HSC 2013 MX2 Marathon (archive) (2 Viewers)

HAX0R · Nov 14, 2013

Re: HSC 2013 4U Marathon

Sy, to edit the title, press edit, and then go advanced.

Ikki · Nov 16, 2013

Re: HSC 2013 4U Marathon

Wassup guys, i'm part of the 2014 cohort, how do you guys post solutions in here? Latex only?

Anyways, a bit of an intro:
Like I said i'm in class of 2014 and basically gonna start coming on here more often cause this forum is awesome.

Subjects:
4 Unit Maths
Advanced English
Physics
Chemistry
ATAR GOAL: 97+ (But might as well reach for the stars right? 99? LOL)

Let me make my addition to this thread haha
Integrate secxtanx/sqrt(1-sec^2x) between b and a where b>a
BTW: Is it just 4u questions or can interesting 3u ones come in here as well?

Ikki · Nov 16, 2013

Re: HSC 2013 4U Marathon

SpiralFlex said:
Put a new 2014 thread? I dislike years getting mixed up =[

$\fbox{1}\; $Solve the equation$\; z^2-8(1-i)z+63-16i=0$

$\fbox{2}\; $Prove the parallelogram law$\; |z_{1}-z_{2}|^2+|z_{1}+z_{2}|^2 = 2(|z_{1}|^2+|z_{2}|^2) \;$for all complex numbers$\;z_{1}, z_{2}. \;$You may want to consider a parallelogram in$\; \mathbb{C} \;$with the vertices$\; \mathbf{0}, z_{1}, z_{2}, z_{1}+z_{2}$

$\fbox{3*}\; $Let$\; x, y, z \;$be distinct complex numbers such that$\; y=tx+(1-t)z \;$where$\; 0<t<1.$

$$Prove that$\; \frac{|z|-|y|}{|z-y|} \geq \frac{|z|-|x|}{|z-x|} \geq \frac{|y|-|x|}{|y-x|}$

(Q1)

5-12i and 3+4i.
----------
z=4(1-i)+-sqrt(-63-16i) (After simplification) (Quadratic Formula)
----------
Delta=-63-16i
Root Delta= +-(1-8i)
-----------------------------

Therefore,
z=4-4i+-(1-8i)
z=5-12i and 3+4i

(Q2) Drew up parallelogram with z1 being on real axis and z2 being on imaginary axis. Then using head to tail etc etc.
Formed a square -> right angled triangles using diagonals.
In other words: (using pythagorus)
|z1|^2+|z2|^2=|z1+z2|^2 <-(1)
and
|z1|^2+|z2|^2=|z1-z2|^2 <-(2)

Now, (1)+(2):
|z1+z2|^2 + |z1-z2|^2 = 2(|z1|^2+|z2|^2)
Therefore, proven.

(Q3)-- Having problems knowing where to start, hint maybe? LOL

Anyways, thanks guys, this forum is awesome.
Keep the hard complex questions coming, 4u assessment is pure complex

obliviousninja · Nov 16, 2013

Re: HSC 2013 4U Marathon

Ikki said:
Wassup guys, i'm part of the 2014 cohort, how do you guys post solutions in here? Latex only?

Anyways, a bit of an intro:
Like I said i'm in class of 2014 and basically gonna start coming on here more often cause this forum is awesome.

Subjects:
4 Unit Maths
Advanced English
Physics
Chemistry
ATAR GOAL: 97+ (But might as well reach for the stars right? 99? LOL)

Let me make my addition to this thread haha
Integrate secxtanx/sqrt(1-sec^2x) between b and a where b>a
BTW: Is it just 4u questions or can interesting 3u ones come in here as well?

there's a separate 3u thread, but if its under harder 3u, post here.

Ikki · Nov 17, 2013

Re: HSC 2013 4U Marathon

Alright. this might pass as harder 3u, not sure lol. It's pretty interesting for 4u anyways:
Just learning latex off others haha:

$\fbox{1} \int \frac{secxtanx}{\sqrt{1-sec^2x} } \,\mathrm{d}x $ between b and a, where $ b>a$

LOL My noob latex skillz.
Could someone tell me how to put boundaries on integration and seperate the dx from the fraction to make it look more nice
Thankyou

Kurosaki · Nov 17, 2013

Re: HSC 2013 4U Marathon

Ikki said:
Alright. this might pass as harder 3u, not sure lol. It's pretty interesting for 4u anyways:
Just learning latex off others haha:

$\fbox{1} \int \frac{secxtanx}{\sqrt{1-sec^2x} } \mathrm{d}x $ between b and a, where $ b>a$

$$Hint: Use the substitution $ u=secx$

LOL My noob latex skillz.
Could someone tell me how to put boundaries on integration and seperate the dx from the fraction to make it look more nice
Thankyou

$\int_{a}^{b}\frac{secxtanx}{\sqrt{1-sec^2x}}dx$
Is this what you mean?

seanieg89 · Nov 17, 2013

Re: HSC 2013 4U Marathon

"\," is a common spacing string used to separate integrand from the "dx".

