Power of Maths Paper 1 OL

Power of Maths: Paper 1 – Section 6

In general, the following results can be deduced from Example 10: 1.

2. _____ 3. 4.

_ z = z

5. | _ 1. Find _

_____ z + w = _

z − w = _ _

z | = | z | EXERCISE 7 z for the following in the form a + bi:

(a) z = 2 + 5i (b) z = 7 + 11i (c) z = 3 + 7i (d) z = √

__ 3 + i

(e) z = a + ci, a, c ∈R

(f) z = 3 − 2i (g) z = −5 − 2i (h) z = 4 + 6i (i) z = 3 (j) z = −2 (k) z = 5i

(a)

_____ z + w

(b) _ (c)

(f) _ (g)

z + __

__ 2z

(d) 2 _ (e)

________ z + w + u

z z + __

____ z + u − __

u

w + _ w

w

(l) z = −7i (m) z = 3i − 2 (n) z = 5i − 1 (o) z = −6i + √

3. If z = 5 + 7i and w = −2 − 8i, fi nd for the following:

__ 3

(p) z = 5 − 1i + 3 − 2i (q) z = x + 2i − 7, x ∈R

(r) z = x + 3i − 2 + yi, x, y ∈R

(s) z = 3 + x − yi, x, y ∈R

(t) z = 4x − 3i + y − 2, x, y ∈R

2. If z = 3 − 11i, w = −3 + 5i and u = 2i, fi nd for the following:

(h)

___________ 2z + 3w + 4u

(i) 2 _ _

_ z

z + 3 __

(j) _____ (k) _ (l)

(m) __

__ w

z − w z − __

w w + 4 _ u

(a) 3 _ (b) 4 __ (c)

z

_______ 3z + 4w

w (d) 2 _ z − __ w (e)

_____ w

(f) 2 _ (g) (h)

(ii) Join z1 and __

__ 2 + z

_____ z + __

4. (a) If z1 = 3 + 4i, fi nd __ (i) Plot z1 and __

______ 3 _

z + w w

z − __ z 1 .

w and an arrow from z1 to __

z 1 on an Argand diagram. z 1 with a line segment

z 1 .

Repeat the process in part (i) and (ii) for the following and plot all complex numbers and their conjugates on the same Argand diagram:

(b) z2 = −3 − 2i (c) z3 = 1 − 3i (d) z4 = −2 + i (e) Make a conclusion regarding z and _

geometric terms.

5. (a) If z = −15 + 8i, fi nd _ (b) If z = √

__ 3 + i, fi nd _

z . Show that |z| = | _ z . Show that |z| = | _

z |. z |. (j) 3

(i) 8 _ _

z + 7 __ w

_ z + __

w

z + __ z − __

______ kz + lw = k _

w w

z + l __ w , k, l ∈R

z in

14.5 Multiplication

To multiply complex numbers, multiply out the brackets term by term and put i 2 = −1 when it occurs.

i(5 − 7i) = 5i − 7i 2 = 5i + 7 = 7 + 5i −3i(4 + 11i) = −12i − 33i 2 = −12i + 33 = 33 − 12i (4 − 2i)(5 + 3i) = 4 × 5 + 4 × 3i − 2i × 5 − 2i × 3i = 20 + 12i − 10i − 6i 2 = 20 + 2i + 6 = 26 + 2i

216

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