Cambridge Prelim MX1 Textbook Marathon/Q&A (1 Viewer)

[InteGrand]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

How would you solve:

Given that z1 = 1 + i , z2 = 2 + 6i , z3 = -1 + 7i , find the three possible values of z4 so that the points representing z1, z2, z3 and z4 form a parallelogram.

 

[appleibeats]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

I only see two possible solution, using the two opposites sides that are equal and equating to find z4. Not sure about a third.

 

[InteGrand]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

I only see two possible solution, using the two opposites sides that are equal and equating to find z4. Not sure about a third.

http://i.stack.imgur.com/vLkjh.png

 

[appleibeats]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Cheers i understand

 

[davidgoes4wce]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Hi, not sure how to go about question 16 from chapter 4J. The question is

the side of a triangle are n2 + n + 1 , 2n + 1 and n2 - 1, where n>1. Find the largest angle of the triangle.

( n2 being --> n squared )

Thanks.

Saw that this was one of the unanswered questions earlier when browsing through this thread. (We know that n^2 +n +1 is the largest length)

 

[davidgoes4wce]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Does anyone know how to answer question 21 from 6A:

a) if x = 2 ^ 1/3 + 4 ^ 1/3, show that x^3 = 6(1+x)

b) if x = 1/2 + 1/2root 5, show that x^2 + x^-2 / x - x^-1 = 3

c) show that pq^-1 - p^-1q / p^2q^-2 - p^-2q^2 = pq/p^2 + q^2


Thanks for your help.

This is the answer to Q21b (Seeing that no one else answered it):

 

[davidgoes4wce]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

This is a bigger version of the text:

 

[davidgoes4wce]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Also this is how to do question 21C in 6A

 

[appleibeats]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

A triangle in the Argand diagram has vertices at the points representing the complex numbers z1,z2,z3. If (z2 - z1) / (z3 - z1) = cos ( pi/3) + isin (pi/3) , show that the triangle is equilaterial.

 

[leehuan]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

As an example, note that with the parallelogram formed by 0+0i, z1, z2 and z1+z2, the vector joining z1 and z2 is z2-z1.
Also note that arg(z/w) = arg(z) - arg(w), which means if you divide two complex numbers, the argument of the quotient is the difference of the arguments.

pi/3 = 60deg = angle in equilateral triangle

I'll let someone else do the full solution

 

[InteGrand]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

A triangle in the Argand diagram has vertices at the points representing the complex numbers z1,z2,z3. If (z2 - z1) / (z3 - z1) = cos ( pi/3) + isin (pi/3) , show that the triangle is equilaterial.

 

[appleibeats]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

I understand your logic. But can you use the information that the vector joining z 1 to z2 is the same length as vector z2 to z3. I.E. using the fact that triangle is equilateral ( equal sides) when you are asked to SHOW ??

 

[InteGrand]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

I understand your logic. But can you use the information that the vector joining z 1 to z2 is the same length as vector z2 to z3. I.E. using the fact that triangle is equilateral ( equal sides) when you are asked to SHOW ??

 

[InteGrand]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

I understand your logic. But can you use the information that the vector joining z 1 to z2 is the same length as vector z2 to z3. I.E. using the fact that triangle is equilateral ( equal sides) when you are asked to SHOW ??

Also, the fact that those vectors are equal in length is not "using the fact that triangle is equilateral", since equilateral means three equal sides. So we use the fact that two of the vectors are equal length, and that their angle between them is 60º to show that this means the triangle is equilateral (easy to show by considering the equal base angles of the isosceles triangle and the angle sum of a triangle).

 

[appleibeats]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Thanks I get it now

 

[appleibeats]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

If z1 and z2 are complex numbers such that | z1| = |z2|, show that:

arg(z1z2) = arg((z1 + z2)^2))

 

[leehuan]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread



A represents z1
B represents z2
C represents z1+z2
D represents (z1+z2)^2
O represents 0+0i

E is just a point on the real axis for the sake of labelling angles.

Since |z1|=|z2| we know that parallelogram OACB must therefore be a rhombus (two adjacent sides equal)

(Note, whenever 3 letters appear, assume I mean angle, not triangle)
So AOC=COB (diagonals of a rhombus bisect its angles)

Let BOE = alpha and COB=AOC=beta
arg(z1)=alpha+2beta
arg(z2)=alpha
So arg(z1z2)
=arg(z1)+arg(z2)
=alpha+alpha+2beta
=2(alpha+beta)

arg(z1+z2)=alpha+beta
So with the aid of De Moivre's theorem (which implies arg(z^n)=n*arg(z))
arg((z1+z2)^2)=2(alpha+beta)
=arg(z1z2)

With these questions, it seriously helps to have a diagram
-----------------
Why not post these questions in the 4u section of the forum?

 

[appleibeats]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

For a question, when one of the parametric equation is a constant , why does the locus equal this constant??

 

[InteGrand]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

For a question, when one of the parametric equation is a constant , why does the locus equal this constant??

Because as the parameter varies through its domain, one of the coordinates stays fixed. You still need to check to see whether the other coordinate has any restrictions on it.

 

[appleibeats]

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

how would you check for restrictions??