VCE Specialist 3/4 Question Thread!

[barydos]

Okay this was in VCAA 2010 Exam 2 Section 2 Question 3:

Spoiler

Okay so, is it wrong to verify the differential equation by subbing in k=800, and then integrating (and using the initial condition) to solve for P which happens to equal the equation for P on the second last line?

Am I supposed to verify the differential equation by diffing the equation for P, and sub in P and k for the differential equation, then showing that they are the same?

Edit: In that same exam for question 3e) i)
It asks for

Spoiler

My question here is, it doesn't specify to what decimal place the answer should be provided. If that's the case, wouldn't you lose marks for not giving it in exact values? Just out of curiosity, since the assessment report has it to 4 d.p.

And for 3e)ii), if k=0, why would arrivals = departures if for increasing values of t, P increases still?
solution:

Spoiler

This was the equation I believe:

Spoiler

[sin0001]

Okay this was in VCAA 2010 Exam 2 Section 2 Question 3:

Spoiler

Okay so, is it wrong to verify the differential equation by subbing in k=800, and then integrating (and using the initial condition) to solve for P which happens to equal the equation for P on the second last line?

Am I supposed to verify the differential equation by diffing the equation for P, and sub in P and k for the differential equation, then showing that they are the same?

Edit: In that same exam for question 3e) i)
It asks for

Spoiler

My question here is, it doesn't specify to what decimal place the answer should be provided. If that's the case, wouldn't you lose marks for not giving it in exact values? Just out of curiosity, since the assessment report has it to 4 d.p.

And for 3e)ii), if k=0, why would arrivals = departures if for increasing values of t, P increases still?
solution:

Spoiler

This was the equation I believe:

Spoiler

For 3a), I would verify it by diffing the equation given for P (the latter of what you have described). I think this is because you're asked to verify that 'P' satisfies the differential equation. If it was the other way around:show that the differential equation and k=800 satisfy the given equation for P, perhaps this would imply integration.
And I agree that VCAA needs to state the requirement of the k-value we find, but they did the right thing by accepting any reasonable rounding for 3ei)
As for why  the P value is increasing despite k being zero, you're just told that population 'would increase at a rate of one per cent per year' irrespective of the k-value (as shown by dP/dt = P/100), maybe because of birth rate>death rate? haha idk. I think that's what you were asking?
Hope that helped!

[ahat]

I have a pretty good solution for 3a) that explains what you're expected to do if that helps?

[barydos]

For 3a), I would verify it by diffing the equation given for P (the latter of what you have described). I think this is because you're asked to verify that 'P' satisfies the differential equation. If it was the other way around:show that the differential equation and k=800 satisfy the given equation for P, perhaps this would imply integration.
And I agree that VCAA needs to state the requirement of the k-value we find, but they did the right thing by accepting any reasonable rounding for 3ei)
As for why  the P value is increasing despite k being zero, you're just told that population 'would increase at a rate of one per cent per year' irrespective of the k-value (as shown by dP/dt = P/100), maybe because of birth rate>death rate? haha idk. I think that's what you were asking?
Hope that helped!

Thanks!
What I meant with 3e)ii) was: yes I know P is increasing, but how can that be so if arrivals = departures (as provided in the assessment report for k=0)? Doesn't that imply stable population?... Unless I'm missing some underlying point to this question :S

Edit:
Also what information can we look at to deduce that the value of k determines the arrival/departure difference, or rather how can we identify the definition of k?
From the graphs in question 3d), we foujnd that P increases when k=800, P is constant for k=200, and P decreases for k=100, so I would've thought a greater k value means P increases, but clearly not since k=-0.0049 increases P over time..... O.o hopefully my question makes sense.

I have a pretty good solution for 3a) that explains what you're expected to do if that helps?

Ahh sure I'd love to hear it

[sin0001]

Thanks!
What I meant with 3e)ii) was: yes I know P is increasing, but how can that be so if arrivals = departures (as provided in the assessment report for k=0)? Doesn't that imply stable population?... Unless I'm missing some underlying point to this question :S

Edit:
Also what information can we look at to deduce that the value of k determines the arrival/departure difference, or rather how can we identify the definition of k?
From the graphs in question 3d), we foujnd that P increases when k=800, P is constant for k=200, and P decreases for k=100, so I would've thought a greater k value means P increases, but clearly not since k=-0.0049 increases P over time..... O.o hopefully my question makes sense.


