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(FP1) Given k = q + 3i and z = w^2 - 8w* - 18q^2 i, and if w is purely imaginary, show that there is only one possible non-zero value of z

k = q + 3i and z = k 2 - 8k* - 18q 2 i. Plug k into w to give: z = (q + 3i) 2 - 8*(q - 3i) - 18 q 2 i. k means the compliment of k which just means that the sign before the imaginary value is switched to neg...
EM
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The complex number -2sqrt(2) + 2sqrt(6)I can be expressed in the form r*exp(iTheta), where r>0 and -pi < theta <= pi. By using the form r*exp(iTheta) solve the equation z^5 = -2sqrt(2) + 2sqrt(6)i.

Using an argand diagram we can see that this (in fact all) complex numbers create a right angled triangle and using this fact we can find the modulus of this number. The modulus of this number is the 'r' in ...
TG
3759 Views

Show that G = {1, -1} is a group under multiplication.

We must prove the four axioms of a group are satisfied (i.e Identity, Associativity, Closure, and Invertibility). Identity: The identity element is an element e in G such that e x = x e = x for all x in G. C...
RT
20635 Views

Cube roots of 8?

8 traditionally has 1 cube root. 2. This is only the real root. It has 2 more complex roots!How can we see this?Consider a vector on the argand diagram. If we square it. What happens to it's magnitude and ar...
VJ
4674 Views

Differentiate: y=x^x

First take log’s each side as it would turn our complicated function into something differentiable by chain rule. ln y = x*ln x Then differentiate y with respect to x: d(ln y)/dx = ln x + 1 1/y * dy/dx = ln ...
MV
3009 Views