# HSC 2017-2018 Maths Marathon (1 Viewer)

#### dan964

##### MOD
Moderator
Welcome to the 2017 HSC 2U Marathon,

1. Only post questions within the level and scope of 2U HSC Mathematics.

2. Provide neat working out when possible

4. Challenging questions that a 2U student can pick up easily are okay, but try to stay within the 2U syllabus content.

Here is a question I've pulled for you (modified from Ascham 2011 2U trial HSC)

For the curve $y=3x^3-6x^2+4x+7$
1. Find the stationary points and determine their nature
2. Find any points of inflexion
3. Sketch the graphs showing info from (1) & (2)
4. When is the curve decreasing with downward concavity?

#### leehuan

##### Well-Known Member
Re: HSC 2017 Maths (Advanced) Marathon

$\text{Can a function }f(x)\text{ be both odd and even at the same time? Justify your answer.}$

#### Green Yoda

##### Hi Φ
Moderator
Re: HSC 2017 Maths (Advanced) Marathon

$\text{Can a function }f(x)\text{ be both odd and even at the same time? Justify your answer.}$
If f(x)=0
then f(x)=f(-x)
and f(-x)=-f(x)
thus being both odd and even

#### frog1944

##### Member
Re: HSC 2017 Maths (Advanced) Marathon

How would you find all the functions that are both odd and even?

#### InteGrand

##### Well-Known Member
Re: HSC 2017 Maths (Advanced) Marathon

How would you find all the functions that are both odd and even?
Any (real-valued) function (of a real variable) defined on a symmetric interval (about 0) that is both odd and even must satisfy f(x) = f(-x) = -f(-x) for all x in the domain, which implies f(-x) = 0 for all x in the domain, whence f(x) = 0 for all x in the domain (as the domain is a symmetric interval).

So the only functions with domain and codomain being subsets of R that are both odd and even are those that are identically 0 on a symmetric domain about 0. (If the domain is not symmetric about 0, we can't really talk about f(x) being equal to f(-x) for all x, which is something we need to do for the function to be even. I guess though you still could by saying that it only matters for x such that x and -x are in the domain. But for HSC purposes, the domain is usually symmetric.)

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#### eyeseeyou

##### Well-Known Member
Re: HSC 2017 Maths (Advanced) Marathon

How would you find all the functions that are both odd and even?
What bout even?

#### dan964

##### MOD
Moderator
Re: HSC 2017 Maths (Advanced) Marathon

For the curve $y=3x^3-6x^2+4x+7$
1. Find the stationary points and determine their nature
2. Find any points of inflexion
3. Sketch the graphs showing info from (1) & (2)
4. When is the curve decreasing with downward concavity?
The current question...

#### JackPatel

##### New Member
Re: HSC 2017 Maths (Advanced) Marathon

y = 3x³-6x²+4x+7
y’ = 9x²-12x+4

For S.P.
y’=0
9x²-12x+4=0
x= 2/3 y=71/9
Therefore S.P. at (2/3 , 71/9)

For I.P
y′′=0
y′′=18x-12
18x-12=0
x=2/3 y= 71/9
Therefore I.P. at (2/3 , 71/9)

Soz don't have time

#### InteGrand

##### Well-Known Member
Re: HSC 2017 Maths (Advanced) Marathon

$\noindent Given n real numbers x_{1},x_{2},\ldots, x_{n} (n a positive integer), a \textsl{weighted average} of these is a combination of them of the form \sum_{i=1}^{n}\alpha_{i}x_{i}, where the \alpha_{i} are non-negative numbers that sum to 1. E.g. \frac{1}{3}x_{1} + \frac{2}{3}x_{2} is an example of a weighted average of \left\{x_{1},x_{2}\right\}.$

$\noindent Let \overline{x}_{1},\overline{x}_{2},\ldots, \overline{x}_{K} each be weighted averages of \left\{x_{1},x_{2},\ldots, x_{n}\right\} (where K is a positive integer). Show that any weighted average of \left\{\overline{x}_{1},\overline{x}_{2},\ldots, \overline{x}_{K}\right\} is a weighted average of \left\{x_{1},x_{2},\ldots, x_{n}\right\}.$

#### Commando007

##### Member
Re: HSC 2017 Maths (Advanced) Marathon

$\noindent Given n real numbers x_{1},x_{2},\ldots, x_{n} (n a positive integer), a \textsl{weighted average} of these is a combination of them of the form \sum_{i=1}^{n}\alpha_{i}x_{i}, where the \alpha_{i} are non-negative numbers that sum to 1. E.g. \frac{1}{3}x_{1} + \frac{2}{3}x_{2} is an example of a weighted average of \left\{x_{1},x_{2}\right\}.$

$\noindent Let \overline{x}_{1},\overline{x}_{2},\ldots, \overline{x}_{K} each be weighted averages of \left\{x_{1},x_{2},\ldots, x_{n}\right\} (where K is a positive integer). Show that any weighted average of \left\{\overline{x}_{1},\overline{x}_{2},\ldots, \overline{x}_{K}\right\} is a weighted average of \left\{x_{1},x_{2},\ldots, x_{n}\right\}.$
Which topic?

