 2015

Re: HSC 2016 3U Marathon

Exercise 1.9 Q7b, there was also a small mistake:

InteGrand · Nov 26, 2015

Re: HSC 2016 3U Marathon

$\underline{NEW QUESTION}$

$(i) Show via mathematical induction that $\cos \left(n\pi\right) = (-1)^n$ for all non-negative integers $n$.$

$(ii) Deduce that the result in (i) holds for all integers $n$.$

davidgoes4wce · Nov 27, 2015

Re: HSC 2016 3U Marathon

There was a mistake in Grove Exercise 2.2 Q 16

Drsoccerball · Nov 27, 2015

Re: HSC 2016 3U Marathon

Firstly it's a typo n can often be confused with h. Also why are you posting this on 2 unit/general question on a 3U marathon? Lastly who cares if they got it wrong...

davidgoes4wce · Nov 27, 2015

Re: HSC 2016 3U Marathon

I'm one of these guys that does every question. Some people call me 'mental', 'crazy' but I just like pointing mistakes out even if noone cares.

There is another mistake E2.7 in Prelim Extension 1 by Grove, Q 30.

Factorise 24a^2b^5+16ab^2

The answer should be: 8ab^2(3b^3+2).

Grove didn't fully simplify it she had: 4ab^2(6ab^3+4)

I'd rather spend Friday nights finding out the smallest Grove error than going out. That's me. I've been called 'weird' in the past and I don't mind or care what other people think.

Drsoccerball · Nov 27, 2015

Re: HSC 2016 3U Marathon

davidgoes4wce said:
I'm one of these guys that does every question. Some people call me 'mental', 'crazy' but I just like pointing mistakes out even if noone cares.

There is another mistake E2.7 in Prelim Extension 1 by Grove, Q 30.

Factorise 24a^2b^5+16ab^2

The answer should be: 8ab^2(3b^3+2).

Grove didn't fully simplify it she had: 4ab^2(6ab^3+4)

I'd rather spend Friday nights finding out the smallest Grove error than going out. That's me. I've been called 'weird' in the past and I don't mind or care what other people think.

Maybe you should make a thread named "Pointing out mistakes in textbooks" where you an derail that as much as you want?

leehuan · Nov 27, 2015

Re: HSC 2016 3U Marathon

Tbh, yeah, if you want to correct textbooks you really should make your own thread. You can post SOS questions on the marathons but that's about the extent lim marathon->post (knowledge) goes.

Failed maths joke I know.

Drsoccerball · Nov 27, 2015

Re: HSC 2016 3U Marathon

leehuan said:
Tbh, yeah, if you want to correct textbooks you really should make your own thread. You can post SOS questions on the marathons but that's about the extent lim marathon->post (knowledge) goes.

Failed maths joke I know.

You should definitely make another thread for yourself as well.

leehuan · Nov 27, 2015

Re: HSC 2016 3U Marathon

Drsoccerball said:
You should definitely make another thread for yourself as well.

I don't always make maths jokes. But when I do, I'm so asymptotic to creative.

Nah I usually don't drop a maths joke on BOS anyway lol

Drsoccerball · Nov 27, 2015

Re: HSC 2016 3U Marathon

leehuan said:
I don't always make maths jokes. But when I do, I'm so asymptotic to creative.

Nah I usually don't drop a maths joke on BOS anyway lol

I don't always make jokes about complex numbers...but when I do, it ends in an argument.

leehuan · Nov 27, 2015

Re: HSC 2016 3U Marathon

Drsoccerball said:
I don't always make jokes about complex numbers...but when I do, it ends in an argument.

Can you modulate yourself and stop with me right now

Ekman · Nov 27, 2015

Re: HSC 2016 3U Marathon

Drsoccerball said:
I don't always make jokes about complex numbers...but when I do, it ends in an argument.

Can confirm, we sometimes argue the mathematical validity of his jokes

calamebe · Nov 29, 2015

Re: HSC 2016 3U Marathon

leehuan said:
Basically all I meant was
sin(x)/cos(x) = (O/H)/(A/H) = O/A = tan(x)
-----------------------------------
NEXT QUESTION
$\\ $a) Explain the geometrical significance of the result $\frac { dy }{ dx } \frac { dx }{ dy } =1 \\ $b) Show that $\frac { d }{ dx } \arcsin { x } =\frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } }$

Instead of 1/root(1-x^2), I got 1/root(1-sin(x)^2). Where did I go wrong?
http://m.imgur.com/EO9hlOt

InteGrand · Nov 29, 2015

Re: HSC 2016 3U Marathon

calamebe said:
Instead of 1/root(1-x^2), I got 1/root(1-sin(x)^2). Where did I go wrong?
http://m.imgur.com/EO9hlOt

$The first line of the working is incorrect. It is not true that if $f$ and $g$ are inverse functions, then $f^\prime (x)\cdot g^\prime (x) = 1$ (which is what you implied in your answer to (a)). What (a) states is that if $f$ is an invertible continuously differentiable function with inverse $g$, and $y = f(x) \Longleftrightarrow x = g(y)$, then $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{\frac{\mathrm{d}x}{\mathrm{d}y}}$, provided the denominator here is not 0. That is, if $b= f(a)$ (so $a = g(b)$), then $f^\prime (a) = \frac{1}{g^\prime (b)}$ provided $g^\prime (b)\neq 0$, where $f^\prime$ is the derivative of the function $f$ and similarly for $g^\prime$. This theorem is called the \textsl{inverse function theorem}.$

calamebe · Nov 29, 2015

Re: HSC 2016 3U Marathon

InteGrand said:
$\underline{NEW QUESTION}$

$(i) Show via mathematical induction that $\cos \left(n\pi\right) = (-1)^n$ for all non-negative integers $n$.$

$(ii) Deduce that the result in (i) holds for all integers $n$.$

http://m.imgur.com/f9NmJqS

InteGrand · Nov 29, 2015

Re: HSC 2016 3U Marathon

$For example, $\mathrm{e}^x$ and $\ln x$ are inverse functions, but clearly their derivatives multiplied together don't give 1 in general. What \textsl{is} true though is that if $b =\mathrm{e}^a \Longleftrightarrow a = \ln b$, and $f(x) = \mathrm{e}^x$ and $g(y)=\ln y$ so that $f^\prime (x) =\mathrm{e}^x$ and $g^\prime (y) =\frac{1}{y}$, we will have $f^\prime (a) =\frac{1}{g^\prime (b)}$, as you can check using the computed derivatives. So this is an example illustrating what the theorem says about general continuously differentiable invertible functions.$

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