Login | Register

Welcome, Guest. Please login or register.

September 19, 2020, 12:34:27 am

Author Topic: VCE Specialist 3/4 Question Thread!  (Read 1320890 times)  Share 

0 Members and 1 Guest are viewing this topic.

TrueTears

  • TT
  • Honorary Moderator
  • Great Wonder of ATAR Notes
  • *******
  • Posts: 16369
  • Respect: +659
VCE Specialist 3/4 Question Thread!
« on: November 26, 2011, 10:07:42 pm »
+13
VCE SPECIALIST MATHS Q&A THREAD

To go straight to posts from 2020, click here.

What is this thread for?
If you have general questions about the VCE Specialist Maths course or how to improve in certain areas, this is the place to ask!


Who can/will answer questions?
Everyone is welcome to contribute; even if you're unsure of yourself, providing different perspectives is incredibly valuable.

Please don't be dissuaded by the fact that you haven't finished Year 12, or didn't score as highly as others, or your advice contradicts something else you've seen on this thread, or whatever; none of this disqualifies you from helping others. And if you're worried you do have some sort of misconception, put it out there and someone else can clarify and modify your understanding! 

There'll be a whole bunch of other high-scoring students with their own wealths of wisdom to share with you, including TuteSmart tutors! So you may even get multiple answers from different people offering their insights - very cool.


To ask a question or make a post, you will first need an ATAR Notes account. You probably already have one, but if you don't, it takes about four seconds to sign up - and completely free!

OTHER SPESH RESOURCES

Original post.
similar to the methods one Methods [3/4] Summer Holidays Question Thread! post away your questions from your summer holidays self-studying, everyone can discuss and benefit! I'll try answer as much questions as possible too ^^
« Last Edit: February 26, 2020, 03:22:03 pm by PhoenixxFire »
Currently studying: PhD in economics at MIT.

Interested in financial economics, econometrics, and asset pricing.

Special At Specialist

  • Victorian
  • Part of the furniture
  • *****
  • Posts: 1558
  • Respect: +86
  • School: Flinders Christian Community College (Tyabb)
  • School Grad Year: 2012
Re: Specialist 3/4 Question Thread!
« Reply #1 on: November 27, 2011, 12:00:02 pm »
0
Let w = 2cis(θ) and z = w + 1/w
Show that z lies on the ellipse with equation (x^2)/25 + (y^2)/9 = 1/4
2012 ATAR - 86.75
2013 ATAR - 88.50
2014: BSci (Statistics) at RMIT
2015 - 2017: BCom at UoM

brightsky

  • Victorian
  • ATAR Notes Legend
  • *******
  • Posts: 3133
  • Respect: +200
Re: Specialist 3/4 Question Thread!
« Reply #2 on: November 27, 2011, 12:28:30 pm »
+4
z = w + 1/w = 2cist + 1/(2cist) = 2cist + 1/(2(cost + isint)) = 2cist + (cost - i sint)/(2) = 5/2 cost + 3/2 i sint
parameters:
x = 5/2cost
y =3/2 sint
convert into cartesian form and you get the equation of the ellipse.
2020 - 2021: Master of Public Health, The University of Sydney
2017 - 2020: Doctor of Medicine, The University of Melbourne
2014 - 2016: Bachelor of Biomedicine, The University of Melbourne
2013 ATAR: 99.95

Currently selling copies of the VCE Chinese Exam Revision Book and UMEP Maths Exam Revision Book, and accepting students for Maths Methods and Specialist Maths Tutoring in 2020!

brightsky

  • Victorian
  • ATAR Notes Legend
  • *******
  • Posts: 3133
  • Respect: +200
Re: Specialist 3/4 Question Thread!
« Reply #3 on: November 27, 2011, 08:47:43 pm »
+1
what do you mean by complicated derivatives? the method of 'antideriving through derivatives' is called integration by recognition, which is just a more abstract form of integration by parts. not sure if this answers your question, but try wikipedia-ing integration by parts.
2020 - 2021: Master of Public Health, The University of Sydney
2017 - 2020: Doctor of Medicine, The University of Melbourne
2014 - 2016: Bachelor of Biomedicine, The University of Melbourne
2013 ATAR: 99.95

Currently selling copies of the VCE Chinese Exam Revision Book and UMEP Maths Exam Revision Book, and accepting students for Maths Methods and Specialist Maths Tutoring in 2020!