##### -insert title here-
Re: HSC 2017 Maths (Advanced) Marathon

A cube is inscribed in a right circular cone of height h and radius r so that one side is coplanar with the base and all four opposite vertices are in contact with the curved surface of the cone.

Find the side length of the cube in terms of h and r.

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#### leehuan

##### Well-Known Member
Re: HSC 2017 Maths (Advanced) Marathon

Which topic?
When you see terms and sums and what not you should be thinking series.

#### dragon658

##### Member
Re: HSC 2017 Maths (Advanced) Marathon

Hey guys, how do I go about doing this: find the values of l that will make the quadratic y=(l+6)x^2-2lx+3 a perfect square. And can someone explain what 1/alpha + 1/beta is when doing the roots of a quadratic.

#### Green Yoda

##### Hi Φ
Moderator
Re: HSC 2017 Maths (Advanced) Marathon

Hey guys, how do I go about doing this: find the values of l that will make the quadratic y=(l+6)x^2-2lx+3 a perfect square. And can someone explain what 1/alpha + 1/beta is when doing the roots of a quadratic.
The expression is a perfect when the discriminant is rational.

1/alpha +1/beta = alpha+beta/alpha*beta which is sum of roots/product of roots

#### dragon658

##### Member
Re: HSC 2017 Maths (Advanced) Marathon

Thanks bro

#### Green Yoda

##### Hi Φ
Moderator
Re: HSC 2017 Maths (Advanced) Marathon

Insert three geometric means between 8 and 1/32.

#### Lugia101101

##### New Member
Re: HSC 2017 Maths (Advanced) Marathon

If I'm interpreting your question correctly:
$\text{So we're looking for three numbers }T_1\text{, }T_2\text{, and }T_3\text{ such that }8,T_1,T_2,T_3,\frac{1}{32}\text{ is a geometric series.}$
$\text{We can express }8\text{ as }2^3\text{ and }\frac{1}{32}\text{ as }2^{-5}\text{.}$
\begin{align*}\text{Now if we let }&a=2^3\text{, then:}\\ &T_1=ar=2^3r\\ &T_2=ar^2=2^3r^2\\ &T_3=ar^3=2^3r^3\\ &2^{-5}=ar^4=2^3r^4\end{align*}
$\text{This lets us solve for }r\text{:}$
\begin{align*}2^{-5}&=2^3r^4\\ 2^{-8}&=r^4\\ r&=\pm2^{-2}\end{align*}
\begin{align*}\text{So:}\\ &T_1=2^3(2^{-2})\text{ or }2^3(-2^{-2})=2\text{ or }-2\\ &T_2=2^3(2^{-2})^2\text{ or }2^3(-2^{-2})^2=\frac{1}{2}\\ &T_3=2^3(2^{-2})^3\text{ or }2^3(-2^{-2})^3=\frac{1}{8}\text{ or }-\frac{1}{8}\end{align*}
\begin{align*}\text{Resulting in the series:}\\ 8,2,&\frac{1}{2},\frac{1}{8},\frac{1}{32}\\ 8,-2,&\frac{1}{2},-\frac{1}{8},\frac{1}{32}\end{align*}

#### boredofstudiesuser1

##### Active Member
Re: HSC 2017 Maths (Advanced) Marathon

Welcome to the 2017 HSC 2U Marathon,

1. Only post questions within the level and scope of 2U HSC Mathematics.

2. Provide neat working out when possible

4. Challenging questions that a 2U student can pick up easily are okay, but try to stay within the 2U syllabus content.

Here is a question I've pulled for you (modified from Ascham 2011 2U trial HSC)

For the curve $y=3x^3-6x^2+4x+7$
1. Find the stationary points and determine their nature
2. Find any points of inflexion
3. Sketch the graphs showing info from (1) & (2)
4. When is the curve decreasing with downward concavity?
I took a photo but apparently it exceeds the size allowed

##### -insert title here-
Re: HSC 2017 Maths (Advanced) Marathon

I took a photo but apparently it exceeds the size allowed
multiple photos *muahaha*

#### LC14199

##### Member
Re: HSC 2017 Maths (Advanced) Marathon

I took a photo but apparently it exceeds the size allowed
Just upload it to imgur or another image hosting site and use the IMG tags