Tamil Nadu Board of Secondary EducationHSC Arts Class 12th

# If z1 = 2 + 5i, z2 = – 3 – 4i, and z3 = 1 + i, find the additive and multiplicative inverse of z1, z2 and z3 - Mathematics

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If z1 = 2 + 5i, z2 = – 3 – 4i, and z3 = 1 + i, find the additive and multiplicative inverse of z1, z2 and z3

#### Solution

(i) z1 = 2 + 5i

⇒ – (2 + 5i) = – 2 – 5i

Multiplicative inverse zis (z1)–1

We know

z1z1–1 = 1

⇒ (2 + 5i)(u + iv) = 1 ......[∵ z–1 = u + iv]

2u + 2iv + 5iu – 5v = 1

(2u – 5v) + i(5u + 2v) = 1 + i0

Equating real and imaginary parts

2u – 5v = 1

5u + 2v = 0

Solving them, we get

u = 2/29

v = - 5/29

∴ (z1)–1 = 2/29 + (- 5/29)"i"

(z1)–1 = 1/29(2 - 5"i")

(ii) z2 = – 3 – 4i

⇒ – (3 – 4i) = 3 + 4i

Multiplicative inverse zis (z2)–1

We know

z2 z2–1 = 1

⇒ (– 3 – 4i)(u + iv) = 1  ......[∵ z2–1 = u + iv]

– 3u – 3iv – 4iu + 4v = 1

(– 3u + 4v) + i(– 4u – 3v) = 1 + i 0

We get – 3u + 4v = 1

– 4u – 3v = 0

Solving them, u = -3/25

v = 4/25

∴ (z2)–1 = 1/25(- 3 + 4"i")

(iii) z3 = 1 + i

= – z3

= – (1 + i)

= – 1 – i

(b) Multiplicative inverse of z3

= 1/"z"_3 1/(1 + "i") xx (1 - "i")/(1 - "i")

= (1 - "i")/2

Concept: Basic Algebraic Properties of Complex Numbers
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