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March 29, 2024, 07:28:26 am

Author Topic: VCE Specialist 3/4 Question Thread!  (Read 2164354 times)  Share 

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Ansaki

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9435 on: March 13, 2019, 10:42:42 pm »
0
hey guys! i got a few questions
if z=1+i, then Arg(i^3z) is: 
A) 7π/4
B) 5π/4
C) -3π/4
D) 3π/4
E) -π/4

The complex relation Re(z-1)*Im(z-1)=1 can be represented on a Cartesian plane as a:
A) hyperbola with asymptotes x=1,y=0
B) hyperbola with asymptotes y= plus-minus x.
C) hyperbola with asymptotes x=0,y=1
D) straight line with equation y=1-x
E) hyperbola with equation y=1//x-1 + 1

Question:
(a) Consider u=a-asqrt3 i, where a<0, and w=bcis(π/4)
i. Express u in polar form.
ii. Express w in Cartesian form

(b) Express each of the following as a complex number.
i. u+w in Cartesian form
ii. uw in polar form
iii. u^6 in cartesian form

(c) Find the square roots of w in polar form.

(d) If a=-1/2 and b=sqrt2, plot u^12 and w^-2 on the Argand diagram below.

AlphaZero

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9436 on: March 14, 2019, 09:10:02 am »
0
...

Please show any attempts you have made / any relevant understanding you have, so that we can figure out the point(s) of confusion.
« Last Edit: March 14, 2019, 02:35:23 pm by AlphaZero »
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yerikim

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9437 on: March 19, 2019, 07:23:48 pm »
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Hi guys,

I need help with this question

Let sec a = b where b ∈ (π/2, π)/ Find, in therms of a, two values of x in [-π, π] which satisfy each of the following equations.
b. cosec x = b


lzxnl

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9438 on: March 19, 2019, 10:05:10 pm »
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Hi guys,

I need help with this question

Let sec a = b where b ∈ (π/2, π)/ Find, in therms of a, two values of x in [-π, π] which satisfy each of the following equations.
b. cosec x = b



How are sine and cosine related? Can you write \(\sin(x)\) as a cosine? That will help.
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lol0967

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9439 on: April 03, 2019, 06:09:33 pm »
+1
Hi guys!
I am struggling to study for Specialist Maths and want to know what is the best way to study for SAC's; I know this sounds cliche but seriously don't know what to exactly do. The Spec. SAC is an application SAC, so should I practise Exam 2 style questions, practise SAC's (BTW my teacher doesn't give out past SAC's for some reason) or just continue with textbook questions (Cambridge textbook). I'm hoping that someone who has experienced Spec. Maths can guide me to what should I do. How did you high achieving student who completed Unit 3/4 Spec. do to get a high study score. Thanks.

AlphaZero

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9440 on: April 03, 2019, 10:08:13 pm »
+1
Hi guys!
I am struggling to study for Specialist Maths and want to know what is the best way to study for SAC's; I know this sounds cliche but seriously don't know what to exactly do. The Spec. SAC is an application SAC, so should I practise Exam 2 style questions, practise SAC's (BTW my teacher doesn't give out past SAC's for some reason) or just continue with textbook questions (Cambridge textbook). I'm hoping that someone who has experienced Spec. Maths can guide me to what should I do. How did you high achieving student who completed Unit 3/4 Spec. do to get a high study score. Thanks.

Hey there! Glad to see you've reached out for help. In addition to my advice below, please do see your teacher for their advice since they obviously know you as a learner better than I do.

Indeed, practice is probably the best way to improve at maths. In preparing for assessments, try to become a member of this cycle: \[\begin{matrix}\text{practice}&\longrightarrow & \text{check}\\
\uparrow & \ & \downarrow\\
\text{implement}& \longleftarrow & \text{correct}\end{matrix}\]
Practice. Then, check your work against solutions and/or a teacher. Correct any issues you have by asking questions. Then, implement your new knowledge in responses. Repeat.

Since your upcoming assessment is an application SAC, you should probably do Exam 2 extended response questions relevant to the topic(s). There are plenty of questions out there, especially in past exams. I would be very surprised if you ran out of questions to do before your SAC. Personally, I didn't do a single question from the textbook the entire year. They are skills questions that you generally won't see on SACs and/or the exams. If you find you need to work on a certain area, then by all means, do them to build your understanding and confidence.

