Power of Maths Paper 1 OL

Power of Maths: Paper 1 – Section 6

EXAMPLE 8

(a) Find | x + yi − 2 − 3i | , x, y ∈R.

(b) What curve on the Argand diagram is represented by

| x + yi − 2 − 3i | = 4?

Solution (a) | x + yi − 2 − 3i | = | (x − 2) + ( y − 3)i |

(b) √

_______________ (x − 2 ) 2 + ( y − 3) 2 = 4

= √

_______________ (x − 2 ) 2 + ( y − 3) 2

(x − 2)2 + ( y − 3)2 = 16 This is a circle with centre (2, 3) and radius 4.

EXAMPLE 9 If | x + 3i | = √

Solution | x + 3i | = √

x 2 + 9 = 13 x 2 = 4

∴ x = ± 2 EXERCISE 6

1. Find | z | for the following: (a) z = 3 + 4i (b) z = 1 + 2i (c) z = 4 + 5i (d) z = 2 + 5i (e) z = 6i + 2 (f) z = 1 − i (g) z = −7 + 24i

(h) z = −2 − 3i (i) z = − √

(j) z = − 1 _

(k) z = √

(l) z = 1 _

___ 2 − 1

__ 3

2 + 1 _

4 i

2. If z = 3 + 4i and w = 5 − 12i, fi nd | z | and |w| and then calculate each of the following:

(a) | z + w | and | z | + | w | (b) | z − w | and | z | − | w | (c) | 2z | and 2 | z | (d) | z + 3 | and | z | + 3 (e) | w + i | and | w | + | i | (f) | 2z − w | and 2 | z | − | w |

(g)

| z __

5

| and 1 __

(h) 1 ___

5 | z | 13 | w | and 13

| w ___

|

3 + 2 _

__ 2 + √

3 i

__ 2 i

__ 2 i

3. Plot 0 = 0 + 0i, z = 3 + 4i and w = 4 − 3i on the same Argand diagram as points O, A and B, respectively.

(a) Show that the triangle formed by these points is right-angled by fi nding the slopes of OA and OB.

(b) Find | z | and | w | .

(c) Find the length of side [AB] and the area of ∆OAB.

4. (a) If z = 3i + 7 − xi + y, x, y ∈R, fi nd | z | . (b) If z = x + yi − 2i + 3, x, y ∈R, fi nd | z | .

(c) If z = x + yi, fi nd the equation and centre of the circle given by | z | = 5, x, y ∈R.

(d) If z = −yi + 7 − 5i + x, fi nd the equation and centre of the circle given by | z | = √

x, y ∈R.

5. Solve the following for k ∈R: (a) | k + 7i | = 25

(b) | 2 + ki | = √ (c) | 2k − 3i | = 3 √

(d) | 5 √

__ 3 − 3ki

___ 13

__ 5

| = 10

__ 6 ,

___ 13 ⇒ √

___ 13 , fi nd x ∈R.

______ x 2 + 9 = √

___ 13

214

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