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Sunday, 15 January 2023

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.





3 comments:

  1. For new book in Ex.1.6 the value is z1 = 1+i and z2= 1-i ..
    kindly help as no solution is given over here

    ReplyDelete
    Replies
    1. We will upload its solution this week and inform you
      JAZAKALLAH

      Delete
  2. Kindly upload math others chapter Ex.

    ReplyDelete