HSC 2016 MX1 Marathon (archive) (1 Viewer)

Flop21 · Feb 11, 2016

Re: HSC 2016 3U Marathon

InteGrand said:
$\noindent To find the other roots, we can use polynomial long division (or synthetic division, or inspection) to find $\left(x^3 -5x^2 +4\right)\div \left(x-1\right)$. This will come out to be a quadratic (it should be $x^2 -4x-4$). Then we can find the roots of this quadratic using the quadratic formula or completing the square etc. The roots of this quadratic are then the other roots of the original cubic.$

Ohhhhhhh right thanks!

davidgoes4wce · Feb 11, 2016

Re: HSC 2016 3U Marathon

I did it using the 'long division' method

Flop21 · Feb 11, 2016

Re: HSC 2016 3U Marathon

Does anyone know where to find a good explanation video on this:

"use mathematical induction to prove the following results" or "prove by induction that 5^n -1 is a multiple of 4"

I can't find one anywhere, maybe there's a different term used for this in the 3u syllabus??

InteGrand · Feb 11, 2016

Re: HSC 2016 3U Marathon

Flop21 said:
Does anyone know where to find a good explanation video on this:

"use mathematical induction to prove the following results" or "prove by induction that 5^n -1 is a multiple of 4"

I can't find one anywhere, maybe there's a different term used for this in the 3u syllabus??

It's just proof by induction. There should be lots of videos about this online, though maybe not for the particular Q. you had in that post.

Here's one example of a YouTube video on this:

.

davidgoes4wce · Feb 14, 2016

Re: HSC 2016 3U Marathon

Find the derivative of

$y=log(1-e^x)$

Drsoccerball · Feb 14, 2016

Re: HSC 2016 3U Marathon

davidgoes4wce said:
Find the derivative of

$y=log(1-e^x)$

Isn't this 2 unit? $y'= \frac{-e^x}{1-e^x}$

davidgoes4wce · Feb 14, 2016

Re: HSC 2016 3U Marathon

Drsoccerball said:
Isn't this 2 unit? $y'= \frac{-e^x}{1-e^x}$

its from Cambridge 3U text, i thought it only applied for natural logs.

davidgoes4wce · Feb 14, 2016

Re: HSC 2016 3U Marathon

Speaking of Cambridge, I shouldn't have put the question on this thread, it should have been on the other one.

leehuan · Feb 14, 2016

Re: HSC 2016 3U Marathon

davidgoes4wce said:
its from Cambridge 3U text, i thought it only applied for natural logs.

Log, refers to the natural log in maths, i.e. Ln.

If you meant Log base 10 you had to actually include that.

$\log _{ 10 }{ f(x) }$

To which you use change of base to convert it to ln(f(x))/ln10 then noting 1/ln10 is just a coefficient, take the derivative as though nothing happened

InteGrand · Feb 14, 2016

Re: HSC 2016 3U Marathon

davidgoes4wce said:
its from Cambridge 3U text, i thought it only applied for natural logs.

The symbol log is the natural log (well, one of the notations for it, the others being ln and log_e).

davidgoes4wce · Feb 15, 2016

Re: HSC 2016 3U Marathon

This is from Grove's Maths in Focus text,

In terms of explaining the AAS, for the side explanation is it also good enough to say,

BC=CE (given)?

porcupinetree · Feb 15, 2016

Re: HSC 2016 3U Marathon

davidgoes4wce said:
This is from Grove's Maths in Focus text,

In terms of explaining the AAS, for the side explanation is it also good enough to say,

BC=CE (given)?

We're not explicitly given that BC=CE. It is a true statement, however, not one that we're given, in fact, it's one that we're essentially trying to prove (by proving that ABC is congruent to DEC). So, by using it in our proof, we'd be assuming the solution before we even reach it. (We are, however, given that AC=CD in the diagram.)

davidgoes4wce · Feb 15, 2016

Re: HSC 2016 3U Marathon

This is from the 2005 HSC exam paper, Q4d worth 3 marks, would this be good enough to give me the full marks for Induction? (I'm aware that there are many different variations in the induction explanation)