RealiseNothing · Nov 17, 2013

Re: HSC 2013 4U Marathon

Ikki said:
Alright. this might pass as harder 3u, not sure lol. It's pretty interesting for 4u anyways:
Just learning latex off others haha:

$\fbox{1} \int \frac{secxtanx}{\sqrt{1-sec^2x} } \mathrm{d}x $ between b and a, where $ b>a$

$$Hint: Use the substitution $ u=secx$

LOL My noob latex skillz.
Could someone tell me how to put boundaries on integration and seperate the dx from the fraction to make it look more nice
Thankyou

Hints aren't welcomed here.

hit patel · Nov 17, 2013

Re: HSC 2013 4U Marathon

A string is attached to a mass and its other end attached to the pivot of a circular disk of radius R. The disk in on a shaft (assume the shaft and the disk is weightless) and the length of the string is h metres. The shaft rotates anticlockwise and as the speed of shaft increases the mass swings outward. Let the Mass be m and the disk rotates about its axle with a constant angular velocity. This constant angular velocity w produces an angle θ with the vertical.
a) Find the relation of w, h, R, g in terms of theta.
b) Use graphical means to show that for a given w there is just one value of theta in the range of 0<θ<pi/2 (less than or equal to symbol), which satisfies this relation.
c) Given R=6, h=2, θ=60 and g=10, find w relative to ground.

I cant seem to get part b). a nd c is done. Please help 2013 ers. Question open to all .

Thanks

asianese · Nov 17, 2013

Re: HSC 2013 4U Marathon

Ikki said:
Alright. this might pass as harder 3u, not sure lol. It's pretty interesting for 4u anyways:
Just learning latex off others haha:

$\fbox{1} \int \frac{secxtanx}{\sqrt{1-sec^2x} } \mathrm{d}x $ between b and a, where $ b>a$

$$Hint: Use the substitution $ u=secx$

LOL My noob latex skillz.
Could someone tell me how to put boundaries on integration and seperate the dx from the fraction to make it look more nice
Thankyou

seanieg89 said:
"\," is a common spacing string used to separate integrand from the "dx".

Alternatively you can use the \mathop{dx} command (mathematical operation) which spaces things nicely.

Ikki · Nov 18, 2013

Re: HSC 2013 4U Marathon

Kurosaki said:
$\int_{a}^{b}\frac{secxtanx}{\sqrt{1-sec^2x}}dx$
Is this what you mean?

Yeah, thanks

RealiseNothing said:
Hints aren't welcomed here.

Step back bro, too pro LOL

seanieg89 said:
"\," is a common spacing string used to separate integrand from the "dx".

Thanks, works great.

hit patel said:
A string is attached to a mass and its other end attached to the pivot of a circular disk of radius R. The disk in on a shaft (assume the shaft and the disk is weightless) and the length of the string is h metres. The shaft rotates anticlockwise and as the speed of shaft increases the mass swings outward. Let the Mass be m and the disk rotates about its axle with a constant angular velocity. This constant angular velocity w produces an angle θ with the vertical.
a) Find the relation of w, h, R, g in terms of theta.
b) Use graphical means to show that for a given w there is just one value of theta in the range of 0<θ<pi/2 (less than or equal to symbol), which satisfies this relation.
c) Given R=6, h=2, θ=60 and g=10, find w relative to ground.

I cant seem to get part b). a nd c is done. Please help 2013 ers. Question open to all .

Thanks

Shii, is this mechanics already?

bottleofyarn · Nov 18, 2013

Re: HSC 2013 4U Marathon

hit patel said:
A string is attached to a mass and its other end attached to the pivot of a circular disk of radius R. The disk in on a shaft (assume the shaft and the disk is weightless) and the length of the string is h metres. The shaft rotates anticlockwise and as the speed of shaft increases the mass swings outward. Let the Mass be m and the disk rotates about its axle with a constant angular velocity. This constant angular velocity w produces an angle θ with the vertical.
a) Find the relation of w, h, R, g in terms of theta.
b) Use graphical means to show that for a given w there is just one value of theta in the range of 0<θ<pi/2 (less than or equal to symbol), which satisfies this relation.
c) Given R=6, h=2, θ=60 and g=10, find w relative to ground.

I cant seem to get part b). a nd c is done. Please help 2013 ers. Question open to all .