Special At Specialist

  • Victorian
  • Part of the furniture
  • *****
  • Posts: 1558
  • Respect: +86
  • School: Flinders Christian Community College (Tyabb)
  • School Grad Year: 2012
Re: Specialist 3/4 Question Thread!
« Reply #4 on: November 27, 2011, 08:48:49 pm »
+2
sorry for hijacking this thread.

heys you know the spesh derivatives that are complicated, it can done be through 'anitderiving through derivatives right'?
is it true that it is possible to antiderive through complicated derivatives? is it way of vce level?

if its possible can any of you guys show me an example of it?

I'm not sure what you mean by anti-deriving through derivatives...
Perhaps you mean is it possible to find an anti-derivative when given a similar derivative? If that's your question, then the answer is yes. Here is an example:

Given that dy/dx = cos(2x), d/dx (sin(2x)) = 2cos(2x) and that x = pi/4 when y = 5, solve for y.
So first you convert the integral of cos(2x) into 1/2 * Integral of 2cos(2x)
Then you use the information they gave you to say:
1/2 * Integral of 2cos(2x) = 1/2 * sin(2x) + C
Now you use the initial conditions to solve for C:
y = 1/2 * sin(2x) + C
5 = 1/2 * sin(pi/2) + C
5 = 1/2 * 1 + C
C = 5 - 1/2
C = 9/2
y = 1/2 * sin(2x) + 9/2

That is called a "differential equation" if you're interested.
I hope this answers your question.

what do you mean by complicated derivatives? the method of 'antideriving through derivatives' is called integration by recognition, which is just a more abstract form of integration by parts. not sure if this answers your question, but try wikipedia-ing integration by parts.

Ye know too much for a year 10 student!
2012 ATAR - 86.75
2013 ATAR - 88.50
2014: BSci (Statistics) at RMIT
2015 - 2017: BCom at UoM

Special At Specialist

  • Victorian
  • Part of the furniture
  • *****
  • Posts: 1558
  • Respect: +86
  • School: Flinders Christian Community College (Tyabb)
  • School Grad Year: 2012
Re: Specialist 3/4 Question Thread!
« Reply #5 on: November 27, 2011, 09:11:07 pm »
+1
Someone please tell me where I went wrong in this question!

Given that z = -3/2 sinθ + 5/2 i cosθ, show that |z - 2i| + |z + 2i| = 5

z - 2i = (-3/2 sinθ) + (5/2 cosθ - 2)i
|z - 2i| = sqrt((-3/2 sinθ))^2 + (5/2 cosθ - 2)^2)
|z - 2i| = sqrt(9/4 sin^2(θ) + 25/4 cos^2(θ) - 10cos(θ) + 4)
|z - 2i| = sqrt(9/4(sin^2(θ) + cos^2(θ)) + 16/4 cos^2(θ) - 10cos(θ) + 4)
|z - 2i| = sqrt(9/4 + 4cos^2(θ) - 10cos(θ) + 16/4)
|z - 2i| = sqrt(4cos^2(θ) - 10cos(θ) + 25/4)
|z - 2i| = sqrt((2cos(θ) - 5/2)^2)
|z - 2i| = 2cos(θ) - 5/2
Similar process for |z + 2i|:
z + 2i = (-3/2 sinθ) + (5/2 cosθ + 2)i
|z + 2i| = sqrt(9/4 sin^2(θ) + 25/4 cos^2(θ) + 10cos(θ) + 4)
|z + 2i| = sqrt(9/4 + 16/4cos^2(θ) + 10cos(θ) + 16/4)
|z + 2i| = sqrt(4cos^2(θ) + 10cos(θ) + 25/4)
|z + 2i| = 2cos(θ) + 5/2
|z - 2i| + |z + 2i| = 2cos(θ) - 5/2 + 2cos(θ) + 5/2
|z - 2i| + |z + 2i| = 4cos(θ)