Paradoxica · Feb 16, 2016

Re: HSC 2016 3U Marathon

$$\noindent$1. f(x) = \frac{x^2-6x+6}{2x-4} \,\,\, g(x) = \frac{ax^2+bx+c}{x-d} \\ $The following information is known about $g(x). \\ g(x) $ and $ f(x) $ have the same vertical asymptotes.$ \\ $The oblique asymptotes are perpendicular, and intersect on the $y$-axis. $ \\ f(x) = g(x) $ has a solution at $x = -2. \\ $Find $g(x)$ and hence find the other solution to $f(x) = g(x)\\$2. $x+\frac{1}{x}$ is an integer. Show by induction, or any other means, that $x^n + \frac{1}{x^n}$ is an integer $ \forall n \in \mathbb{N}\\ $3. a) Show $\tan{\theta}-\cot{\theta} \equiv 2\tan{2\theta}\\ $b) Prove the identity: $8\cot{8x}+4\tan{4x}+2\tan{2x}+\tan{x}-\cot{x} \equiv 0 \\$ c) Prove by induction: $ \forall n \in \mathbb{N}\\2^n \cot{2^n x} + 2^{n-1} \tan{2^{n-1} x} + 2^{n-2} \tan{2^{n-2} x} +\cdots 2\tan{2x}+\tan{x}-\cot{x} \equiv 0$

leehuan · Feb 16, 2016

Re: HSC 2016 3U Marathon

Paradoxica said:
$$\noindent$1. f(x) = \frac{x^2-6x+6}{2x-4} \,\,\, g(x) = \frac{ax^2+bx+c}{x-d} \\ $The following information is known about $g(x). \\ g(x) $ and $ f(x) $ have the same vertical asymptotes.$ \\ $The oblique asymptotes are perpendicular, and intersect on the $y$-axis. $ \\ f(x) = g(x) $ has a solution at $x = -2. \\ $Find $g(x)$ and hence find the other solution to $f(x) = g(x)\\$2. $x+\frac{1}{x}$ is an integer. Show by induction, or any other means, that $x^n + \frac{1}{x^n}$ is an integer $ \forall n \in \mathbb{N}\\ $3. Prove the following identity: $8\cot{8x}+4\tan{4x}+2\tan{2x}+\tan{x}-\cot{x} \equiv 0$

I'm guessing Q3 does not have to be brute forced?

Paradoxica · Feb 16, 2016

Re: HSC 2016 3U Marathon

leehuan said:
I'm guessing Q3 does not have to be brute forced?

No. The final two terms are paired together for a reason.

leehuan · Feb 16, 2016

Re: HSC 2016 3U Marathon

Paradoxica said:
No. The final two terms are paired together for a reason.

Yeah ok just got it out.

It was surprisingly easy.

Flop21 · Feb 17, 2016

Re: HSC 2016 3U Marathon

"How many different ways can 6 adults and 3 children be arranged in a row if the children are to remain together?"

InteGrand · Feb 17, 2016

Re: HSC 2016 3U Marathon

Flop21 said:
"How many different ways can 6 adults and 3 children be arranged in a row if the children are to remain together?"

$\noindent There are nine people: $*********$. Now, imagine the three children together as one unit: $[C_1 C_2 C_3] ******$, where each $*$ represents an adult. In that orientation, there are $3!\cdot 6!$ arrangements ($3!$ for the children and $6!$ for the adults). But the children can also be like this $*C*****$, where $C$ refers to the block of children. In total, the block of children could go in 7 places (just the no. of ways to place a letter $C$ amongst six $*$'s). Hence the answer is $7\cdot 3!\cdot 6!$.$

Flop21 · Feb 17, 2016

Re: HSC 2016 3U Marathon

InteGrand said:
$\noindent There are nine people: $*********$. Now, imagine the three children together as one unit: $[C_1 C_2 C_3] ******$, where each $*$ represents an adult. In that orientation, there are $3!\cdot 6!$ arrangements ($3!$ for the children and $6!$ for the adults). But the children can also be like this $*C*****$, where $C$ refers to the block of children. In total, the block of children could go in 7 places (just the no. of ways to place a letter $C$ amongst six $*$'s). Hence the answer is $7\cdot 3!\cdot 6!$.$

Thanks!!

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