CHAPTER Complex Numbers Solutions Revision Section A 28

CHAPTER Complex Numbers Solutions: Revision – Section A 28

28 1. (i) Revision and Exam Style Questions: Section A The complex numbers p and q are represented on the Argand diagram below. Write p and q in the form a + bi. p = − 1 + 2 i q = 2 + 3 i

28 1. (ii) Revision and Exam Style Questions: Section A The complex numbers p and q are represented on the Argand diagram below. Find |p| and |q|.

28 1. (iii) Revision and Exam Style Questions: Section A The complex numbers p and q are represented on the Argand diagram below. Find and , the conjugates of p and q respectively.

28 1. (iv) Revision and Exam Style Questions: Section A The complex numbers p and q are represented on the Argand diagram below. If r = 2 p − 3 q, find r and draw it on an Argand diagram. r = 2 p – 3 q = 2(− 1 + 2 i) – 3(2 + 3 i) = − 2 + 4 i – 6 – 9 i = − 2 – 6 + 4 i – 9 i = − 8 – 5 i

28 2. (i) Revision and Exam Style Questions: Section A Given that i 2 = − 1 and z = − 5 − 2 i: find the value of i 11 = i(i 2)5 = i(− 1) =−i

28 2. (ii) Revision and Exam Style Questions: Section A Given that i 2 = − 1 and z = − 5 − 2 i: find the value of i 11 multiplied by z zi 11 = z(−i) = (− 5 – 2 i)(−i) = (− 5)(−i) – (2 i)(−i) = 5 i – (− 2 i 2) = 5 i + 2(− 1) = 5 i – 2 = − 2 + 5 i

28 2. (iii) Revision and Exam Style Questions: Section A Given that i 2 = − 1 and z = − 5 − 2 i: what of the effect of multiplication by i 11 on z? Multiplying by i 11 is the same as multiplying by − i. It rotates z anticlockwise by 270.

28 3. Revision and Exam Style Questions: Section A p and q are complex numbers, where p = 3 + 4 i and q = 2 − 3 i. Find q, expressing your answer in the form a + bi.

28 4. Revision and Exam Style Questions: Section A Given the complex number z = 2 + 3 i, represent the complex numbers z, 2 z, and iz on one Argand diagram. Comment on their relative positions. z = 2 + 3 i 2 z = 2(2 + 3 i) = 4 + 6 i Position: Twice the distance from 0 + 0 i as z. Position: Image of z by axial symmetry in real axis.

28 4. Revision and Exam Style Questions: Section A Given the complex number z = 2 + 3 i, represent the complex numbers z, 2 z, and iz on one Argand diagram. Comment on their relative positions. Position: Rotation of z anticlockwise, by 90.

28 5. Revision and Exam Style Questions: Section A Given that i 2 = − 1, simplify 5 + 4(6 + 3 i) − i(2 − 5 i) and write your answer in the form x + yi, where x, y R. 5 + 4(6 + 3 i) – i(2 – 5 i) = 5 + 24 + 12 i – 2 i + 5 i 2 = 5 + 24 + 10 i + 5(− 1) = 29 – 5 + 10 i = 24 + 10 i

28 6. (i) Revision and Exam Style Questions: Section A Let w = 2 − 3 i and u = 5 − 7 i. Plot w and u on an Argand diagram.

28 6. (ii) Revision and Exam Style Questions: Section A Let w = 2 − 3 i and u = 5 − 7 i. Verify that |u − w| = 5. u–w = 5 – 7 i – (2 – 3 i) = 5 – 7 i – 2 + 3 i = 5 – 2 – 7 i + 3 i = 3 – 4 i

28 7. (i) Revision and Exam Style Questions: Section A Let z = 4 − 2 i and w = . Plot z and w on an Argand diagram.

28 7. (ii) Revision and Exam Style Questions: Section A Let z = 4 − 2 i and w = . Investigate whether z × w = |z|2. L. H. S: z w = (4 – 2 i) (4 + 2 i) = 4(4 + 2 i) – 2 i(4 + 2 i) = 16 + 8 i – 4 i 2 = 16 – 4(− 1) = 20 Therefore, z w = |z|2 R. H. S: |z|2

Revision and Exam Style Questions: Section A 28 8. The Argand diagram shows the complex number z = 3 + 2 i. (i) Identify the other points P, Q and R as iz, − z and 2 z. iz = i(3+2 i) = 3 i + 2 i 2 = 3 i + 2(− 1) = − 2 +3 i Q = − 2 + 3 i = iz 2 z = 2(3+2 i) = 6 + 4 i R = 6 + 4 i = 2 z − z = −(3+2 i) = − 3− 2 i P = − 3– 2 i = − z

28 8. (ii) Revision and Exam Style Questions: Section A The Argand diagram shows the complex number z = 3 + 2 i. Explain the relationship between z and each of the points P, Q and R. P is the image of z through central symmetry in 0 + 0 i. P is also a rotation of z by 180 and therefore is z multiplied by i 2. Q is an anticlockwise rotation of z by 90. z is doubled to make R. R is z scaled by 2.

28 9. (i) Revision and Exam Style Questions: Section A Let z = + i and w = 3 i. Find |z| and |w|, the modulus of z and w respectively.

28 9. (ii) Revision and Exam Style Questions: Section A Let z = + i and w = 3 i. Simplify the expressions zw and , giving the answers in the form a + bi.

28 9. (ii) Revision and Exam Style Questions: Section A Let z = + i and w = 3 i. Simplify the expressions zw and , giving the answers in the form a + bi.

28 9. (iii) Revision and Exam Style Questions: Section A Let z = + i and w = 3 i. Investigate whether:

28 9. (iii) Revision and Exam Style Questions: Section A Let z = + i and w = 3 i. Investigate whether: from part (i) Therefore, and the statement is true.

28 10. (i) Revision and Exam Style Questions: Section A Let z 1 = 1 + i and z 2 = 2 − i. Find the imaginary part of z 1 z 2. z 1 z 2 = (1 + i) (2 – i) = 1(2 – i) + i(2 – i) = 2 – i + 2 i – i 2 = 2 + i – (− 1) =3+i Imaginary part = 1

28 10. (ii) Revision and Exam Style Questions: Section A Let z 1 = 1 + i and z 2 = 2 − i. Find the value of |z 1 − z 2|. z 1 – z 2 = 1 + i – (2 – i) =1+i– 2+i =1– 2+i+i = − 1 + 2 i

28 10. (iii) Revision and Exam Style Questions: Section A Let z 1 = 1 + i and z 2 = 2 − i. Find the value of .

28 11. (i) Revision and Exam Style Questions: Section A Let and . Show that z 1 = i.

28 11. (ii) Revision and Exam Style Questions: Section A Let and . Draw on an Argand diagram z 1 and z 2.

28 11. (iii) Revision and Exam Style Questions: Section A Let and . Show that |z 1| = |z 2|. Therefore, |z 1| = |z 2|.

28 12. Revision and Exam Style Questions: Section A Solve the equation 2 x 2 + 98 = 0. a=2 b=0 c = 98 Alternatively, 2 x 2 + 98 = 0 2 x 2 = − 98 x 2 = − 49 x= x = ± 7 i