HSC 2016 MX1 Marathon (archive) (1 Viewer)

wu345 · Apr 11, 2016

Re: HSC 2016 3U Marathon

nvm,
$a)\quad { (\sin { x } +\cos { x } ) }^{ 2 }\\ =\sin ^{ 2 }{ x } +\cos ^{ 2 }{ x } +2\sin { x } \cos { x } \\ =1+2\sin { x } \cos { x } \\ b)\quad \int { \frac { \cos { x } -\sin { x } }{ 1+\sin { x } \cos { x } } } dx\\ =\int { \frac { 2(\cos { x } -\sin { x } ) }{ 2+2\sin { x } \cos { x } } } dx\\ =\int { \frac { 2(\cos { x } -\sin { x } ) }{ { 1+(\sin { x } +\cos { x } ) }^{ 2 } } } dx\\ =2\tan ^{ -1 }{ (\sin { x } +\cos { x } )\quad +C } \quad ($by\quad reverse\quad chain\quad rule$)$

Jeff_ · Apr 11, 2016

Re: HSC 2016 3U Marathon

$Solve: sin^{-1}(x)-cos^{-1}(x)=sin^{-1}(3x+1)$

Drsoccerball · Apr 11, 2016

Re: HSC 2016 3U Marathon

Jeff_ said:
$Solve: sin^{-1}(x)-cos^{-1}(x)=sin^{-1}(3x+1)$

$$ Take note that $ \sin^{-1}x + cos^{-1}x = \frac{\pi}{2} \\ \therefore 3x+1= \sin(2\cdot \sin^{-1}x - \frac{\pi}{2}) \\ 3x+1 =-\cos{(2(\sin^{-1}x))} $ Which can be found by drawing a triangle with angle $ \sin^{-1}x.$

Paradoxica · Apr 11, 2016

Re: HSC 2016 3U Marathon

Jeff_ said:
$$\noindent Solve: $\sin^{-1}(x) - \cos^{-1}(x) = \sin^{-1}(3x+1)$

By inspection of the graphs, there is no solution. ■

Jeff_ · Apr 11, 2016

Re: HSC 2016 3U Marathon

Drsoccerball said:
$$ Take note that $ \sin^{-1}x + cos^{-1}x = \frac{\pi}{2} \\ \therefore 3x+1= \sin(2\cdot \sin^{-1}x - \frac{\pi}{2}) \\ 3x+1 =-\cos{(2(\sin^{-1}x))} $ Which can be found by drawing a triangle with angle $ \sin^{-1}x.$

A bit confusing there (at least to me), why is $sin^{-1}(x)+cos^{-1}(x)$ used when the question is about $sin^{-1}(x)-cos^{-1}(x)$ ?

Paradoxica said:
By inspection of the graphs, there is no solution. ■

Argh probably a typo, maybe it's meant to be a plus instead of a minus.

Paradoxica · Apr 11, 2016

Re: HSC 2016 3U Marathon

Jeff_ said:
A bit confusing there (at least to me), why is $sin^{-1}(x)+cos^{-1}(x)$ used when the question is about $sin^{-1}(x)-cos^{-1}(x)$ ?

Argh probably a typo, maybe it's meant to be a plus instead of a minus.

If it's a plus, the left hand side is constant, and there is only one solution, which is at the end point of the right hand side.

parad0xica · Apr 11, 2016

Re: HSC 2016 3U Marathon

Jeff_ said:
A bit confusing there (at least to me), why is $sin^{-1}(x)+cos^{-1}(x)$ used when the question is about $sin^{-1}(x)-cos^{-1}(x)$ ?

Simultaneous equations!

btw [tex.]\sin^{-1} (x) + \cos^{-1} (x) [/tex.] without the red dots yield $\sin^{-1} (x) + \cos^{-1} (x)$

Jeff_ · Apr 11, 2016

Re: HSC 2016 3U Marathon

Paradoxica said:
If it's a plus, the left hand side is constant, and there is only one solution, which is at the end point of the right hand side.

Yep, if it's a plus I'm fine with that.

parad0xica said:
Simultaneous equations!

btw [tex.]\sin^{-1} (x) + \cos^{-1} (x) [/tex.] without the red dots yield $\sin^{-1} (x) + \cos^{-1} (x)$

Fix up the formatting

Paradoxica · Apr 11, 2016

Re: HSC 2016 3U Marathon

Jeff_ said:
Yep, if it's a plus I'm fine with that.

Let me elaborate on that.

If it's a plus, the equation is trivial. By inspection, when x = 0, the right hand side is the same as the left hand side.

And since the right hand side is strictly increasing, there is only one solution. (The solution is also the terminal point of the graph)

parad0xica · Apr 11, 2016

Re: HSC 2016 3U Marathon

Jeff_ said:
Fix up the formatting

o__O

I saw how you used

Jeff_ · Apr 11, 2016

Re: HSC 2016 3U Marathon

Paradoxica said:
Let me elaborate on that.

If it's a plus, the equation is trivial. By inspection, when x = 0, the right hand side is the same as the left hand side.

And since the right hand side is strictly increasing, there is only one solution. (The solution is also the terminal point of the graph)

Gotcha, and I meant I knew how to do this

parad0xica said:
o__O

I saw how you used to get the LaTeX'd equations. Just providing an easier way to do it lol[/QUOTE]

Are you and the above same people o_O but lets not derail the thread! ;)

parad0xica · Apr 11, 2016

Re: HSC 2016 3U Marathon

Jeff_ said:
Are you and the above same people but lets not derail the thread!

No. You can examine our past posts and make this deduction and it's highly unlikely there is one person being active on both accounts and who is that careful with their word choice, thoughts, activity, etc.

Question.

When six fair coins are tossed simultaneously, in how many of the outcomes will at most three of the coins turn up as heads?

Paradoxica · Apr 11, 2016

Re: HSC 2016 3U Marathon

Jeff_ said:
Are you and the above same people but lets not derail the thread!

No, he's just a digital stalker...

Which gives me a slight ego boost, as I have enough of a presence to have a stalker.

Paradoxica · Apr 11, 2016

Re: HSC 2016 3U Marathon

parad0xica said:
Question.

When six fair coins are tossed simultaneously, in how many of the outcomes will at most three of the coins turn up as heads?

Doesn't this depend on whether the coins are distinguishable or not?

parad0xica · Apr 11, 2016

Re: HSC 2016 3U Marathon

Paradoxica said:
No, he's just a digital stalker...

Which gives me a slight ego boost, as I have enough of a presence to have a stalker.

I'm an ambitious detective

Drsoccerball · Apr 11, 2016

Re: HSC 2016 3U Marathon

I'm pretty sure Parad0xica is a uni student.

Paradoxica · Apr 11, 2016

Re: HSC 2016 3U Marathon

Drsoccerball said:
I'm pretty sure Parad0xica is a uni student.

Since when did a person have to be in Uni to know high levels of mathematics?

Plenty of people are self taught.

Drsoccerball · Apr 12, 2016

Re: HSC 2016 3U Marathon

Paradoxica said:
Since when did a person have to be in Uni to know high levels of mathematics?

Plenty of people are self taught.

He did the limit question which only uni students learn :L

leehuan · Apr 12, 2016

Re: HSC 2016 3U Marathon

Paradoxica said:
Since when did a person have to be in Uni to know high levels of mathematics?

Plenty of people are self taught.

There actually is a strong vibe that the other parad0xica is a uni student.

Paradoxica · Apr 12, 2016

Re: HSC 2016 3U Marathon

Drsoccerball said:
He did the limit question which only uni students learn :L

"only"

evidence please.

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