Dekoyl's Questions

[dekoyl]

I think I should have a questions thread like my methods one from last year =]
Excuse me if some questions are obvious. Sometimes I need a little nudge for things to click.
Thanks in advance to everyone.

1. Let . A tangent to the graph of where makes an angle of with the positive direction of the x-axis. Find the possible values of .
Answer:


2. Find (to 2 d.p.), the angle between the line and the tangent to the graph of at the point of (2,2).
I got the tangent which is, I think,
Answer:

[TrueTears]

Ok so we know that the angle the tangent makes with the positive direction of the x axis is (where m is the gradient)

but we also know that the derivative of tan(2x) also gives the gradient at any point on the graph.

so

so equate      (70 degerees in radians is )

so (just solve for a)

o crap lunch is over i g2g lol... anyways ill type up the rest when i get home, but that should help you if you see this

[dekoyl]

Ok so we know that the angle the tangent makes with the positive direction of the x axis is (where m is the gradient)

but we also know that the derivative of tan(2x) also gives the gradient at any point on the graph.

so

so equate      (70 degerees in radians is )

so (just solve for a)

o crap lunch is over i g2g lol... anyways ill type up the rest when i get home, but that should help you if you see this

Ah yeah =_= Forgot about trying to convert to radians.

Thanks TrueTears

[TrueTears]

k here it is



as you can see in the diagram, is the angle we are looking for and this is equal to

so (just form a triangle)



and

so

so

[dekoyl]

Thanks for the diagram TrueTears =]

Another one.. I think I'm misinterpreting the question =\
3. Given that . The points are collinear. Which one of the following is not true?

is parallel to
is parallel to
is parallel to
is parallel to
is parallel to

Don't they all lie on the same, straight line?

[TrueTears]

Is it is parallel to ?

[dekoyl]

Is it is parallel to ?

I'm not sure myself =P No answer given to this question.

[TrueTears]

wait nvm i think its the last one

[/0]

There is no guarantee that and are parallel. It can be true under certain conditions however.

The first 4 will give vectors along the same line.

[dekoyl]

^Thanks TT and /0.

I can't visualise this very well =| (Unsure about where point D goes)
4. Let m > n >0 and a, b be the position vectors of two points A and B respectively. Point C, D divide the line segment AB in a ratio of m:n internally and externally respectively. i.e. C is between A, B and AC:CB = m:n. D is not between A,B and AD:DB = m:n.

I don't have a problem with what the question asks me to do just drawing the diagram/visualising part.

Thanks.

[TrueTears]

Okie so it means this, here is the diagram first



1. C is between A, B and AC:CB = m:n

so from this we can conclude

so cross multiply yields nAC = mCB

therefore

2. D is not between A,B and AD:DB = m:n.

doing the same as part 1. yields

But we know m > n > 0 so for example to get an idea of the diagram let m = 4 and n = 2

this means which is same as saying AC = 2CB, so this means AC is TWICE the length of CB, and in the diagram you can see this

the same goes for AD, so , so this means AD is TWICE the length of DB, again in the diagram you can see this.

Now that just gives an idea of the diagram, obviously we don't know the values of m and n, but at least now you can have a good idea of the diagram to do the question.

[dekoyl]

Ahh thanks TT. Didn't know it was a straight line.

[edit]

Yeah just that "internally,externally" made me visualise it as something that was enclosed =\
v
v

[TrueTears]

nps XD

everytime you see something like AD = kDB where (see how you have D at the 'end' of the LHS and D at the 'start' of the RHS), it means that they are collinear

[dekoyl]

I just got one question that I'm not too sure about what it's asking.
This is the exact wording of the question.

Let and . If the vector is equal to the unit vector in the positive direction of the x-axis, find the constants of and .

I got    

Thanks

[/0]

I think this is what it is asking: