HSC 2016 MX1 Marathon (archive) (1 Viewer)

ifireballx · Jan 28, 2016

Re: Flop math question thread

Factorise
Top: (x-1)(x^2 +x + 1)
Bottom: (x-1)(x+1)

Cancel out (x-1) and then sub in x=1

leehuan · Jan 28, 2016

Re: Flop math question thread

Just to put it into LaTeX:

$\lim _{ x\rightarrow 1 }{ \frac { { x }^{ 3 }-1 }{ { x }^{ 2 }-1 } } \\ =\lim _{ x\rightarrow 1 }{ \frac { \left( x-1 \right) \left( { x }^{ 2 }+x+1 \right) }{ \left( x-1 \right) \left( x+1 \right) } } \\ =\lim _{ x\rightarrow 1 }{ \frac { { x }^{ 2 }+x+1 }{ x+1 } } \\ =\frac { { 1 }^{ 2 }+1+1 }{ 1+1 } \\ =\frac { 3 }{ 2 }$

If I were to guess where you went wrong you factorised the top incorrectly

Nailgun · Jan 28, 2016

Re: Flop math question thread

Are limits in 2U? I remember doing them before, but it might of been 3U..

InteGrand · Jan 28, 2016

Re: Flop math question thread

Nailgun said:
Are limits in 2U? I remember doing them before, but it might of been 3U..

Yeah, they are in 2U.

leehuan · Jan 28, 2016

Re: Flop math question thread

Nailgun said:
Are limits in 2U? I remember doing them before, but it might of been 3U..

3u limit is just the lim x->inf 1/x = 0

Drsoccerball · Jan 28, 2016

Re: Flop math question thread

leehuan said:
3u limit is just the lim x->inf 1/x = 0

And also $\lim_{x \rightarrow 0 }\frac{\sin{x}}{x} = 1$

leehuan · Jan 28, 2016

Re: Flop math question thread

Drsoccerball said:
And also $\lim_{x \rightarrow 0 }\frac{\sin{x}}{x} = 1$

That's actually 2u. But in terms of the HSC it only appears to get examined in 3u.

mcchicken · Jan 28, 2016

Re: Flop math question thread

guysss u r doing this all wrong

here lemme show u how da pros do it:

Drsoccerball · Jan 29, 2016

Re: Flop math question thread

leehuan said:
That's actually 2u. But in terms of the HSC it only appears to get examined in 3u.

If it doesn't appear in the HSC then its not 2u

Flop21 · Jan 29, 2016

Re: Flop math question thread

Thanks all.

-snip-

sorry I'm thinking of inequalities, removed what I said before

leehuan · Jan 29, 2016

Re: Flop math question thread

This is an example of a 3u limit question

$\lim _{ x\rightarrow \infty }{ \frac { 4{ x }^{ 3 }-9{ x }^{ 2 }+16x-25 }{ 2{ x }^{ 3 }-8x+100 } }$

The sin(x)/x goes to 1 limit isn't typically examined in the 2U course due to lack of purpose. It has been clearly stated in the still-old-unchanged syllabus that it's a part of the 2u section.

Flop21 · Jan 29, 2016

Re: Flop math question thread

leehuan said:
This is an example of a 3u limit question

$\lim _{ x\rightarrow \infty }{ \frac { 4{ x }^{ 3 }-9{ x }^{ 2 }+16x-25 }{ 2{ x }^{ 3 }-8x+100 } }$

The sin(x)/x goes to 1 limit isn't typically examined in the 2U course due to lack of purpose. It has been clearly stated in the still-old-unchanged syllabus that it's a part of the 2u section.

Would that be 2 the answer?

InteGrand · Jan 29, 2016

Re: Flop math question thread

Flop21 said:
Would that be 2 the answer?

Yes

leehuan · Jan 29, 2016

Re: Flop math question thread

Flop21 said:
Would that be 2 the answer?

Indeed. From dividing top and bottom by x^3 the limit can then be substituted so that almost every term goes to 0, leaving you with (4-0+0-0)/(2-0+0) = 2

Alternatively this can actually be done from inspection by noticing that the degree of the numerator and denominator and equal. Hence the quotient of the coefficients automatically gives the answer.
______________

Yep so 3U inequalities would be something like

$\\ $Solve$ \\ \frac { 3{ x }^{ 2 } }{ 7x+4 } \le -2$

kenrokzz · Jan 30, 2016

Re: HSC 2016 3U Marathon

Can someone help with questions 4 and 5

InteGrand · Jan 30, 2016

Re: HSC 2016 3U Marathon

kenrokzz said:
View attachment 32838

Can someone help with questions 4 and 5

I'll show how to do Q.4 (Q.5 isn't too hard, just tedious).

$\noindent \underline{QUESTION 4}$

$\noindent The equation of the chord of contact is $xx_1 =2a(y+y_1)$. Since $R$ lies on this line and lies on the directrix $y=-a$, at $R$ we have $x_R x_1 = 2a(-a+y_1)\Rightarrow x_R = \frac{2a(-a+y_1)}{x_1}$ (note $x_1\neq 0$, since if $x_1=0$, $T$ lies on the $x$-axis and there is no chord of contact possible. Hence this expression for $x_R$ is well-defined). Also, $y_R = -a$ as $R$ lies on the directrix. Hence $R=\left( \frac{2a(-a+y_1)}{x_1},-a\right)$. Also, the focus is $S(0,a)$. Let the slope of $ST$ and $RS$ be $m_{ST}$ and $m_{RS}$ respectively. Then$

$\begin{align*}m_{ST}&= \frac{y_S - y_T}{x_S - x_T} \\ &= \frac{a-y_1}{0-x_1} \\ &= \frac{y_1 - a}{x_1},\end{align*}$

$\noindent and$

$\begin{align*}m_{RS}&= \frac{y_R - y_S}{x_R - x_S} \\ &= \frac{-a-a}{\frac{2a(-a+y_1)}{x_1}-0} \\ &= \frac{-2a}{\frac{2a(-a+y_1)}{x_1}} \\ &= \frac{x_1}{y_a - a} \\ &= -\frac{1}{m_{ST}},\end{align*}$

$\noindent which implies that $RS$ and $ST$ are perpendicular, as required.$

wu345 · Jan 30, 2016

Re: HSC 2016 3U Marathon

These two please if anyone is willing

http://i.imgur.com/dYvwTlF.png

http://i.imgur.com/fKj2m6q.png

InteGrand · Jan 30, 2016

Re: HSC 2016 3U Marathon

wu345 said:
These two please if anyone is willing
http://i.imgur.com/dYvwTlF.png

http://i.imgur.com/fKj2m6q.png

A proof of Snell's Law is on this page: https://proofwiki.org/wiki/Snell's_Law

wu345 · Jan 30, 2016

Re: HSC 2016 3U Marathon

InteGrand said:
A proof of Snell's Law is on this page: https://proofwiki.org/wiki/Snell's_Law

cool thanks, on second thought I probably should have just looked it up hahaha
here's another http://i.imgur.com/Okk5fO1.png

Drsoccerball · Jan 31, 2016

Re: HSC 2016 3U Marathon

InteGrand said:
$\noindent \textbf{NEW QUESTION}$

$\noindent Let $n$ and $k$ be positive integers with $k\leqslant n$. Suppose there are a total of $n$ marbles in an urn and exactly one of these is red, and the others are all green. Alice randomly selects $k$ marbles from the urn without replacement. What is the probability (in terms of $n$ and $k$) that one of her selected marbles is the red one?$

Are the green marbles all identical ? Also does order matter ?

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