Unit 1: Real And Complex Numbers - Mathematics For Class IX (Science Group) - Solved Exercise 1.6

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Unit 1: Real And Complex Numbers
Exercise 1.6

1. Perform the indicated operations of the following complex numbers.
(i) (3, 2) + (9, 3)
Solution:
Formula:
∴ (a, b) + (c, d) = (a + c, b + d)
⇒ (3, 2) + (9, 3)
⇒ (3 + 9, 2 + 3)
⇒ (12, 5) Ans.

(iv) (15, 12) - (10, -9)
Solution:
Formula:
∴ (a, b) - (c, d) = (a - c, b - d)
⇒ (15, 12) + (10, -9)
⇒ [15 - 10, 12-(-9)]
⇒ (5, 12+9)
⇒ (5, 21)Ans.

(vi) (1, 2) (1, -2)
Solution:
Formula:
∴ (a, b)(c, d) = (ac - bd, ad + bc)
⇒[(1)(1) - (2)(-2) , (1)(-2) + (2)(1)]
⇒[1 + 4, - 2 + 2]
⇒[5, 0]Ans.

OR
Solution:
(1, 2) (1, -2)
⇒ (1 + 2i) (1 - 2i)
⇒ [(1)(1) - 2i + 2i + (2i)(-2i)]
⇒ [1² + (-2 + 2)i - 4i²]
⇒ [1 + 0i- 4(-1) ] (∴ i² = - 1)
⇒ (1 + 4 + 0i)
⇒ (5 + 0i)
⇒ (5, 0) Ans

(vii) (4, -5) (5, -4)
Solution:
Formula:
∴ (a, b)(c, d) = (ac - bd, ad + bc)
⇒[(4)(5) - (-5)(-4) , (4)(-4) + (-5)(5)]
⇒[20 - 20, - 16 - 25] ⇒[0, 41]Ans.

OR
Solution:
(4, -5) (5, -4)
⇒ (4 -5i) (5 - 4i)
⇒ [(4)(5) +(4)(-4i) + (-5i)(-4i) + (-5i)(5)]
⇒ [20 -16i + 20i² - 25i]
⇒ [20 -16i - 25i + 20(-1)] (∴ i² = - 1)
⇒ (20 -20 -41i) = (0 - 41i
⇒ (0, -41) Ans.
⇒ (5, 0) Ans

OR

EXAMPLES

Example No.1: If z₁ = 6 + 9i and z₂ = -1 + 2i, find z₁ + z₂
Solution:
Given that
z₁ = 6 + 9i = (6, 9) and
z₂ = -1 + 2i = (-1, 2)
We know that z₁ + z₂ = (a + c) + i(b + d) = (a + c, b + d)
∴ z₁ + z₂ = (6, 9) + (-1, 2)
⇒ z₁ + z₂ = (6 - 1, 9 + 2)
⇒ z₁ + z₂ = (5, 11) Ans.


Example No.2: If z₁ = -7 + 2i and z₂ = 4 - 9i, find z₁ - z₂
Solution:
Given that
z₁ = -7 + 2i = (-7, 2) and
z₂ = 4 - 9i = (4, -9)
We know that z₁ + z₂ = (a - c, b - d)
∴ z₁ + z₂ = (6, 9) + (-1, 2)
⇒ z₁ - z₂ = (-7 - 4, 2 + 9)
⇒ z₁ + z₂ = (-11, 11) Ans.


Example No.3: If z₁ = 3 + 4i = (3, 4) and z₂ = -3 - 4i = (-3, -4), find the product z₁z₂
Solution:
Given that
z₁ = 3 + 4i = (3, 4) and
z₂ = -3 - 4i = (-3, -4)
We know that z₁ + z₂ = (ac - bd, ad + bc)
∴ z₁ . z₂ = (3, 4) . (-3, -4)
⇒ z₁ . z₂ = (-9 + 16, -12 -12)
⇒ z₁ . z₂ = (7, -24) Ans.