Speedy's Methods 3/4 Questions

[speedy]

Anyone have a quick way to solve these on the classpad?

Spoiler

This isn't a particularly hard example - I'm more so interested in solving the ones which have the unknown variable in the "c" part of the equation too.


I don't quite understand this question - and why the (cos(x)+2) is discluded:

Spoiler

Also how would you do composite functions via inspection (I graphed similar functions on my calculator to find the answer, but in the case of it not being that simple):

Spoiler

[GeniDoi]

Can't help you with the first one because I don't have a classpad, but:

Question 21:

The question is essentially asking you to solve for x.

Let u = cos(x)
u^2 + 2u = 0
u(u + 2) = 0
u = 0 or u + 2 = 0

Therefore:

1) cos(x) = 0 (good)

2) cos(x) + 2 = 0
    => cos(x) = -2
No solution exists here because the range of cos(x) is [-1,1], and 2 is outside that set.
Since none of the answers have x given explicitly it is ok to leave it as cos(x).
Therefore, A)

Question 22:

You know for sure it will be symmetrical around the Y-axis as your subbing a modulus into a function, ie f(|x|) (its f(-|x|) in this case but the symmetry still applies), leaving only B and E from inspection.

Now, any function "f(x)" is essentially substituting the line y = x into f (Just as subbing in f(x^2) subs in a parabola, subbing in f(sin(x)) subs in values of a sine wave, etc).
Well for the domain (-infinity, 0], g(x) is identical to y = x, so it must logically follow that the part left of the Y axis (ie, (-infinity,0]) of the composite function should be identical to the left part of the original function f(x), which only leaves out E.

[speedy]

Forgot to thank you for this, amazing explanations If you have done the 2007 VCAA exam, would you be able to explain the very last probability question? If not I can send you the question but you'd have to do it all so it's doesn't matter if you cbf.


How do you find expected value for dependant events with matrices? Question from 2010 Exam 2 uses matrices as one of the solutions:

Question:

Spoiler

VCAA method:

Spoiler