As for preparing for assessments, that's probably the most general advice I can give to students of all levels. Depending on your ability and what you're aiming for, the advice I can give varies drastically. (Every student is also different).

Personally, I really only had two goals the entire year.
1) Understand all concepts at a very high level, and
2) Understand the inner workings and motivation(s) behind every question.

Does this mean you should have the same goals as me? Absolutely not. They're mine. Make your own ;)
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Re: VCE Specialist 3/4 Question Thread!
« Reply #9441 on: April 14, 2019, 02:12:31 am »
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Hey all
I'm having trouble figuring out part b of the question below and am confused whether d/dx * dy/dx =d^2y/dx^2

Let a curve be defined by the relation
a) Find the equation of the tangent to the curve at the point y=0
b) Find the value of at the point where x=0

(Apologies for the horrible TeX formatting, used Mathematica to convert all working out into TeX)
This is what I tried to do (this is for part b):
Spoiler







subbing in dy/dx,

However, when subbing in (0,1) the double derivative is cancelled so I'm not sure what to do. Cheers in advance for any help.
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S_R_K

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9442 on: April 14, 2019, 11:13:48 am »
+1
Hey all
I'm having trouble figuring out part b of the question below and am confused whether d/dx * dy/dx =d^2y/dx^2

Let a curve be defined by the relation
a) Find the equation of the tangent to the curve at the point y=0
b) Find the value of at the point where x=0

(Apologies for the horrible TeX formatting, used Mathematica to convert all working out into TeX)
This is what I tried to do (this is for part b):
Spoiler







subbing in dy/dx,

However, when subbing in (0,1) the double derivative is cancelled so I'm not sure what to do. Cheers in advance for any help.

d^2y/dx^2 can be found by differentiating dy/dx with respect to x. In this case, once you get an equation involving dy/dx (which you have from part a) then imp-diff again to get d^y/dx^2. This is what you appear to have tried.

I get the following when I imp-diff to get the first derivative, same as your result:



Imp-diffing again gives me (writing it out before simplification, so you can see how each term is derived)



Then substituting in (0, 1) gives:

.

I'm not entirely sure, but I think there are two sources of error in your working. The first is that you may be assuming that d^2y/dx^2 = (dy/dx)^2, which is not true. The second is that you may have forgotten to apply (or incorrectly applied) the product rule when differentiating dy/dx(xe^y+e^x) (the last expression in the equation for the first derivative).
« Last Edit: April 14, 2019, 11:18:46 am by S_R_K »

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9443 on: April 14, 2019, 04:34:10 pm »
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d^2y/dx^2 can be found by differentiating dy/dx with respect to x. In this case, once you get an equation involving dy/dx (which you have from part a) then imp-diff again to get d^y/dx^2. This is what you appear to have tried.

I get the following when I imp-diff to get the first derivative, same as your result:



Imp-diffing again gives me (writing it out before simplification, so you can see how each term is derived)



Then substituting in (0, 1) gives:

.

I'm not entirely sure, but I think there are two sources of error in your working. The first is that you may be assuming that d^2y/dx^2 = (dy/dx)^2, which is not true. The second is that you may have forgotten to apply (or incorrectly applied) the product rule when differentiating dy/dx(xe^y+e^x) (the last expression in the equation for the first derivative).
Oh, so you treat dy/dx and everything in the brackets that it's multiplied to like 2 separate functions and use the product rule to find the double derivative? Also did you sub back in dy/dx into the relation with both d^2y/dx^2 and dy/dx so all that was left is d^2y/dx^2, then rearranged for d^2y/dx^2? Thanks!
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Re: VCE Specialist 3/4 Question Thread!
« Reply #9444 on: April 14, 2019, 07:06:14 pm »
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Oh, so you treat dy/dx and everything in the brackets that it's multiplied to like 2 separate functions and use the product rule to find the double derivative?

Yes.

Quote
Also did you sub back in dy/dx into the relation with both d^2y/dx^2 and dy/dx so all that was left is d^2y/dx^2, then rearranged for d^2y/dx^2? Thanks!

