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March 29, 2024, 10:37:39 pm

Author Topic: Specialist 1/2 Question Thread!  (Read 120000 times)  Share 

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jazzycab

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Re: Specialist 1/2 Question Thread!
« Reply #225 on: April 25, 2018, 05:02:10 pm »
+2
Hey guys,

Sorry if this was the wrong area to post this, but I was given the cambridge senior maths chapter 18 (matrices) because my methods tutor (i'm in year 11) said it would give us a bigger overall picture on matrices. I'm having a little bit of trouble with exercise 18C. I know how to do the basic multiplying, but don't understand a few of the trickier questions, and I'd be super grateful if you guys could assist me with the following!!

Question 2: http://puu.sh/A9Yqb/283fa45c3e.png
Question 4 and Question 5: http://puu.sh/A9YrO/0bea0789de.png
Question 12: http://puu.sh/A9Yts/f21438a27d.png

Even if you guys can help me with even one of the questions it'd help a lot!!

Cheers!


Question 2 is testing your understanding of the matrix multiplication process, that is:
A product of 2 matrices is defined only if the number of columns in the first matrix is equal to the number of rows in the second matrix.
The matrices have the following dimensions:
\(\mathbf{X}:\ 2\times1,\ \mathbf{Y}:\ 2\times1,\ \mathbf{A}:\ 2\times2,\ \mathbf{B}:\ 2\times2,\ \mathbf{C}:\ 2\times2,\ \mathbf{I}:\ 2\times2\).
Note that the dimensions are listed as "number of rows"\(\times\)"number of columns"
Therefore the following matrix products are defined (of the ones listed):
\(\mathbf{A}\mathbf{Y},\ \mathbf{C}\mathbf{I}\).

For question 4, consider the matrix product \(\mathbf{AB}\) for two arbitrary \(2\times2\) matrices:

If \(\mathbf{AB}=\mathbf{O}\) then \(ae+bg=0,\ af+bh=0,\ ce+dg=0,\ cf+dh=0\).
All we need to determine is if there is any possible combination of these such that not all of \(a,\ b,\ c\text{ and }d\) are 0 and not all of \(e,\ f,\ g\text{ and }h\) are 0.
We have \(ae=-bg\Rightarrow\frac{a}{b}=-\frac{g}{e}\) and \(af=-bh\Rightarrow\frac{a}{b}=-\frac{h}{f}\). Therefore, \(\frac{g}{e}=\frac{h}{f}\).
We also have \(ce=-dg\Rightarrow\frac{c}{d}=-\frac{g}{e}\) and \(cf=-dh\Rightarrow\frac{c}{d}=-\frac{h}{f}\).
Combining all four of these gives us \(\frac{a}{b}=\frac{c}{d}=-\frac{g}{e}=-\frac{h}{f}\).
If we can find an example where this is true, for which none of \(a,\ b,\ c,\ d,\ e,\ f\text{ and }g\) are 0, then we have disproven the statement.
Consider the simple case where \(a=1,\ b=1,\ c=1,\ d=1,\ g=1,\ e=-1,\ h=1\text{ and }f=-1\). In this case, we get:

Thus, the statement is disproved (i.e. the null factor law is not true for matrices - although the product \(\mathbf{OX}=\mathbf{O}\), this doesn't necessarily mean that the product \(\mathbf{AB}=\mathbf{O}\) implies that \(\mathbf{A}=\mathbf{O}\) or \(\mathbf{B}=\mathbf{O}\) (or both).