Where did I go wrong? I was supposed to get the answer 5 but instead I got 4cos(θ) :(
2012 ATAR - 86.75
2013 ATAR - 88.50
2014: BSci (Statistics) at RMIT
2015 - 2017: BCom at UoM

brightsky

  • Victorian
  • ATAR Notes Legend
  • *******
  • Posts: 3133
  • Respect: +200
Re: Specialist 3/4 Question Thread!
« Reply #6 on: November 27, 2011, 09:42:39 pm »
+4
|z - 2i| = sqrt((2cos(θ) - 5/2)^2)
|z - 2i| = 2cos(θ) - 5/2
here's the error. cosθ =< 1, which means  2cosθ - 5/2 =< -1/2
so sqrt((2cos(θ) - 5/2)^2) = abs(2cos(θ) - 5/2) = 5/2 - 2cosθ
2020 - 2021: Master of Public Health, The University of Sydney
2017 - 2020: Doctor of Medicine, The University of Melbourne
2014 - 2016: Bachelor of Biomedicine, The University of Melbourne
2013 ATAR: 99.95

Currently selling copies of the VCE Chinese Exam Revision Book and UMEP Maths Exam Revision Book, and accepting students for Maths Methods and Specialist Maths Tutoring in 2020!

Special At Specialist

  • Victorian
  • Part of the furniture
  • *****
  • Posts: 1558
  • Respect: +86
  • School: Flinders Christian Community College (Tyabb)
  • School Grad Year: 2012
Re: Specialist 3/4 Question Thread!
« Reply #7 on: November 27, 2011, 09:51:46 pm »
0
I never knew you had to do that...
What if it was sqrt((cosθ - 1/2)^2) ?
How would you know whether to take the positive or negative solution?
2012 ATAR - 86.75
2013 ATAR - 88.50
2014: BSci (Statistics) at RMIT
2015 - 2017: BCom at UoM

TrueTears

  • TT
  • Honorary Moderator
  • Great Wonder of ATAR Notes
  • *******
  • Posts: 16369
  • Respect: +659
Re: Specialist 3/4 Question Thread!
« Reply #8 on: November 27, 2011, 09:55:27 pm »
0
oh absolute values!

if it were ur alternative case, then itd be cos(t) <=1 hence cos(t) -1/2 <= -1/2
Currently studying: PhD in economics at MIT.

Interested in financial economics, econometrics, and asset pricing.

brightsky

  • Victorian
  • ATAR Notes Legend
  • *******
  • Posts: 3133
  • Respect: +200
Re: Specialist 3/4 Question Thread!
« Reply #9 on: November 27, 2011, 09:56:32 pm »
0
I never knew you had to do that...
What if it was sqrt((cosθ - 1/2)^2) ?
How would you know whether to take the positive or negative solution?

depends on the restrictions of θ. if none are specified, then you'll have two solutions.
2020 - 2021: Master of Public Health, The University of Sydney
2017 - 2020: Doctor of Medicine, The University of Melbourne
2014 - 2016: Bachelor of Biomedicine, The University of Melbourne
2013 ATAR: 99.95

Currently selling copies of the VCE Chinese Exam Revision Book and UMEP Maths Exam Revision Book, and accepting students for Maths Methods and Specialist Maths Tutoring in 2020!

Special At Specialist

  • Victorian
  • Part of the furniture
  • *****
  • Posts: 1558
  • Respect: +86
  • School: Flinders Christian Community College (Tyabb)
  • School Grad Year: 2012
Re: Specialist 3/4 Question Thread!
« Reply #10 on: November 27, 2011, 10:00:12 pm »
0
That makes sense... actually I think I remember a similar problem when taking an integral which ended up with an absolute value and I was unsure whether to write ln(5 - x) or ln(x - 5) so I had to use the initial conditions to see which solution it was.
This is basically the same concept.
Thanks for your help :)
2012 ATAR - 86.75
2013 ATAR - 88.50
2014: BSci (Statistics) at RMIT
2015 - 2017: BCom at UoM

