problem sheet 1
DESCRIPTION
MathematicsTRANSCRIPT
University of NottinghamSchool of Mathematical Sciences
HG1M11 Engineering Mathematics 1
Complex numbers Problem Sheet 1
1. Solve (a) 2x2 − x + 2 = 0, (b) x2 + 2x + 2 = 0.
2. Express the following complex numbers in the form x + iy, where x and y are real.
(a) (3 + 4i)(4− 5i); (b)18− i
2− 3i; (c)
(1 + i)3
(1− i); (d)
(1− i)(1 + 3i)
(2 + i)2.
Show each answer on an Argand diagram.
3. Let
Z =1 + z
1− z�
where z = x + iy and x and y are real. Write Z in the form X + iY where X and Y arereal. That is, find expressions for X and Y in terms of x and y.
4. Let
Z1 = R� Z2 = iωL and Z3 =1
iωC�
where ω, R, L and C are all real constants and let Z be defined by the equation
1
Z=
1
Z1
+1
Z2 + Z3
.
Show that the real part of Z is
Re Z =R
�ωL− 1
ωC
�2
R2 +�ωL− 1
ωC
�2
and find a similar expression for its imaginary part.
5. Express the following in the form reiθ = r(cos θ + i sin θ)� −π < θ ≤ π.
(a) −i (b) 1 + i (c) −8�i√
3− 1�
(d)�1 + i
√3�8
Show each number on the Argand diagram.
6. Complete the following tablex + iy modulus argument polar form exponential form
(a) −2 + 2i√
3
(b) 3 −3π/4
(c) 6(cos π
6+ i sin π
6)
(d) πei
18
7. Find all roots of the following equations.(a) z4 = 1 (b) z4 = 16 (c) z3 = −27i
(d) z3 = −4 (e) z6 = 64(1 + i)/√
2
8. (a) Use de Moivre’s theorem to show that
cos nθ = 1
2(wn + w−n)� (dM)
where w = cos θ + i sin θ.
(b) By writing cos4 θ = ((w + w−1)/2)4, hence confirm that
cos4 θ = 1
8cos 4θ + 1
2cos 2θ + 3
8.
9. The equations for a mechanical vibration problem are found to have the followingmathematical solution
z(t) =eiωt
ω20 − ω2 + iγ
�
where t represents time and ω, ω0 and γ are all positive real physical constants. Althoughz(t) is complex and cannot directly represent a physical solution, it turns out that the realand imaginary parts x(t) and y(t) in z(t) = x(t) + iy(t) can. Polar notation can be usedto extract this physical information efficiently as follows.
(a) Put the numerator in the form
ω2
0− ω2 + iγ = aeiδ�
where you should give explicit expressions for a and δ in terms of ω, ω0 and γ.
(b) Hence find contants b and ϕ such that
x(t) = b cos(ωt + ϕ)
and write a similar expression for y(t).
19