Note that the way question 5 is asked gives a big hint as to the answer to question 4.
Firstly, if \(\mathbf{A}^2\) is defined, then \(\mathbf{A}\) must be square. Also, if the dimensions of \(\mathbf{A}\) are \(m\times m\), then the dimensions of \(\mathbf{A}^2\) and therefore \(\mathbf{O}\) are also \(m\times m\).
If \(\mathbf{A}\times\mathbf{A}=\mathbf{O}\), then \(\mathbf{A}^{-1}\mathbf{A}\mathbf{A}=\mathbf{A}^{-1}\mathbf{O}\), which gives \(\mathbf{I}\mathbf{A}=\mathbf{A}^{-1}\mathbf{O}\), finally giving \(\mathbf{A}=\mathbf{A}^{-1}\mathbf{O}\). Note that this seems to indicate that \(\mathbf{A}=\mathbf{O}\), however, this is ignoring the case where the inverse doesn't exist (i.e. when \(\text{det}\left(\mathbf{A}\right)=0\).)
Let's consider the \(2\times2\) case, to keep the arithmetic as simple as possible.

Now we have \(a^2+bc=0,\ ab+bd=0,\ ac+cd=0,\ bc+d^2=0\text{ and }ad=bc\), which gives \(a^2=-bc,\ ab=-bd,\ ac=-cd,\ bc=-d^2\text{ and }ad=bc\).
Consider the simple case where \(c=0\). This gives: \(a^2=-b\times0,\ ab=-bd,\ a\times0=-0\times d,\ b\times0=-d^2\text{ and }ad=b\times0\), which gives \(a^2=0,\ ab=-bd,\ 0=0,\ 0=-d^2\text{ and }ad=0\), finally giving \(a=0, c=0, d=0\). This will be true for ANY real value of \(b\), so simply choose a non-zero \(b\) and you have a matrix \(mathbf{A}\) for which this property is true (note that there are other solutions to)


For question 12a, note that the top row of the \(2\times2\) matrix contains the time, in minutes, that it takes John to drink a milkshake and eat a banana split, respectively and its bottom row contains the cost of a milkshake and a banana split respectively.
The matrix product gives us

The intermediate multiplication step gives us an insight into what the product represents. We have, in the top row, the time it takes John to drink a milkshake multiplied by 1, added to the time it takes John to eat a banana split multiplied by 2 and, in the bottom row, the cost of a milkshake multiplied 1, plus the cost of a banana split multiplied by 2.
Hence, the product tells us the time it would take John to drink a milkshake and eat 2 banana splits (row 1) and the cost of one milkshake and two banana splits (row 2).
The interpretation for part b, follows from that of part a:

Assuming that the speed that John's friends eat and drink at is exactly the same as John, we can interpret the information in this matrix as follows:
The time it takes John to drink a milkshake and eat 2 banana splits (row 1, column 1)
The cost of John's milkshake and 2 banana splits (row 2, column 1)
The time it takes one of John's friends to drink 2 milkshakes and eat a banana split (row 1, column 2)
The cost of John's first friend's banana split and 2 milkshakes (row 2, column 2)
The time it takes the second of John's friends to eat a banana split (row 1, column 3)
The cost of John's second friend's banana split (row 2, column 3)
Given that we weren't actually given any information about how fast John's friends eat and drink, you may interpret this matrix slightly differently (i.e. maybe John ordered different things on different days), but the information about his friends indicates that this is the logical way to answer it.

An additional note: The knowledge of these matrix ideas is well beyond what is required, even at Unit 3&4 Methods. A basic understanding of how matrices are multiplied is sufficient for what can be assessed on the VCAA exams.
« Last Edit: April 25, 2018, 06:18:58 pm by jazzycab »

jacquieg

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Re: Specialist 1/2 Question Thread!
« Reply #226 on: September 01, 2018, 10:57:58 am »
0
Hi everyone,
I've come across something strange in my antidiff homework, I've attached the question for those who want to check.
I know it's the integral of the equation between x=4 and x=5 so I long divided it then used partial fractions etc and i checked this part on the CAS so i'm sure it's all fine, and one part of equation that I had to anti differentiate was 1/3(x-3) so i just took out the 1/3 and differentiated it as usual, so I got 1/3loge|x-3|. But I got an incorrect answer for the question as a whole.
Then I thought if i just expand those brackets in the denominator I get 1/3x-9 and that antidifferentiated is 1/3loge|3x-9|.... so two different answers from the same equation.... how do I know which one is right and what's the difference between these 2? Is there only one way to correctly antidifferentiate 1/3x-9 or are both acceptable?
I'm so confused!
Jacquie G

