Complex Analysis Answers

We have a sample of 50 micro-drills for drilling holes in low-carbon alloy steel. The average lifetime, expressed as the number of holes drilled before failure, is 12.68. The standard deviation of the sample of 50 is 6.83 holes. Find the 95% confidence interval for this sample.

https://i.stack.imgur.com/1Xj0C.png
https://i.stack.imgur.com/i15Wr.png

Hopefully those images work, since the question is a bit long. I think I've got (a) and (d), I'm just confused on (b) and (c)

QUESTION 9

9.1 State De Moivre’s Theorem.
(2)

9.2 Express cos 5θ and sin 4θ as polynomials in terms of sin θ and cos θ.
(8)

9.3 Let w be a negative real number, z a 6th root of w.

(a) Show that z (k) = ρ 6

6th roots of w.

cos

( π+2kπ )

+ i sin

( π+2kπ )

, k = 0, 1, 2, 3, 4, 5 is a formula for the



Show all your working.
(8)

(b) Hence determine the 6th roots of −729.
(2)

(c) Given z = cos θ + i sin θ and u + iv = (1 + z)(1 + z2). Prove that v = u tan( 3θ ) and

u2 + v2 = 16 cos2( θ ) cos2(θ)
(10)

[30]

If 4xy−kx2y−5y3 is the imaginary part of the analytic function f(z)=u+iv then, the value of k

(d^2 y)/(dt^2) - 2 dy/dt + 1=0 with y(t=0)=5 and y ̇(t=0)=-9
use laplace transform

Use 9.2 to evaluate sin π
5
,sin 2π
5
and cos π
5
. (9)
10.2 Let z = cosθ + sin θ.
Then z
n = cos nθ + isin θ for all n ∈ N (by de Moivre) and z
−n = cos nθ − isin nθ.
(a) Show that 2 cos nθ = z
n + z
−n
and 2isin nθ = z
n − z
n
. (2)
(b) Show that 2
n
cosn
θ =

Obtain the 6th roots of (-7) , and represent them in an Argand diagram

dy/dt + 4y = 6e ^2t
With y (t=0) = 3, Use laplace transform

F(t)=sint×u(t-2) laplace transform

I need help with my Complex Analysis Quiz on April 16. It will consist of 20 multiple choice for 50 minutes. I can send you more information and details soon. Please get back to me asap. Thank you.

Regards,
Mark Eskandar