IB Math HL Y2
Mr. Jauk
1.
Complex Numbers Test Prep
The complex numbers z1 = 2 – 2i and z2 = 1 – i 3 are represented by the points A
and B respectively on an Argand diagram. Given that O is the origin,
(a)
find AB, giving your answer in the form a b  3 , where a, b 
+
;
(3)
(b)
calculate AÔB in terms of π.
(3)
(Total 6 marks)
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2.
Consider the complex number ω =
(a)
z i
, where z = x + iy and i =
z2
1 .
If ω = i, determine z in the form z = r cis θ.
(6)
(b)
Prove that ω =
( x 2  2 x  y 2  y)  i( x  2 y  2)
( x  2) 2  y 2
.
(3)
(c)
Hence show that when Re(ω) = 1 the points (x, y) lie on a straight line, l1, and
write down its gradient.
(4)
(d)
Given arg (z) = arg(ω) =
π
, find │z│.
4
(6)
(Total 19 marks)
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3.
If z is a non-zero complex number, we define L(z) by the equation
L(z) = ln│z│ + i arg (z), 0 ≤ arg (z) < 2π.
(a)
Show that when z is a positive real number, L(z) = ln z.
(2)
(b)
Use the equation to calculate
IB Math HL Y2
Mr. Jauk
Complex Numbers Test Prep
(i)
L(–1);
(ii)
L(1 – i);
(iii)
L(–1 + i).
(5)
(c)
Hence show that the property L(z1z2) = L(z1) + L(z2) does not hold for all
values
of z1 and z2.
(2)
(Total 9 marks)
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4.
Find, in its simplest form, the argument of (sin + i (1− cos ))2 where  is an acute
angle.
(Total 7 marks)
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8.
Consider w =
z
where z = x + iy, y  0 and z2 + 1  0.
z 1
2
Given that Im w = 0, show that z = 1.
(Total 7 marks)
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5. The roots of the equation z2 + 2z + 4 = 0 are denoted by α and β?
(a)
Find α and β in the form reiθ.
(6)
(b)
Given that α lies in the second quadrant of the Argand diagram, mark α and β
on an Argand diagram.
(2)
(c)
Use the principle of mathematical induction to prove De Moivre’s theorem,
which states that cos nθ + i sin nθ = (cos θ + i sin θ)n for n  +.
(8)
IB Math HL Y2
Mr. Jauk
(d)
Complex Numbers Test Prep
3
Using De Moivre’s theorem find 2 in the form a + ib.

(4)
(e)
Using De Moivre’s theorem or otherwise, show that α3 = β3.
(3)
(f)
Find the exact value of αβ* + βα* where α* is the conjugate of α and β* is the
conjugate of β.
(5)
(g)
Find the set of values of n for which αn is real.
(3)
(Total 31 marks)
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6.
(a)
Solve the equation z3 = –2 + 2i, giving your answers in modulus–argument
form.
(6)
(b)
Hence show that one of the solutions is 1 + i when written in Cartesian form.
(1)
(Total 7 marks)
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7.
Find the values of n such that (1 +
3 i)n is a real number.
(Total 5 marks)
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8.
(a)
Let z = x + iy be any non-zero complex number.
(i)
Express
(ii)
If z 
1
in the form u + iv.
z
1
 k, k 
z
, show that either y = 0 or x2 + y2 = 1.
IB Math HL Y2
Mr. Jauk
(iii)
Complex Numbers Test Prep
Show that if x2 + y2 = 1 then │k│ ≤ 2.
(8)
(b)
Let w = cos θ + i sin θ.
(i)
Show that wn + w–n = 2cos nθ, n 
(ii)
Solve the equation 3w2 – w + 2 – w–1 + 3w–2 = 0, giving the roots in the
form x + iy.
.
(14)
(Total 22 marks)
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