S_R_K

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Re: Specialist 1/2 Question Thread!
« Reply #227 on: September 01, 2018, 03:44:49 pm »
0
Hi everyone,
I've come across something strange in my antidiff homework, I've attached the question for those who want to check.
I know it's the integral of the equation between x=4 and x=5 so I long divided it then used partial fractions etc and i checked this part on the CAS so i'm sure it's all fine, and one part of equation that I had to anti differentiate was 1/3(x-3) so i just took out the 1/3 and differentiated it as usual, so I got 1/3loge|x-3|. But I got an incorrect answer for the question as a whole.
Then I thought if i just expand those brackets in the denominator I get 1/3x-9 and that antidifferentiated is 1/3loge|3x-9|.... so two different answers from the same equation.... how do I know which one is right and what's the difference between these 2? Is there only one way to correctly antidifferentiate 1/3x-9 or are both acceptable?
I'm so confused!

Note that:



Hence:



since ln(3) is a constant.

Purple_Mango

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Re: Specialist 1/2 Question Thread!
« Reply #228 on: September 12, 2018, 05:20:07 pm »
0
Hello everyone,
I don't know if this is the right place to post... but... it's relevant to Spesh 1&2, so it should be okay?
Anyway, I just had a SAC on vectors, and there was this one question I (and a majority of my classmates) had trouble with, and I would really like to know how one would answer it. Here it is, from the top of my memory, under spoiler in case some people don't want to read it.

Spoiler
Quadrilateral ABCD has the points E, F, G, H, P and Q as midpoints of AD, AB, BC, CD, AC, and BD respectively.
Show that EG, FH and PQ bisect each other at a point O.

7 Marks
I tried drawing a diagram of it - hopefully it's accurate !
I do not know how to use LaTex... so that is why it does not look right... I`m sorry

2018: Chemistry [45], Maths Methods [43]
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jacquieg

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Re: Specialist 1/2 Question Thread!
« Reply #229 on: October 07, 2018, 08:44:39 pm »
0
plz help question 8B, dont really get this topic either...
Jacquie G

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Re: Specialist 1/2 Question Thread!
« Reply #230 on: October 07, 2018, 10:21:24 pm »
0
plz help question 8B, dont really get this topic either...

8b is asking for the second derivative of x with respect to t. You've already got the first derivative (it's given in the equation as dx/dt), so just differentiate again to get the second derivative.

anon101

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Re: Specialist 1/2 Question Thread!
« Reply #231 on: November 02, 2018, 09:33:34 pm »
0
For learning:
Khan Academy (and practice)
Paul's Online Math Notes
Better Explained

If you're running out of fresh questions to do, maybe try a methods textbook? Obviously not all the content will be the same but there's plenty of relevant stuff in there. You should ask your teacher too, they might have more resources. Personally, I learned the most during practice (and actual ::) ) SACs. This might also be helpful :)
Is there anywhere I can find any year 11 practise exams free though?
Because as it's the end of the year, I feel what I need to do most is exam style questions.

Sine

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Re: Specialist 1/2 Question Thread!
« Reply #232 on: November 03, 2018, 12:18:46 am »
0
Is there anywhere I can find any year 11 practise exams free though?
Because as it's the end of the year, I feel what I need to do most is exam style questions.
As schools teach and assess different content it's quite difficult to get practice exams online that are indicative of what you will recieve from your teacher. Furthermore, commercial practice exams cannot be shared online due to copyright.
That is why it's recommended that you use the resources your teacher has provided you as that is probably what is going to be most like what you will get from the exam.

