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Exercise - 5.1

Complex Numbers And Quadratic Equations

**Question-1 :-** Express the complex number in the form of a + bi:

(5i)(-3i/5)

We have, (5i)(-3i/5) = -15/5 x i^{2}= -3 x (-1) [i² = -1] = 3

**Question-2 :-** Express the complex number in the form of a + bi:

i^{9} + i^{19}

We have, i^{9}+ i^{19}= (i²)^{4}.i + (i²)^{9}.i = i[(-1)^{4}+ (-1)^{9}] [i² = -1] = i[1 - 1] = 0

**Question-3 :-** Express the complex number in the form of a + bi:

i^{-39}

We have, i^{-39}= 1/i^{39}= 1/(i²)^{19}.i = 1/(-1)^{19}.i [i² = -1] = 1/(-i) = 1/(-i) x (-i)/(-i) = -i/i^{2}= -i/(-1) = i

**Question-4 :-** Express the complex number in the form of a + bi:

3(7 + i7) + i (7 + i7)

We have, 3(7 + i7) + i (7 + i7) = 21 + 21i + 7i + 7i^{2}= 21 + 28i + 7 x (-1) [i² = -1] = 21 - 7 + 28i = 14 + 28i

**Question-5 :-** Express the complex number in the form of a + bi:

(1 – i) – ( –1 + i6)

We have, (1 – i) – ( –1 + i6) = 1 - i + 1 - 6i = 2 - 7i

**Question-6 :-** Express the complex number in the form of a + bi:

We have, (1/5 + 2i/5) - (4 + 5i/2) = 1/5 + 2i/5 - 4 - 5i/2 = (1/5 - 4) + (2i/5 - 5i/2) = (1 - 20)/5 + (4i - 25i)/10 = -19/5 + (-21i)/10 = -19/5 - 21i/10

**Question-7 :-** Express the complex number in the form of a + bi:

We have, [(1/3 + 7i/3) + (4 + i/3)] - (-4/3 + i) = 1/3 + 7i/3 + 4 + i/3 + 4/3 - i = (1/3 + 4/3 + 4) + (7i/3 + i/3 - i) = (1 + 4 + 12)/3 + (7i + i - 3i)/3 = 17/3 + 5i/3

**Question-8 :-** Express the complex number in the form of a + bi:

(1 - i)^{4}

We have, (1 - i)^{4}= (1 - i)² x (1 - i)² By using Property, (z₁ + z₂)^{2}= z₁^{2}+ 2 z₁.z₂ + z₂^{2}= [1² + i² + 2i] x [1² + i² + 2i] [i² = -1] = [1 - 1 + 2i] x [1 - 1 + 2i] = 2i x 2i = 4i² = 4 x (-1) [i² = -1] = -4

**Question-9 :-** Express the complex number in the form of a + bi:

(1/3 + 3i)³

We have, (1/3 + 3i)^{3}By using Property, (z₁ + z₂)^{3}= z₁^{3}+ z₂^{3}+ 3 z₁^{2}.z₂ + 3 z₁.z₂^{2}= (1/3)³ + (3i)³ + 3 x (1/3)² x 3i + 3 x 1/3 x (3i)² = 1/27 + 27i³ + 9i x 1/9 + 9i² = 1/27 + 27 x (-i) + i + 9 x (-1) [i² = -1; i³ = -i] = 1/27 - 27i + i - 9 = (1/27 - 9) - 26i = -242/27 - 26i

**Question-10 :-** Express the complex number in the form of a + bi:

(-2 - i/3)³

We have, (-2 - i/3)³ By using Property, (z₁ + z₂)^{3}= z₁^{3}+ z₂^{3}+ 3 z₁^{2}.z₂ + 3 z₁.z₂^{2}= (-2)³ + (-i/3)³ + 3 x (-2)² x (-i/3) + 3 x (-2) x (-i/3)² = -8 + (-i)³/27 - 4i - 6 x i²/9 = -8 + i/27 - 4i + 2/3 [i² = -1; (-i)³ = i] = (-8 + 2/3) + (i/27 - 4i) = -22/3 - 107i/27

**Question-11 :-** Find the multiplicative inverse of the complex numbers:

4 - 3i

Let z = 4 – 3i

Conjugatez= 4 + 3i |z|^{2}= 4^{2}+ (-3)^{2}= 16 + 9 = 25 Therefore, the multiplicative inverse of 4 - 3i is given by z^{-1}=z/|z|^{2}z^{-1}= (4 + 3i)/25 = 4/25 + 3i/25

**Question-12 :-** Find the multiplicative inverse of the complex numbers:

√5 + 3i

Let z = √5 + 3i

Conjugatez= √5 - 3i |z|^{2}= √5^{2}+ 3^{2}= 5 + 9 = 14 Therefore, the multiplicative inverse of √5 + 3i is given by z^{-1}=z/|z|^{2}z^{-1}= (√5 - 3i)/14 = √5/14 - 3i/14

**Question-13 :-** Find the multiplicative inverse of the complex numbers:

-i

Let z = 0 – 1.i

Conjugatez= 0 + 1.i |z|^{2}= 0 + (-1)^{2}= 0 + 1 = 1 Therefore, the multiplicative inverse of -i is given by z^{-1}=z/|z|^{2}z^{-1}= (0 + i)/1 = i

**Question-14 :-** Express the following expression in the form of a + ib :

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