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March 28, 2024, 08:27:40 pm

Author Topic: 3U Maths Question Thread  (Read 1230184 times)  Share 

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jakesilove

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Re: 3U Maths Question Thread
« Reply #645 on: September 02, 2016, 07:10:16 pm »
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I was wondering how do you do these two questions?

Hey! Just for everyone's else sake,

THIS IS NOT HSC!

Whilst I have done matrices, I don't really understand the terminology used, so won't be able to help you out here. Sorry!

Jake
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anotherworld2b

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Re: 3U Maths Question Thread
« Reply #646 on: September 02, 2016, 10:48:48 pm »
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oh okay :) thanks anyway
Hey! Just for everyone's else sake,

THIS IS NOT HSC!

Whilst I have done matrices, I don't really understand the terminology used, so won't be able to help you out here. Sorry!

Jake

jamonwindeyer

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Re: 3U Maths Question Thread
« Reply #647 on: September 02, 2016, 11:32:12 pm »
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I was wondering how do you do these two questions?

I can handle this one ;D

First, a definition a shear, in case you need it. A shear parallel to a given line through the origin is a transformation in which the component of a vector perpendicular to the line is unchanged, and the component parallel to the line is increased by an amount proportional to the perpendicular component. If the proportionality constant is k, we call it a k–shear.

Okay, so what you are looking at here is a linear transformation that applies both a shear, and a rotation, on the cartesian plane. Recall that we can represent any linear transformation with a matrix multiplication, we just need to find a matrix A such that:



How do we do this? Fortunately, we can actually get each column of the matrix A by considering the transformation of the standard basis vectors in the domain, \(\binom{1}{0}, \binom{0}{1}\). The results in the co-domain will give us the columns of matrix A.

I'll do the first one slow. First, apply the shear to \(\binom{1}{0}\):



Now, apply the rotation:



Let me know if you need help with those individual parts of this question! So, the two vectors we need are:



So the matrix is formed with these as its columns:



There is a fair bit of decently complex theory at play here, let me know if you need anything clarified! :)

I'll leave you to tackle the second one, same principle, it is just applying the rotation first!

anotherworld2b

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Re: 3U Maths Question Thread
« Reply #648 on: September 03, 2016, 01:13:54 am »
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For this question im not sure how to start it

jamonwindeyer

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Re: 3U Maths Question Thread
« Reply #649 on: September 03, 2016, 02:57:06 am »
+1
For this question im not sure how to start it

Hey! You'll form two sets of simultaneous, do the matrix multiplication on the LHS:



Equate corresponding matrix elements to obtain two sets of simultaneous equations, see how that treats you ;)

Disclaimer: Again, non HSC related question

In fact, another world, post your math questions in the relevant board here from now on, I'll be sure to watch it, just to keep this thread for HSC stuff to prevent confusion ;D
« Last Edit: September 03, 2016, 02:59:04 am by jamonwindeyer »

RuiAce

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Re: 3U Maths Question Thread
« Reply #650 on: September 03, 2016, 06:31:59 am »
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For this question im not sure how to start it
Matrices are in no way a part of the HSC course.

I don't mind questions from students outside of NSW however please refer to the syllabus for the HSC before posting questions here.
I can handle this one ;D

First, a definition a shear, in case you need it. A shear parallel to a given line through the origin is a transformation in which the component of a vector perpendicular to the line is unchanged, and the component parallel to the line is increased by an amount proportional to the perpendicular component. If the proportionality constant is k, we call it a k–shear.

How did you even know this anyway  :o

For this question im not sure how to start it

You guys were taught matrix multiplication right? Or does Jamon's answer need clarity
« Last Edit: September 03, 2016, 06:48:26 am by RuiAce »

anotherworld2b

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Re: 3U Maths Question Thread
« Reply #651 on: September 03, 2016, 08:16:28 am »
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Sorry for the inconvenience caused.i wasnt wure where to post my questions

RuiAce

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Re: 3U Maths Question Thread
« Reply #652 on: September 03, 2016, 09:33:23 am »
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Sorry for the inconvenience caused.i wasnt wure where to post my questions
All good

There are threads for every state to post their questions. But I understand perfectly if they feel isolated or hard to find

anotherworld2b

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Re: 3U Maths Question Thread
« Reply #653 on: September 03, 2016, 10:09:46 am »
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I tried to make a new topic for a while but all my photos of questions keep failing security checks so i cant post it?

jamonwindeyer

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Re: 3U Maths Question Thread
« Reply #654 on: September 03, 2016, 10:31:41 am »
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How did you even know this anyway  :o

Actually did an example on shears in a lecture last week, it was timely, I'm doing linear transformations right now ;) (all revision from Math 1B)

jakesilove

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Re: 3U Maths Question Thread
« Reply #655 on: September 04, 2016, 12:01:38 pm »
+1
For anyone looking to absolutely smash their HSC exam, understanding what the question actually expects of you is vital. Check our Rui's beast guide of Maths verbs HERE, and get an edge in your final exam! As always, thanks must go out to the legend himself, RuiAce; Improving Atars Since 2015.
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amandali

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Re: 3U Maths Question Thread
« Reply #656 on: September 08, 2016, 10:45:18 am »
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how to do part b ii) a and b

jakesilove

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Re: 3U Maths Question Thread
« Reply #657 on: September 08, 2016, 11:18:29 am »
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(Image removed from quote.)

how to do part b ii) a and b

For a), we know that the amount of oil being poured into the hemisphere is



Therefore, after 8 minutes, there will be a volume of



Setting this as V,




Just looking at the equation, it's pretty clear that the solution is h=1m

For b), we need to start by finding the rate of change equation.



This should be intuitive by now; set what you want to find on the left hand side, then use what you can find out to produce a right hand side. We know the change in volume over time; we just need to change h over V.





I'll leave you to sub the relevant numbers in
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Ali_Abbas

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Re: 3U Maths Question Thread
« Reply #658 on: September 08, 2016, 11:31:24 am »
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For a), we know that the amount of oil being poured into the hemisphere is



Therefore, after 8 minutes, there will be a volume of



Setting this as V,




Just looking at the equation, it's pretty clear that the solution is h=1m

For b), we need to start by finding the rate of change equation.



This should be intuitive by now; set what you want to find on the left hand side, then use what you can find out to produce a right hand side. We know the change in volume over time; we just need to change h over V.





I'll leave you to sub the relevant numbers int

Remark: The resultant degree-3 polynomial in h will, by a consequence of the Fundamental Theorem of Algebra, yield exactly three roots. Although h = 1 was obtained as a solution by inspection, there was no justification for why the remaining two roots are to be rejected. Of course, the simple answer is that intuitively they must not be fitting to the physical context of our scenario. However, it is my opinion that this should be shown. We demonstrate it as follows:

Clearly, one root is negative and the other is greater than three (which is the radius of the semi-sphere). Thus, h = 1 is the only answer which is physically possible.

QED

Note: The Fundamental Theorem of Algebra belongs exclusively to the 4U course. But yes, justification of why certain roots are to be rejected is important.

(Also if the factorisation method is confusing for a 3U student reading this, you can do it by long division.)
« Last Edit: September 08, 2016, 12:01:07 pm by RuiAce »

amandali

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Re: 3U Maths Question Thread
« Reply #659 on: September 09, 2016, 07:02:43 am »
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how to do part a  iii)