anon101

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Re: Specialist 1/2 Question Thread!
« Reply #233 on: November 10, 2018, 01:09:35 pm »
0
The thing is.
Our teacher doesn't really give us anything at all!
I've just got the textbook, and it doesn't have exam-like questions.
Even for Methods 1&2 though, I got like 5 practise exams off a friend.
So why isn't it possible to find any for 1&2 Specialist. I know that the content's slightly different but still.
Shouldn't there be like topic tests for like Geometry and Circle Geometry with like exam styled questions?
Thanks a load dude

S_R_K

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Re: Specialist 1/2 Question Thread!
« Reply #234 on: November 10, 2018, 02:38:07 pm »
0
The thing is.
Our teacher doesn't really give us anything at all!
I've just got the textbook, and it doesn't have exam-like questions.
Even for Methods 1&2 though, I got like 5 practise exams off a friend.
So why isn't it possible to find any for 1&2 Specialist. I know that the content's slightly different but still.
Shouldn't there be like topic tests for like Geometry and Circle Geometry with like exam styled questions?
Thanks a load dude

The difference is that the Methods course is completely prescribed (ie. every school teaches exactly the same content). Specialist 1 & 2 is not; there is a lot of flexibility in what topics are chosen and how they are taught. So while there are commercial exams available for Specialist 1 & 2, schools may choose not to purchase them because the course they have run doesn't align with the questions on the commercial exams.

This also puts teachers in a difficult position because, typically, there is only one class and hence one teacher, so it's asking a lot for that one teacher to produce a whole lot of exam-like questions.

Don't worry, Units 3 & 4 is much better resourced.

anon101

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Re: Specialist 1/2 Question Thread!
« Reply #235 on: November 10, 2018, 07:19:32 pm »
0
The difference is that the Methods course is completely prescribed (ie. every school teaches exactly the same content). Specialist 1 & 2 is not; there is a lot of flexibility in what topics are chosen and how they are taught. So while there are commercial exams available for Specialist 1 & 2, schools may choose not to purchase them because the course they have run doesn't align with the questions on the commercial exams.

This also puts teachers in a difficult position because, typically, there is only one class and hence one teacher, so it's asking a lot for that one teacher to produce a whole lot of exam-like questions.

Don't worry, Units 3 & 4 is much better resourced.

It's just that I need to do well in this exam if I'm going to do Specialist 3&4.
Because otherwise, I'm not quite sure how I'll go in year 12.
Thanks,
Kamron

accountingpro

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Re: Specialist 1/2 Question Thread!
« Reply #236 on: November 21, 2018, 09:37:49 pm »
0
Hey y'all can someone give us a hand with this one. Been trying for a while but I think I'm just missing 1 step repeatedly lol.

RuiAce

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Re: Specialist 1/2 Question Thread!
« Reply #237 on: November 21, 2018, 10:22:53 pm »
+2

Sine

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Re: Specialist 1/2 Question Thread!
« Reply #238 on: November 22, 2018, 12:06:52 am »
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(Image removed from quote.) Hey y'all can someone give us a hand with this one. Been trying for a while but I think I'm just missing 1 step repeatedly lol.
For these types of questions (only MCQ so you always have your CAS and don't need to show working out) if you are struggling to convert the question into an answer a not so eloquent solution would be to assume x = some random number (you can go to heaps of decimal places) and then substitute that same number for all 6 expressions (The question + 5 answers) then you can select the one that matches.

EDIT: the above is more of an exam technique when you can't do the question rather than something you should always rely on :)
« Last Edit: November 22, 2018, 12:12:39 am by Sine »

accountingpro

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Re: Specialist 1/2 Question Thread!
« Reply #239 on: November 22, 2018, 11:39:30 am »
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I've checked all 5 answers on WolframAlpha and none of them are true.
A
B
C
D
E

It may be worth mentioning that I got it down to \( \frac12 \tan x \cos 2x \), which Wolfram does agree with.

I think I got the same. Just was told today that none of these are correct lol.