1. Complex Numbers Exam Questions

3. 2(3 – i) + i(4 + 5i)
= 6 – 2i + 4i + 5(-1)
= 6 – 2i + 4i - 5
= 1 + 2i

4. z = 5 + 4i
z = 5 – 4i
z = 5 + 4i
z = 5 – 4i

5. z = 5 + 4i
z = 5 – 4i

6. Imaginary parts are equal
p(2 + i) + 8 – ki = 5k i
2p + pi + 8 – ki = 5k i
2p pi – ki = 5k i
Real parts are equal
Imaginary parts are equal
2p i(p – k) = 5k i
2p + 8 = 5k and p – k = - 1

7. k = 3 and p = 2 2p + 8 = 5k - 3 and p – k = - 1
Put 3 for k in (2)
2p – 5k = (1)
p – (3) = - 1
p – k = (2)
p – 3 = - 1
2p – 5k = (1)
- p + k = (2) x - 1
p = = 2
2p – 5k =
(1) x 1
k = 3 and p = 2
- 2p + 2k =
(2) x 2
-3k = - 9
(1) + (2)
k = 3

9. 7(2 + i) + i(11 + 9i)
= 14
+ 7i
+ 11i
+ 9(-1)
= 14
+ 18i
- 9
= 5
+ 18i

10. w = 3 - i w + 6i = (3 – i) + 6i = 3 + 5i |w + 6i| = w + 6i = 3 + 5i

12. z = 2 + 4i
z2 = (2 + 4i)2 = (2 + 4i)(2 + 4i)
= 2(2 + 4i) + 4i(2 + 4i)
= 4 + 8i
+ 8i + 16(-1)
= i - 16
= i
z2 + 28
= i + 28
= i

13. Imaginary Parts are Equal
z = 2 + 4i
|z| = |2 + 4i| =
k(z2 + 28) =|z|(1 + i)
Real Parts are Equal
Imaginary Parts are Equal

15. z = 5 + 4i
z – 4i = 5 + 4i - 41 = 5 + 0i = 5
z = 5 + 4i
z – 4i = 5

16. u = 3 – 6i
|u| =

17. u + 3i = 3 – 6i + 3i = 3 – 3i

18. w = i - 2
w2 = (-2 + i)2 = (-2 + i)(-2 + i)
= -2(-2 + i) + i(-2 + i)
= 4 - 2i
- 2i -1
= 3 - 4i

19. kw2 = 2w ti
k(3 – 4i) = 2(-2 + i) ti
3k – 4ki = i +1 + ti
Real Parts are Equal
Imaginary Parts are Equal
3k – 4ki = -3 + i(2 + t)
-4k = 2 + t
3k = -3
-4(-1) = 2 + t
k = -1
2 + t = 4
t = 2