Thanks

Mechanics already!
I'm not sure if I'm missing a force or my diagram is wrong, but I get $\tan\theta=\frac{\omega^2R}{g}$ which is missing h. Graphing this for w>0, it has an inverse function (autonomous increase is the phrase I think) and so there is only one value for theta for any w.
My answer for c) is $\omega=\frac{\sqrt5}{\sqrt[4]3}$ .
Also, is the pivot of the disc the centre?

RealiseNothing · Nov 18, 2013

Re: HSC 2013 4U Marathon

bottleofyarn said:
Mechanics already!
I'm not sure if I'm missing a force or my diagram is wrong, but I get $\tan\theta=\frac{\omega^2R}{g}$ which is missing h. Graphing this for w>0, it has an inverse function (autonomous increase is the phrase I think) and so there is only one value for theta for any w.
My answer for c) is $\omega=\frac{\sqrt5}{\sqrt[4]3}$ .
Also, is the pivot of the disc the centre?

Monotone increasing would be a better phrase.

bottleofyarn · Nov 18, 2013

Re: HSC 2013 4U Marathon

RealiseNothing said:
Monotone increasing would be a better phrase.

That's the one.

hayabusaboston · Nov 18, 2013

Re: HSC 2013 4U Marathon

Show that for n greater than 2, there are no positive integer solutions for a^n+b^n=c^n

seanieg89 · Nov 18, 2013

Re: HSC 2013 4U Marathon

hayabusaboston said:
Show that for n greater than 2, there are no positive integer solutions for a^n+b^n=c^n

"Positive integer solutions" means a,b,c must be positive integers. You have not restricted n though beyond saying that it is larger than 2.

So

a=b=3, c=4 solves the equation with n=log(1/2)/log(3/4).

Hence the claim is untrue.

Ikki · Nov 18, 2013

Re: HSC 2013 4U Marathon

hayabusaboston said:
Show that for n greater than 2, there are no positive integer solutions for a^n+b^n=c^n

I'm not sure where you got this from but it is a very famous problem:

Fermat's Last Theorem

No way, any of us on here could probably do it...

Anyways, Check it out,
http://en.wikipedia.org/wiki/Fermat's_Last_Theorem
and
https://www.youtube.com/watch?v=qiNcEguuFSA

UNLESS:

seanieg89 said:
"Positive integer solutions" means a,b,c must be positive integers. You have not restricted n though beyond saying that it is larger than 2.

So

a=b=3, c=4 solves the equation with n=log(1/2)/log(3/4).

Hence the claim is untrue.

LOL nice one, caught on technicality? But pretty sure it's referring to fermat's last theorem. Where n must also be an integer.

hit patel · Nov 18, 2013

Re: HSC 2013 4U Marathon

bottleofyarn said:
Mechanics already!
I'm not sure if I'm missing a force or my diagram is wrong, but I get $\tan\theta=\frac{\omega^2R}{g}$ which is missing h. Graphing this for w>0, it has an inverse function (autonomous increase is the phrase I think) and so there is only one value for theta for any w.
My answer for c) is $\omega=\frac{\sqrt5}{\sqrt[4]3}$ .
Also, is the pivot of the disc the centre?

Oh okkkkk... Right thanks. Yep mechanics is funn... I literally think mechanics is gonna be my best subject but havent done conics yet so cant say too much about the combination but mechanics with parametrics is yeh good, (physics seems like a kid in front of it- apparently i heard they took out mechanics from physsics and putit in four unit) nope the pivot is at the end of the circular disk thereby given radius R; If it was the centre of the disk and therfore the shaft , then i donot see why would it be required am i right? Thanks I got everything except the meaning of autonomous?

RealiseNothing said:
Monotone increasing would be a better phrase.

Oh ok i get that.

hayabusaboston said:
Show that for n greater than 2, there are no positive integer solutions for a^n+b^n=c^n

Whow How do you show that?> Is that harder ext 1?

seanieg89 said:
"Positive integer solutions" means a,b,c must be positive integers. You have not restricted n though beyond saying that it is larger than 2.

So

a=b=3, c=4 solves the equation with n=log(1/2)/log(3/4).

Hence the claim is untrue.

I dont get it.

RealiseNothing · Nov 18, 2013

Re: HSC 2013 4U Marathon

hit patel said:
Whow How do you show that?> Is that harder ext 1?

Don't worry about it. It was an unsolved problem for about 300 years until Wiles solves it in the 90's after progress and links from many previous mathematician's work.

You'll be able to do it by half yearlies.

hit patel · Nov 18, 2013

Re: HSC 2013 4U Marathon

RealiseNothing said:
Don't worry about it. It was an unsolved problem for about 300 years until Wiles solves it in the 90's after progress and links from many previous mathematician's work.

You'll be able to do it by half yearlies.

haha i feel dumb,

. But hope so. What topic is it in?

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