HSC 2016 Maths Marathon (archive) (1 Viewer)

calamebe · Nov 1, 2015

Re: HSC 2016 2U Marathon

Drsoccerball said:
ahah my bad... Why are you on the 2U marathon :L ? You should spend heeps of time on extension 2

Oh I'm just doing whatever right now. I'm doing some ext 2 past papers, studying for year 10 exams, and why not do some 2U as well.

leehuan · Nov 1, 2015

Re: HSC 2016 2U Marathon

I prefer calambe's proof slightly, because that's more of a means to actually prove the result instead of "verify" it. However, going from cos(theta)=x -> sin(theta)=sqrt(1-x^2) is always nice.

However, despite this, if we swap Drsoccerball's line 3 and 4 around he could cheat his way around it because he reverse engineered it in his head.

leehuan · Nov 1, 2015

Re: HSC 2016 2U Marathon

$\\ $The locus of P$(x,y)$ is defined as the set of all points that are equidistant from the point S$(0,4)$ and the line $y=x-4$. Explain briefly why this curve will be a parabola and find it's equation. \\ Challenge: Find the axis of symmetry of this parabola.$

calamebe · Nov 1, 2015

Re: HSC 2016 2U Marathon

leehuan said:
$\\ $The locus of P$(x,y)$ is defined as the set of all points that are equidistant from the point S$(0,4)$ and the line $y=x-4$. Explain briefly why this curve will be a parabola and find it's equation. \\ Challenge: Find the axis of symmetry of this parabola.$

http://m.imgur.com/AglK0Mt i feel as though I am wrong though, never done a tilted parabola before.

leehuan · Nov 1, 2015

Re: HSC 2016 2U Marathon

$An unbiased coin is tossed three times. Without the use of the Ext 1 technique of binomial probability, find the probability that the coin will land on heads twice.$

calamebe · Nov 1, 2015

Re: HSC 2016 2U Marathon

I'll let someone else do this question, but may I ask was my answer to the previous question correct?

Green Yoda · Nov 1, 2015

Re: HSC 2016 2U Marathon

leehuan said:
$An unbiased coin is tossed three times. Without the use of the Ext 1 technique of binomial probability, find the probability that the coin will land on heads twice.$

I hope I am correct.

calamebe · Nov 1, 2015

Re: HSC 2016 2U Marathon

Rathin said:
View attachment 32627 I hope I am correct.

Yep looks all good, don't know whether he meant exactly two or at least two, but you found the probability it would get heads exactly two times.

leehuan · Nov 1, 2015

Re: HSC 2016 2U Marathon

calamebe said:
I'll let someone else do this question, but may I ask was my answer to the previous question correct?

It was, that's why I liked it.

Green Yoda · Nov 1, 2015

Re: HSC 2016 2U Marathon

Next Question:
Sketch y = log x, for 0 ≤ x ≤ e.

leehuan · Nov 1, 2015

Re: HSC 2016 2U Marathon

Rathin said:
Next Question:
Sketch y = log x, for 0 ≤ x ≤ e.

Has to be 0 < x ≤ e, as log 0 is undefined.

Drsoccerball · Nov 1, 2015

Re: HSC 2016 2U Marathon

Rathin said:
Next Question:
Sketch y = log x, for 0 ≤ x ≤ e.

On these marathons I think algebra based questions would be easier than sketch types.

davidgoes4wce · Nov 2, 2015

Re: HSC 2016 2U Marathon

Mary paid $1600 for her holiday birthday package to California. This payment included a 2.5% late entry fee. What was the original price of the package?

A) $1560 B) $1560.98 C) $1597.50 D) $1640

calamebe · Nov 2, 2015

Re: HSC 2016 2U Marathon

B

davidgoes4wce · Nov 2, 2015

Re: HSC 2016 2U Marathon

that is the correct answer . Any idea on the formula used to calculate this?

InteGrand · Nov 2, 2015

Re: HSC 2016 2U Marathon

davidgoes4wce said:
that is the correct answer . Any idea on the formula used to calculate this?

$Basic percentages. Let the original price be $x$. Then we have $1.025 x = \$ 1600$. Therefore, $x=\frac{\$ 1600}{1.025}=\$ 1560.97561.... \approx \$ 1560.98$ (nearest cent). Hence the answer is (B).$

leehuan · Nov 2, 2015

Re: HSC 2016 2U Marathon

NEXT QUESTION:

$\\ $Prove that the derivative of an even function gives an odd function.$$

Green Yoda · Nov 2, 2015

Re: HSC 2016 2U Marathon

leehuan said:
NEXT QUESTION:

$\\ $Prove that the derivative of an even function gives an odd function.$$

might of missed a couple of steps. am i correct?

InteGrand · Nov 2, 2015

Re: HSC 2016 2U Marathon

Rathin said:
View attachment 32636 might of missed a couple of steps. am i correct?

$I didn't check the details of what you did (just glanced at the general method), but there is a faster method: just use the chain rule. If $f$ is even, then $f(x)=f(-x)$ for all $x$ in the domain. Differentiating both sides with respect to $x$, we have$

$\begin{align*} f^\prime (x) &=f^\prime (-x)\cdot \frac{\mathrm{d}}{\mathrm{d}x}(-x) \quad (\text{chain rule})\\ &=f^\prime (-x)\cdot (-1)\\ &=-f^\prime (-x), \end{align*}$

$and so $f^\prime $ is odd, as required.$

Drsoccerball · Nov 2, 2015

Re: HSC 2016 2U Marathon

leehuan said:
NEXT QUESTION:

$\\ $Prove that the derivative of an even function gives an odd function.$$

$Let $ F(x) $ be an even function$

$\therefore F(-x)= F(x)$

$(-1)F'(-x)=F'(x)$

$\Therefore $ Derivative of an even function give and odd function, since $ -g(-x)=g(x) $ for odd functions $$

EDIT: beaten again

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