TrueTears

  • TT
  • Honorary Moderator
  • Great Wonder of ATAR Notes
  • *******
  • Posts: 16369
  • Respect: +659
Re: Specialist 3/4 Question Thread!
« Reply #11 on: November 27, 2011, 10:00:23 pm »
+4
sorry for hijacking this thread.

heys you know the spesh derivatives that are complicated, it can done be through 'anitderiving through derivatives right'?
is it true that it is possible to antiderive through complicated derivatives? is it way of vce level?

if its possible can any of you guys show me an example of it?
yup exactly what brightsky said, this is just integration by parts, you don't need to know this for spesh either, however it's not too hard to learn, it's just derived from the product rule, wikipedia is nice like brightsky said, however just for heck of it, i'll provide an example

basically to derive it, check wiki, you simply just antiderive the product rule, then after rearranging this is the result:



basically u and v are functions, ill show u an example and this will make more asense, note that here im stressing more about the application of this "formula" rather than going through rigorous details etc

eg

first we need to guess which function is u and which is dv, ie what i'm doing is basically "equating equations" (again im not stressing formality, simply showing you the mechanics) so what im doing is this:

we have the integration by parts rule  then we got our "equation" that we need to integrate, ie

so its like saying

so we have to guess, what is u and what is dv, ie, lets guess u = x and dv = sin(x)dx

yes if you're wondering, there's another guess we cudda taken, ie, u = sin(x) and dv = x dx [as you will see only one "pair" would work]

so if we let u = x and dv = sin(x) dx

then du = dx and v = -cos(x)

then all we gotta do is sub this into the formula!

so it becomes

but we know how to do

and so the rest is trivial

you go to try the other "guess" and you will see that the 2nd integral is not something we can integrate easily, so picking the first guess is better.



hopefully that makes a tiny bit more sense, note integration by parts just takes practise, there's no set method, you gain experience as you do more, so lets say

[note this is actually ]

here u can either do let and dv = 1dx or u = 1 and but it is very clear why the 2nd guess doesnt work!

« Last Edit: November 27, 2011, 10:05:50 pm by TrueTears »
Currently studying: PhD in economics at MIT.

Interested in financial economics, econometrics, and asset pricing.

kensan

  • Victorian
  • Forum Leader
  • ****
  • Posts: 693
  • Do you even lift?
  • Respect: +20
  • School: L.C.
  • School Grad Year: 2012
Re: Specialist 3/4 Question Thread!
« Reply #12 on: November 27, 2011, 10:12:51 pm »
+5
I'm doing specialist next year, and I'm looking at these questions having no idea what's going on hahaha. How do you guys know this stuff already, was it in the year 11 course?? feelsbadman.jpeg
2013: BSc at UoM

TrueTears

  • TT
  • Honorary Moderator
  • Great Wonder of ATAR Notes
  • *******
  • Posts: 16369
  • Respect: +659
Re: Specialist 3/4 Question Thread!
« Reply #13 on: November 27, 2011, 10:19:26 pm »
0
I'm doing specialist next year, and I'm looking at these questions having no idea what's going on hahaha. How do you guys know this stuff already, was it in the year 11 course?? feelsbadman.jpeg
heh, no need to worry, just do some self study and this stuff will be easy ^^
Currently studying: PhD in economics at MIT.

Interested in financial economics, econometrics, and asset pricing.

kensan

  • Victorian
  • Forum Leader
  • ****
  • Posts: 693
  • Do you even lift?
  • Respect: +20
  • School: L.C.
  • School Grad Year: 2012
Re: Specialist 3/4 Question Thread!
« Reply #14 on: November 27, 2011, 10:25:04 pm »
0
I'm doing specialist next year, and I'm looking at these questions having no idea what's going on hahaha. How do you guys know this stuff already, was it in the year 11 course?? feelsbadman.jpeg
heh, no need to worry, just do some self study and this stuff will be easy ^^
From the textbook? I don't think I'm getting that till late Jan next year.. because I'm going overseas to Japan!! I'm thinking of taking some material over there though, are there any online places where I can learn the basics?
2013: BSc at UoM