5.8 PROBLEMS

Back

1. Express the following complex numbers in the form x + iy; x; y real:

(i) (¡3 + i)(14 ¡ 2i); (ii)

2 + 3i

1 ¡ 4i

; (iii)

(1 + 2i)2

1 ¡ i

.

[Answers: (i) ¡40 + 20i; (ii) ¡10

17 + 11

17 i; (iii) ¡7

2 + i

2 .]

2. Solve the following equations:

112 CHAPTER 5. COMPLEX NUMBERS

(i) iz + (2 ¡ 10i)z = 3z + 2i;

(ii) (1 + i)z + (2 ¡ i)w = ¡3i

(1 + 2i)z + (3 + i)w = 2 + 2i.

[Answers:(i) z = ¡ 9

41 ¡ i

41 ; (ii) z = ¡1 + 5i; w = 19

5 ¡ 8i

5 :]

3. Express 1 + (1 + i) + (1 + i)2 + : : : + (1 + i)99 in the form x + iy; x; y

real. [Answer: (1 + 250)i.]

4. Solve the equations: (i) z2 = ¡8 ¡ 6i; (ii) z2 ¡ (3 + i)z + 4 + 3i = 0.

[Answers: (i) z = §(1 ¡ 3i); (ii) z = 2 ¡ i; 1 + 2i:]

5. Find the modulus and principal argument of each of the following

complex numbers:

(i) 4 + i; (ii) ¡3

2 ¡ i

2 ; (iii) ¡1 + 2i; (iv) 1

2 (¡1 + ip3).

[Answers: (i) p17; tan¡1 1

4 ; (ii)

p10

2 ; ¡¼ + tan¡1 1

3 ; (iii) p5; ¼ ¡

tan¡1 2.]

6. Express the following complex numbers in modulus-argument form:

(i) z = (1 + i)(1 + ip3)(p3 ¡ i).

(ii) z =

(1 + i)5(1 ¡ ip3)5

(p3 + i)4

.

[Answers:

(i) z = 4p2(cos 5¼

12 + i sin 5¼

12 ); (ii) z = 27=2(cos 11¼

12 + i sin 11¼

12 ).]

7. (i) If z = 2(cos ¼

4 +i sin ¼

4 ) and w = 3(cos ¼

6 +i sin ¼

6 ), ¯nd the polar

form of

(a) zw; (b) z

w; (c) w

z ; (d) z5

w2 .

(ii) Express the following complex numbers in the form x + iy:

(a) (1 + i)12; (b)

³

1¡i p2

´

¡6

.

[Answers: (i): (a) 6(cos 5¼

12 + i sin 5¼

12 ); (b) 2

3 (cos ¼

12 + i sin ¼

12 );

(c) 3

2 (cos ¡ ¼

12 + i sin ¡ ¼

12 ); (d) 32

9 (cos 11¼

12 + i sin 11¼

12 );

(ii): (a) ¡64; (b) ¡i.]

5.8. PROBLEMS 113

8. Solve the equations:

(i) z2 = 1 + ip3; (ii) z4 = i; (iii) z3 = ¡8i; (iv) z4 = 2 ¡ 2i:

[Answers: (i) z = §(p3+i)

p2

; (ii) ik(cos ¼

8 + i sin ¼

8 ); k = 0; 1; 2; 3; (iii)

z = 2i; ¡p3¡i; p3¡i; (iv) z = ik2

3

8 (cos ¼

16 ¡i sin ¼

16 ); k = 0; 1; 2; 3.]

9. Find the reduced row{echelon form of the complex matrix

2

4

2 + i ¡1 + 2i 2

1 + i ¡1 + i 1

1 + 2i ¡2 + i 1 + i

3

5 :

[Answer:

2

4

1 i 0

0 0 1

0 0 0

3

5.]

10. (i) Prove that the line equation lx + my = n is equivalent to

pz + pz = 2n;

where p = l + im.

(ii) Use (ii) to deduce that re°ection in the straight line

pz + pz = n

is described by the equation

pw + pz = n:

[Hint: The complex number l + im is perpendicular to the given

line.]

(iii) Prove that the line jz¡aj = jz¡bj may be written as pz+pz = n,

where p = b ¡ a and n = jbj2 ¡ jaj2. Deduce that if z lies on the

Apollonius circle jz¡aj

jz¡bj

= ¸, then w, the re°ection of z in the line

jz ¡ aj = jz ¡ bj, lies on the Apollonius circle jz¡aj

jz¡bj

= 1

¸.

11. Let a and b be distinct complex numbers and 0 < ® < ¼.

(i) Prove that each of the following sets in the complex plane rep-

resents a circular arc and sketch the circular arcs on the same

diagram:

114 CHAPTER 5. COMPLEX NUMBERS

Arg

z ¡ a

z ¡ b

= ®; ¡®; ¼ ¡ ®; ® ¡ ¼.

Also show that Arg

z ¡ a

z ¡ b

= ¼ represents the line segment joining

a and b, while Arg

z ¡ a

z ¡ b

= 0 represents the remaining portion of

the line through a and b.

(ii) Use (i) to prove that four distinct points z1; z2; z3; z4 are con-

cyclic or collinear, if and only if the cross{ratio

z4 ¡ z1

z4 ¡ z2

=

z3 ¡ z1

z3 ¡ z2

is real.

(iii) Use (ii) to derive Ptolemy's Theorem: Four distinct points A; B; C; D

are concyclic or collinear, if and only if one of the following holds:

AB ¢ CD + BC ¢ AD = AC ¢ BD

BD ¢ AC + AD ¢ BC = AB ¢ CD

BD ¢ AC + AB ¢ CD = AD ¢ BC:

Навигация