Argand diagram Each marking on the axes represents one unit 2 Plot the following complex numbers on an Argand diagram i 3 2 i ii 1 5 i iii 4 i ID: 1030637
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2. 1.Identify each of the complex numbers shown on the Argand diagram. Each marking on the axes represents one unit.
3. 2.Plot the following complex numbers on an Argand diagram: (i) 3 + 2i (ii) −1 − 5i (iii) −4 + i (iv) 2 − 4i (v) 3i (vi) −6
4. 3.Given that z = 3 − 2i and w = −4 + 3i, plot the following on an Argand diagram: (i)(iii)z z = 3 − 2i w 3z 3z = 3(3 − 2i) w = −4 + 3i = 9 − 6i (ii)−2w − 2w = −2(−4 + 3i)= 8 − 6i (iv)
5. 3.Given that z = 3 − 2i and w = −4 + 3i, plot the following on an Argand diagram: (v)(vi)− z − z = −(3 − 2i) = −3 + 2i
6. 4.Find the modulus of each of the following: (i)7 − i 7 − i a = 7 b = −1
7. 4.Find the modulus of each of the following: (ii)10 − 2i 10 − 2i a = 10 b = −2
8. 4.Find the modulus of each of the following: (iii)3 − 6i 3 − 6i a = 3 b = −6
9. 4.Find the modulus of each of the following: (iv)−5 − 5i−5 − 5i a = −5 b = −5
10. 4.Find the modulus of each of the following: (v)−4i −4i a = 0 b = –4
11. 4.Find the modulus of each of the following: (vi)−4 − 3i −4 − 3i a = −4 b = −3
12. 5.z1 = 2 + 5i, z2 = −3 − 2i, z3 = −4 + 3i and z4 = 1 − i. (i)Plot z1, z2, z3 and z4 on an Argand diagram.
13. 5.z1 = 2 + 5i, z2 = −3 − 2i, z3 = −4 + 3i and z4 = 1 − i. (ii)Calculate |z1|, |z2|, |z3| and |z4|. z1 = 2 + 5i a = 2 b = 5 z2 = −3 − 2i a = −3 b = −2| z2 || z1 |
14. 5.z1 = 2 + 5i, z2 = −3 − 2i, z3 = −4 + 3i and z4 = 1 − i. (ii)Calculate |z1|, |z2|, |z3| and |z4|. z3 = −4 + 3i a = −4 b = 3 | z3 |
15. 5.z1 = 2 + 5i, z2 = −3 − 2i, z3 = −4 + 3i and z4 = 1 − i. (ii)Calculate |z1|, |z2|, |z3| and |z4|. z4 = 1 − i a = 1 b = −1 | z4 |
16. 6.(i) Plot the following on an Argand diagram: (a)(b)
17. 6.(i) Plot the following on an Argand diagram: (c)(d)
18. 6.(ii) Find the modulus of each of the complex numbers above.(a)−3 + 7i a = −3 b = 7(b)5 − 2i a = 5 b = −2
19. 6.(ii) Find the modulus of each of the complex numbers above.(c)−7 + 11i a = −7 b = 11
20. 6.(ii) Find the modulus of each of the complex numbers above.(d)6 − 14i a = 6 b = −14
21. 7.Given |p + 4i| = find the value of p. p2 + 16 = 41p2 = 41 − 16p2 = 25p = ±5
22. 8.Given |1 + 3i| = |−3 + bi|, find the value of b.|1 + 3i| = |−3 + bi| a = 1, b = 3 a = −3, b = b
23. 9.(i) Plot z = −5 − 4i on an Argand diagram and find |z|. z = −5 − 4i a = −5 b = −4
24. 9.(ii) Plot z1, the image of under axial symmetry in the real axis, on an Argand diagram and find the modulus of z1. z1 = −5 + 4i a = −5 b = 4 | z1 |
25. 9.Plot z2, the image of under axial symmetry in the imaginary axis, on an Argand diagram and find the modulus of z2. z2 = 5 − 4i a = 5 b = −4 | z2 |
26. 9.Plot 3, the image of z under central symmetry in 0 + 0i, on an Argand diagram and find |z3|. z3 = 5 + 4i a = 5 b = 4| z3 |
27. 9.(v) What effect does reflection of a complex number in the real axis, imaginary axis or in the point 0 + 0i have on the modulus of the complex number? The modulus of the complex numbers stays the same.This is because as the modulus represents a length, it will always be positive. The formula will always return a positive result. Therefore even if the points on the original diagram have negative numbers, the modulus will always be positive.The modulus represents the distance from 0 + 0i to the complex number and this distance is not changed by reflection in the real, imaginary axis or in the point 0 + 0i.
28. 10.z1 = 1 + i and z2 = 3 + 3i. (i)Plot z1 and z2 on an Argand diagram.
29. 10.z1 = 1 + i and z2 = 3 + 3i. (ii)Calculate |z1| and |z2|. z1 = 1 + i a = 1 b = 1 | z1 |z2 = 3 + 3i a = 3 b = 3 | z2 |
30. 10.z1 = 1 + i and z2 = 3 + 3i. (iii)Comment on the relationship between z1 and z2. z2 is three times z1.z1 and z2 are both points on the same line. z2 could be seen to be a scaling of z1 by a factor of 3, with both the real and imaginary values multiplied by 3.
31. 10.z1 = 1 + i and z2 = 3 + 3i. (iv)Comment on the relationship between |z1| and |z2|. |z2| is three times |z1|. This is also evident from the diagram and it is because z2 is three times further away from the origin than z1.