Ahh sure I'd love to hear it

'k' is already defined in the stem of the question to be: ppl leaving - ppl entering the country (or whatnot). So a greater k-value would imply, and could be interpreted as for 3eii), the number of ppl leaving being substantially greater than the number of ppl entering, therefore 'P' is decreasing *faster* when the k-value increases.
'P' is already shown to be increasing by 1% even if there wasn't a k-value, so k=0 implies that the country is increasing in population at its natural rate of 1%, rather than maintaining a stable population.
Yup, so it's all in the stem of the question, that's how you are required to identify what 'k' is. I hope you can now see how greater k-value=P decreases at a faster rate, if I've managed to explain clearly :S

[barydos]

'k' is already defined in the stem of the question to be: ppl leaving - ppl entering the country (or whatnot). So a greater k-value would imply, and could be interpreted as for 3eii), the number of ppl leaving being substantially greater than the number of ppl entering, therefore 'P' is decreasing *faster* when the k-value increases.
'P' is already shown to be increasing by 1% even if there wasn't a k-value, so k=0 implies that the country is increasing in population at its natural rate of 1%, rather than maintaining a stable population.
Yup, so it's all in the stem of the question, that's how you are required to identify what 'k' is. I hope you can now see how greater k-value=P decreases at a faster rate, if I've managed to explain clearly :S

What a derp moment, yes your explanation is fine! I'm just being stupid haha thanks a lot

[ahat]

Here it is

[barydos]

Here it is

It's beautiful... thanks so much ahat, I really appreciate it

[ahat]

It's beautiful... thanks so much ahat, I really appreciate it

Pleasure

[sin0001]

For VCAA 2005, Exam 1, Q4: http://www.vcaa.vic.edu.au/Documents/exams/mathematics/2005specmath1.pdf
I thought the graph of cot(2x) was translated pi/6 units to the right, but it looks like from the solutions that the translation was actually pi/12 units...
Think I'm forgetting the fundamentals here, can someone please walk me through the transformations
Adding another question:
For 05' Exam 2, Q5b)- why is the frictional force accelerating the package upwards, I thought Fr is always less than or equal to the opposing forces, and never causes the particle to move?

[Stevensmay]

For VCAA 2005, Exam 1, Q4: http://www.vcaa.vic.edu.au/Documents/exams/mathematics/2005specmath1.pdf
I thought the graph of cot(2x) was translated pi/6 units to the right, but it looks like from the solutions that the translation was actually pi/12 units...
Think I'm forgetting the fundamentals here, can someone please walk me through the transformations
Adding another question:
For 05' Exam 2, Q5b)- why is the frictional force accelerating the package upwards, I thought Fr is always less than or equal to the opposing forces, and never causes the particle to move?

Cot(x) will have it's first x-intercept at and asymptotes at
Seeing as the new asymptote is now at , the graph must have been translated by that much to the right.

For the answer given we have but we need to remember that when doing translations we need to have a single positive x.

 

Which shows our translation to the right.

[barydos]

Ughh, for question 5a) in 2011 VCAA: http://www.vcaa.vic.edu.au/Documents/exams/mathematics/2011specmath2-w.pdf
How come we don't take into account the from the concentration coming into the tank?
Apparently the answers is just

[b^3]

, here the amount of chemical in the tank at time is just , so our numerator is . The concentration () of the liquid that is flowing in, is accounted for inside this . Then we just divide by the volume at time , which is changing depending on what our in. We start off with 10 litres, and each second we lose 10 L but gain another 20, so we second we gain another 10 L, hence .

[barydos]

, here the amount of chemical in the tank at time is just , so our numerator is . The concentration () of the liquid that is flowing in, is accounted for inside this . Then we just divide by the volume at time , which is changing depending on what our in. We start off with 10 litres, and each second we lose 10 L but gain another 20, so we second we gain another 10 L, hence .

Ah that makes more sense, thanks !

[ahat]

A man stands on open, level ground flying a kite. A tracking device attached to the kite enables him to monitor its flight on a recording device placed on the ground nearby. From t = 0 seconds, when the monitoring begins, the position of the kite is given by:
r(t) = (4t - 20)i  + (2t)j + (80 - 2t)k

i - east , j - north, k - vertically upward . The origin of this coordinate system is situated at the recording device.

This question:
When is the kite closest to the recording device?
In a last ditch effort, I did r(t).v(t) = 0, which was correct, but am still struggling to understand why. Would someone mind explaining?

Also, When is the kite furthest east of the recording device?

Thank you.

Answer

t=10, t=40 respectively

Also, btw, for these types of questions, if we wanted to find the angle it 'hits the ground', do we always use velocity? Thanks.