No. I found dy/dx at (0, 1) and substituted that in, along with substituting in (0, 1) for the values of x and y, and then solved for d^2y/dx^2. There was no need to first derive an expression for d^2y/dx^2 in terms of x and y, then substitute in (0, 1).
« Last Edit: April 14, 2019, 07:13:31 pm by S_R_K »

S_R_K

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9445 on: April 16, 2019, 04:09:11 pm »
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This is a question of the use of inverse circular functions to find the principal argument of a complex number. Take z = –1 – i, for example. I have encountered students writing things along the following lines.:

.

This is the correct principal argument, however, the working is incorrect, since that value is not in the range of arctan. (arctan(1) = pi/4, not -3pi/4.). Correct working using arctan could be:

.

My question is how pedantic examiners are about this sort of thing, because in Derrick Ha's book, he gives that incorrect working as an example of how to find the principal argument of a complex number, and circumstantial evidence suggests that he didn't lose marks for such things. I also note that the Cambridge book glosses over this complication, whereas Maths Quest treats this explicitly and correctly.

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9446 on: April 16, 2019, 04:46:40 pm »
+1
...

A simple way to avoid having to use the hybrid definition of principal argument is to instead not write \(\arctan()\) anywhere. I always teach this to my students.

For example:

Where  \(\theta=\text{Arg}(-1-i)\),  we have \begin{align*} \tan(\theta)&=\frac{-1}{-1}=1,\quad -\pi<\theta<\frac{-\pi}{2}\\
\implies \theta&=\frac{-3\pi}{4}.\end{align*}
I must say, I would be very surprised if VCAA would let people off for this in questions where more than 1 mark is available. Not only is it wrong, but it is given on the formula sheet that  \(-\pi/2<\arctan(x)<\pi/2\).

Trying to recall all the past exams in my head, I don't remember VCAA ever asking students to show the working behind the principal argument of a complex number that isn't in the first or fourth quadrant. This might be why it is 'glossed' over in some texts.
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Re: VCE Specialist 3/4 Question Thread!
« Reply #9447 on: April 17, 2019, 11:00:54 am »
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Trying to recall all the past exams in my head, I don't remember VCAA ever asking students to show the working behind the principal argument of a complex number that isn't in the first or fourth quadrant. This might be why it is 'glossed' over in some texts.

I did a scan through past papers and didn't find anything. Last year did have a "show that" question for converting cartesian to polar form, but in the first quadrant. The examiners report suggests (although somewhat obliquely) that using the arctan method was accepted (which it should be, in this case).

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9448 on: April 21, 2019, 06:25:36 pm »
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Hi I can't figure out how to do part b from both questions linked. For question 1 I can't figure out how to use recognition to rearrange the equation (which I'm pretty sure is the method they want you to use) and I only got so far as rearranging for the integral of xcos^n(x) from the original equation. Question 2 they most likely expect you to use recognition too. I tried to change all the x's in part a) ii into (pi/2 - x) and then isolate the xcos(x) integral, but got (pi-x)sin(x)-cos(x) as a final answer which is wrong too (I know using integration by parts would make things 20 times less complicated but I need those working marks if I get the wrong answer). As always help is appreciated.
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schoolstudent115

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9449 on: April 21, 2019, 07:19:28 pm »
+1
Hi I can't figure out how to do part b from both questions linked. For question 1 I can't figure out how to use recognition to rearrange the equation (which I'm pretty sure is the method they want you to use) and I only got so far as rearranging for the integral of xcos^n(x) from the original equation. Question 2 they most likely expect you to use recognition too. I tried to change all the x's in part a) ii into (pi/2 - x) and then isolate the xcos(x) integral, but got (pi-x)sin(x)-cos(x) as a final answer which is wrong too (I know using integration by parts would make things 20 times less complicated but I need those working marks if I get the wrong answer). As always help is appreciated.
Question 1: From what you said, you did part (a),  you should have gotten: (using the chain rule and product rule).

So . Rearranging for cos^n(x)


Integrating both sides (and flipping the equation):

Let

Notice that the integrand on the RHS of the equation from before is just
So

So substituting the previous result back in:

Adding to both sides

The left hand side is just , So divide through by to get the final value of the integral:


As for the second question you had, you may have to use a similar technique. If you need further help feel free to reply.

« Last Edit: April 22, 2019, 11:26:50 am by schoolstudent115 »
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