AA HL 1 Unit 7: Further Algebra: Complex IB Practice Problems
Numbers
1. Let z = x + yi. Find the values of x and y if (1 – i)z = 1 – 3i.
(Total 4 marks)
2. Let z1 = a æç cos p + i sin p ö÷ and z2 = b æç cos p + i sin p ö÷.
è 4 4ø è 3 3ø
3
æz ö
Express ç 1 ÷ in the form z = x + yi.
è z2 ø
(Total 3 marks)
3. Find the values of a and b, where a and b are real, given that (a + bi)(2 – i) = 5 – i.
(Total 3 marks)
4. (z + 2i) is a factor of 2z3–3z2 + 8z – 12. Find the other two factors.
(Total 3 marks)
5. Given that z = (b + i)2, where b is real and positive, find the exact value of b when arg z = 60°.
(Total 3 marks)
6. The complex number z satisfies i(z + 2) = 1 – 2z, where i = – 1 . Write z in the form z = a + bi,
where a and b are real numbers.
(Total 3 marks)
7. Find the three cube roots of the complex number 8i. Give your answers in the form x + iy.
(Total 8 marks)
2 3
æ π πö æ π πö
ç cos – i sin ÷ ç cos + i sin ÷
4 4ø è 3 3ø
8. Consider the complex number z = è 4
.
æ π π ö
ç cos – i sin ÷
è 24 24 ø
(a) (i) Find the modulus of z.
(ii) Find the argument of z, giving your answer in radians.
(4)
3
(b) Using De Moivre’s theorem, show that z is a cube root of one, ie z = 1 .
(2)
2
(c) Simplify (l + 2z)(2 + z ), expressing your answer in the form a + bi, where a and b are
exact real numbers.
(5)
(Total 11 marks)
1
9. Consider the equation 2(p + iq) = q – ip – 2 (1 – i), where p and q are both real numbers. Find p
and q.
(Total 6 marks)
10. The complex number z satisfies the equation
2
z= + 1 – 4i.
1– i
Express z in the form x + iy where x, y Î .
(Total 6 marks)
11. Let the complex number z be given by
i
z=1+ .
i– 3
Express z in the form a +bi, giving the exact values of the real constants a, b.
(Total 6 marks)
12. Given that (a + i)(2 – bi) = 7 – i, find the value of a and of b, where a, b Î .
(Total 6 marks)
13. Let P(z) = z3 + az2 + bz + c, where a, b, and c Î . Two of the roots of P(z) = 0 are –2 and
(–3 + 2i). Find the value of a, of b and of c.
(Total 6 marks)
14. Given that | z | = 2 5 , find the complex number z that satisfies the equation
25 - 15 = 1 - 8i.
z z*
(Total 6 marks)
15. A complex number z is such that z = z - 3i .
3
(a) Show that the imaginary part of z is .
2
(2)
(b) Let z1 and z2 be the two possible values of z, such that z = 3.
(i) Sketch a diagram to show the points, which represent z1 and z2 in the complex
plane, where z1 is in the first quadrant.
π
(ii) Show that arg z1 = .
6
(iii) Find arg z2.
(4)
æ z1k ö
(c) Given that arg ç z 2 ÷ = π, find a value of k.
ç 2i ÷
è ø
(4)
(Total 10 marks)
2
16. The complex number z is defined by
æ 2π 2π ö æ π πö
z = 4 ç cos + i sin ÷ + 4 3 ç cos + i sin ÷.
è 3 3 ø è 6 6ø
iq
(a) Express z in the form re , where r and q have exact values.
(b) Find the cube roots of z, expressing in the form reiq, where r and q have exact values.
(Total 6 marks)
17. Let u =1+ 3 i and v =1+ i where i2 = −1.
u 3 +1 3 -1
(a) (i) Show that = + i.
v 2 2
(ii) By expressing both u and v in modulus-argument form show that
u æ π πö
= 2 ç cos + i sin ÷ .
v è 12 12 ø
π
(iii) Hence find the exact value of tan in the form a + b 3 where a, bÎ .
12
(15)
+
(b) Use mathematical induction to prove that for nÎ ,
( n
) æ nπ nπ ö
1 + 3 i = 2 n ç cos + i sin ÷.
è 3 3 ø
(7)
2 v+u
(c) Let z = .
2 v -u
Show that Re z = 0.
(6)
(Total 28 marks)
18. Given that z = cosθ + i sin θ show that
æ 1 ö +
(a) Im ç z n + n ÷ = 0, n Î ;
è z ø
(2)
æ z -1 ö
(b) Re ç ÷ = 0, z ≠ –1.
è z +1ø
(5)
(Total 7 marks)
3
2p 2p
19. Let w = cos + i sin .
5 5
(a) Show that w is a root of the equation z5 − 1 = 0.
(3)
(b) Show that (w − 1) (w4 + w3 + w2 + w + 1) = w5 − 1 and deduce that
w4 + w3 + w2 + w + 1 = 0.
(3)
2p 4p 1
(c) Hence show that cos + cos =- .
5 5 2
(6)
(Total 12 marks)
20. 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 Î +.
(8)
a3
(d) Using De Moivre’s theorem to find in the form a + ib.
b2
(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)
