# Thread: HSC 2017-2018 Maths Marathon

1. ## HSC 2017-2018 Maths 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?

2. ## Re: HSC 2017 Maths (Advanced) Marathon

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

3. ## Re: HSC 2017 Maths (Advanced) Marathon

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

4. ## Re: HSC 2017 Maths (Advanced) Marathon

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

5. ## Re: HSC 2017 Maths (Advanced) Marathon

Originally Posted by frog1944
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.)

6. ## Re: HSC 2017 Maths (Advanced) Marathon

Originally Posted by frog1944
How would you find all the functions that are both odd and even?
What bout even?

7. ## Re: HSC 2017 Maths (Advanced) Marathon

Originally Posted by dan964
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...

8. ## 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

9. ## Re: HSC 2017 Maths (Advanced) Marathon

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

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

10. ## Re: HSC 2017 Maths (Advanced) Marathon

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

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

11. ## 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.

12. ## Re: HSC 2017 Maths (Advanced) Marathon

Originally Posted by Commando007
Which topic?
When you see terms and sums and what not you should be thinking series.

13. ## 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.

14. ## Re: HSC 2017 Maths (Advanced) Marathon

Originally Posted by dragon658
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

Thanks bro

16. ## Re: HSC 2017 Maths (Advanced) Marathon

Insert three geometric means between 8 and 1/32.

17. ## 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*}

18. ## Re: HSC 2017 Maths (Advanced) Marathon

Originally Posted by dan964
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

19. ## Re: HSC 2017 Maths (Advanced) Marathon

Originally Posted by boredofstudiesuser1
I took a photo but apparently it exceeds the size allowed
multiple photos *muahaha*

20. ## Re: HSC 2017 Maths (Advanced) Marathon

Originally Posted by boredofstudiesuser1
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

21. ## Re: HSC 2017 Maths (Advanced) Marathon

$\text{Let }f(x)\text{ and }g(x)\text{ be continuous, monotonic increasing functions.}\\ \text{1. Show that }f(g(x))\text{ is also monotonic increasing.}\\ \text{2. What happens if one and only one of }f(x),g(x)\text{ were monotonic decreasing?}$

22. ## Re: HSC 2017 Maths (Advanced) Marathon

\begin{align*}\text{Let }h(x)&=f(g(x))\\ h^{\prime}(x)&=f^{\prime}(g(x))g^{\prime}(x)\\ \text{So }&f^{\prime}(x)>0,\forall x\in\mathbb{R}\\ &g^{\prime}(x)>0,\forall x\in\mathbb{R}\\ \therefore&h^{\prime}(x)>0,\forall x\in\mathbb{R}\\ \text{So }&f(g(x))\text{ is monotonic increasing.}\end{align*}

If one function, but not both functions, is monotonic decreasing, then the product of their first derivatives will be negative, and so $f(g(x))$ will be monotonic decreasing.

23. ## Re: HSC 2017 Maths (Advanced) Marathon

Correct, but small technicality error: Monotone means the inequality isn't necessarily strict (≥)

24. ## Re: HSC 2017 Maths (Advanced) Marathon

Thanks for clearing that up; none of my teachers were able to tell me definitively.

25. ## Re: HSC 2017 Maths (Advanced) Marathon

\begin{align*}\text{Let }f(x)&=-x^3\\ f^{\prime}(x)&=-3x^2\end{align*}
$\text{For all }x\in\mathbb{R},x\neq0\text{, }x^2>0\text{, so }-3x^2<0$
$\therefore\text{As }f^{\prime}(x)<0\text{ for all }x\text{ in the given domain, }x^3\text{ is monotonic decreasing.}$

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