Page 63 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Answers

Chapter 4 Complex Numbers (d) i 3, –i 3, 2i, –2i, –2, 2
(e) ± 2, ±i 2, ±2i, ±2
Exercise 4.1 1 i 3 1 i 3
1. (a) 2 + 2i (b) 9 – 5i (f) –2, ±1, 1 ± i 3 , ± 2 , – ± 2
2
2
(c) 2 + 2 2i (d) –3 – 7 2i Exercise 4.3
(e) 4 + 4 5i 1. (a) y (b) y
2. (a) 8 + 2i (b) 1 – 5i
(c) 3 + 3i (d) –5 + 3i P(0, 3)
3. (a) –4 + 3i (b) 4 + 2i P(–2, 1)
(c) –5 – 12i (d) 11 + 3i θ
(e) –1 + 2 2i (f) –3 + i θ x 0 x
1
4. (a) 1 – i (b) 2 + i 0 π
5 5 |z| = 3, q = |z| = 5, q = 2.678 rad
2
(c) i (d) i (c) y (d)
(e) 18 – 1 i (f) 7 – 1 i y
13 13 25 25
5. (a) 2 (b) 5 0 θ x P(5, 7)
3
4

(c) – – i (d) –3 + 4i
5
5
(e) –3 + 4i (f) 1
5 θ x
6. (a) 7 – 7i (b) 7 + 7i P(2, –4) 0
7
(c) 25i (d) 24 – 25 i |z| = 2 5, |z| = 74,
q = –1.107 rad
q = 0.951 rad
25
(e) 24 + 7 i
25 25 (e) y (f) y
3
7. (a) x = 9, y = –7 (b) x = – , y = 7 2 x P(–4, 3)
2
(c) x = 3, y = –5 (d) x = 2, y = 4 0 θ
(e) x = –5, y = 3
θ
8. (a) ±(2 – i) (b) ±(4 + 3i) x
(c) ±(3 + 2i) (d) ±(3 – i) P(–4, –4) 0
(e) ±(7 + 5i) 3
|z| = 4 2, q = – π |z| = 5, q = 2.498 rad
4
9. – 11 + 3 i
10 10 2. (a) z = 2 3 cos – i sin 4
π
π
10. a = b = 5 4 4
2 (b) z = 23 cos + i sin 4
π
π
11. z = 3 – 2i 6 6
12. z = 3 – 4i (c) z = 5[cos (0.927) + i sin (0.927)]
(d) z = 13[cos (–1.176) + i sin (–1.176)]
Exercise 4.2 (e) z = 7 [cos (–0.714) + i sin (–0.714)]
1. (a) –1 ± 2i (b) 1 (3 ± 7i) (f) z = 43 [cos (0.867) + i sin (0.867)]
2 3. (a) z = 12 + 5i
1
3
(c) – (7 ± 31i) (d) 1 2 (5 ± 3i) (b) |z | = 13, arg z = 0.395 rad,
4
3
3
3
(e) 1 (2 ± 2i) z = 13 [cos (0.395) + i sin (0.395)
3 4. (a) z = –1 + 3i
2. (a) z – 2 2z + 3 = 0 (b) z – 4z + 7 = 0 (b) |z| = 10, arg z = –1.249 rad,
2
2
(c) z – 3z + 1 = 0 (d) 25z – 30z + 11 = 0 z = 10 [cos (–1.249) + i sin (–1.249)
2
2
2
(e) 7z + 8z + 3 = 0 5. 8 6 8
1
5
3. (a) z = 2, 1 ± i 2 (b) z = – , –2 ± i 3 6. 5 + i, 2, 0.927, 2 [cos (0.927) + i sin (0.927)]
2 π 2
5
2
3
(c) z = 3, ± i 2 (d) z = – , ± i 3 7. (a) 2, 4 (b) 2 , 2 2
2
2
2
2
5
1
(e) z = , 2 ± i 8. –0.977, 2.16
3 Exercise 4.4
1
4. z + 4z + 2; z = – , –2 ± i 2
2
4π
4π
2 1. (a) cos 2π + i sin 2π (b) cos 1 – 2 + i sin 1 – 2
1
2
5. (z – 2z + 5)(z + z + 1); 1 ± 2i, – ± i 3 3 3 3 3
2
2 2 (c) cos 5π + i sin 5π (d) cos 3π – i sin 3π
6. (a) ± 2 , ±2 2i 6 6 7 2 2 –3
1
(b) – , 1, ±i 2. (a) (cos q + i sin q) (b) (cos q + i sin q)
2 q q –5 (d) 1 cos + i sin 2 13
q
q
(c) –3, 2i, 2i, –2i, –2i (c) 1 cos + i sin 2 6 6
3
3
282
Answers STPM Math T S1.indd 282 3/28/18 4:25 PM