VCE Specialist 3/4 Question Thread!

[Alwin]

how do you guys approach questions related to vectors in geometry. Im finding it really difficult

1. Draw a diagram

2. Look at diagram

3. Get equations from diagram

haha sorry for being a bit vague, but that's how I go about it. If you had a specific question or issue, I can explain step-by-step how I would tackle it

But a major part of proofs in spesh is being intuitive (which some people struggle with). That said, practise helps a lot since you are able to recognise a question and know what to do from prior experience

[tote.moore]

From essentials chapter 6g(diff and rational functions):
if x is positive, find the least value of x + (4/x^2)

not sure of the non-calc approach, i worked out the cord of the turning point but that isn't the least value. (answer is 3)

[e^1]

From essentials chapter 6g(diff and rational functions):
if x is positive, find the least value of x + (4/x^2)

not sure of the non-calc approach, i worked out the cord of the turning point but that isn't the least value. (answer is 3)

From using CAS, solving gives the x-value (ie. ) which gives the answer 3. Was there a mistake in you manually differentiating or something?

[tote.moore]

Oh wow, you are correct, late night spesh aha.. I multiplied -1 with 4 instead of -2. Silly me! thank you!

[Homer]

not sure how to

[lzxnl]

You could always derive Pythag with the properties of vectors and then use that.

[Alwin]

not sure how to

Okay, so step 1 is given in the question, you already have a diagram given congrats!
Now, what kind of equations can you get from it?



And you want to prove:
or

Now, lets assume the third equation is true then we can prove the second equation

If you're still stuck, here is a solution:

Spoiler

So, it can be seen that is in fact perpendicular to . Hence, when the vector is drawn perpendicular to (and obviously perpendicular to ) it divides in half since
 

Hope that helps!

[Daenerys Targaryen]

[b^3]

Hint:

For the second part, you will have complex and real factors, expand out the complex factors so that you will obtain a real factor.

[Alwin]

Okay, so you CAN try factorise but it gets a bit messy and you have to solve it anyways
So personally I would just solve: using de Moivre's theorem then express each of the answers, as linear factors eg

For factorising over R, expand the complex factors.

And THANKS b^3.. I was half way thro typing when you replied. wow you're fast..

[Daenerys Targaryen]

Hint:

For the second part, you will have complex and real factors, expand out the complex factors so that you will obtain a real factor.

This is what i get

[lzxnl]

OR you can simply note that in factorizing z^4+81=0 and hence finding the four fourth roots of -81, you can find one of them easily enough; -81=81cis pi, so one fourth root is 3 cis pi/4 = 3/2*(sqrt 2+sqrt 2 i)
Then, just multiply by the other fourth roots of unity, in this case, i, -1 and -i. In this case the multiplication isn't that bad.

[brightsky]

another way of approaching the question:

(z^2 +9)^2 - 18z^2
= (z^2 + 9 - 3sqrt(2)z)(z^2 + 9 + 3sqrt(2)z)
etc
etc

[Daenerys Targaryen]

For factorise over C i got:



But when i expand them I get factors with i's in it.

Do you guys mean expand them in pairs so i get two quadratic ones?

[Alwin]

For factorise over C i got:



But when i expand them I get factors with i's in it.

Do you guys mean expand them in pairs so i get two quadratic ones?

Yeah, expand conjugate factors, I believe that you get , unless I made some error somewhere