Tricky questions

[jane1234]

There's more to this question, if you want:

is defined by
Find and such that is a subset of

and:

and has only one solution.
Find the possible values of 

Have fun

EDIT: I typed a mistake, was meant to be find (not )

EDIT 2: EEP another typo! Find b for the second part, not a lol.

Just curious, where are these questions from?

[funkyducky]

Tutor. Makes up hardcore questions that make me feel like I don't know anything for spesh :S Good challenge though

[luffy]

Could someone tell me if there is an easier way to do these questions? Below is how I did them. (Sorry, in advance, for the huge post and for any errors I may have made )

is defined by . Find . Sketch on an Argand diagram.

Let z = x + yi





Tan both sides.

There's more to this question, if you want:

is defined by
Find and such that is a subset of

Let z = x +yi. You could simply do this question by recognition as you know that complex numbers of this form have the cartesian equation of a circle with centre z0.





Let z = x + yi and = a + bi





i.e. -1.

and has only one solution.
Find the possible values of 

has cartesian equation

has cartesian equation

For one solution, y = x - b is a tangent to the circle.

Hence, gradient of circle is 1.







When dy/dx = 1,



Sub this back into the circle equation.









Therefore, co-ordinates are P
and Q

Reject P as y > 0.

Sub Q into y = x - b



I get the feeling I made a fundamental error in step 1 of one of those problems. So, let me know where all my mistakes are

[Mao]

is defined by . Find . Sketch on an Argand diagram.

Interesting. After some thought, I can arrive at the result via geometry. It is the top half of the circle centered about (-1,0) with a radius of 5, and excludes the points (4,0) and (-6,0).

I shall draw up a picture later.

[funkyducky]

Nicely done, luffy

EDIT: Hang on,

What happened to compound angle formula?

EDIT: wait I get it, cos it =0...nevermind

EDIT: also, b can only equal since y>0 at the point of intersection

[funkyducky]

Here's another:

If the position vectors of A, B and C are , and respectively, determine whether the lines and intersect each other.

[luffy]

EDIT: also, b can only equal since y>0 at the point of intersection

Sorry, this is probably a really noob question. But, why does y have to be greater than 0?

EDIT: Nevermind, I get it. I'll edit the post. Brilliant questions by the way.

Also, sorry for skipping a few steps in my working. In my defense, it was really late at night

is defined by . Find . Sketch on an Argand diagram.

Interesting. After some thought, I can arrive at the result via geometry. It is the top half of the circle centered about (-1,0) with a radius of 5, and excludes the points (4,0) and (-6,0).

I shall draw up a picture later.

Pleasseeeee post up the picture as soon as you get timee I really suck at recognising complex numbers by geometry and by basic inspection - my greatest weakness in specialist I think. Hence, it would really come in handy to learn it before the specialist exam.

EDIT: Nevermind, I just figured out the geometry of the relation.

Sorry, How exactly do you now that a=5 rather than a=5 or a=-5.

Edit: @below. I think you're right. I'll edit my post.

[Lord of Tzeentch]

is defined by . Find . Sketch on an Argand diagram.

Sorry, How exactly do you now that a=5 rather than a=5 or a=-5.

[funkyducky]

^ Yeah, technically it could be either, the question should specify a>0. (Just because that's the consensus for the equation of a circle...)

[Lord of Tzeentch]

Could someone tell me if there is an easier way to do these questions? Below is how I did them. (Sorry, in advance, for the huge post and for any errors I may have made )

is defined by . Find . Sketch on an Argand diagram.

Let z = x + yi

I get the feeling I made a fundamental error in step 1 of one of those problems. So, let me know where all my mistakes are

Is that first step, the fundamental error, 'Let z = x + yi'?
 Don't be so harsh on yourself, you did an amazing job. 

[Mao]

is defined by . Find . Sketch on an Argand diagram.

Interesting. After some thought, I can arrive at the result via geometry. It is the top half of the circle centered about (-1,0) with a radius of 5, and excludes the points (4,0) and (-6,0).

I shall draw up a picture later.

Pleasseeeee post up the picture as soon as you get timee I really suck at recognising complex numbers by geometry and by basic inspection - my greatest weakness in specialist I think. Hence, it would really come in handy to learn it before the specialist exam.

EDIT: Nevermind, I just figured out the geometry of the relation.

Good to know you figured it out, but here's a picture anyways.

I have always been a fan of doing these questions geometrically. You can find a few more geometric descriptions in my bound notes (it's floating around in this board somewhere).

EDIT: I remember in my bound notes, there is a question similar to this, and I tackled it with the cartesian description. It was difficult and confusing. On top of that, you must consider the different quadrants and whether or not the locus will exist there.

[funkyducky]

UPDATE: copied the other parts + second question into the first post. Give the second one a try...

[Mao]

Could someone tell me if there is an easier way to do these questions? Below is how I did them. (Sorry, in advance, for the huge post and for any errors I may have made )

is defined by . Find . Sketch on an Argand diagram.

Let z = x + yi

I get the feeling I made a fundamental error in step 1 of one of those problems. So, let me know where all my mistakes are

Is that first step, the fundamental error, 'Let z = x + yi'?
 Don't be so harsh on yourself, you did an amazing job. 

The fundamental error is when you went from
Since inverse tan has a range of , it is a bit different to Arg. The full description is:



Thus, you need to solve your cartesian equation using this hybrid function. When you have two Arg terms, you will have two Arg terms with which you need to combine into a hybrid function with more pieces in order to preserve all the conditions. After all that, you can obtain the bounds for your final solution.

That process is very complex, confusing, and I do not recommend it. (but you don't really have to worry about it. you won't have to deal with this kind of questions in an exam anyways.)

[luffy]

Could someone tell me if there is an easier way to do these questions? Below is how I did them. (Sorry, in advance, for the huge post and for any errors I may have made )

is defined by . Find . Sketch on an Argand diagram.

Let z = x + yi

I get the feeling I made a fundamental error in step 1 of one of those problems. So, let me know where all my mistakes are

Is that first step, the fundamental error, 'Let z = x + yi'?
 Don't be so harsh on yourself, you did an amazing job. 

The fundamental error is when you went from
Since inverse tan has a range of , it is a bit different to Arg. The full description is:



Thus, you need to solve your cartesian equation using this hybrid function. When you have two Arg terms, you will have two Arg terms with which you need to combine into a hybrid function with more pieces in order to preserve all the conditions. After all that, you can obtain the bounds for your final solution.

That process is very complex, confusing, and I do not recommend it. (but you don't really have to worry about it. you won't have to deal with this kind of questions in an exam anyways.)

Oh - I originally had a similar thing to that hybrid in my answer. However, I edited it out when I was told my domains were wrong. It'll take too long to edit it back in haha. But thanks a lot for the picture and explanation - very much appreciated.

[Mao]

Here's another:

If the position vectors of A, B and C are , and respectively, determine whether the lines and intersect each other.

The easiest thing is to construct vector paths along these two lines. (i.e. if a plane flies with some vector velocity from some initial position vector, blah blah)




If at any point OA(t) = BC(s) for some arbitrary values of t and s, then the two paths do cross. We thus have a system of 3 equations (equating x y and z) and 2 variables (s and t):


Solving the first two simultaneously, we get s=0.5, t=0
Substituting into the last eqn, we see that this does not satisfy the final relationship. Thus the two lines do not cross.