Math, asked by Aryangupta5235, 7 months ago

3)Find a and b ifi) a + 2b + 2ai = 4 + 6iii) (a-b) + (a+b)i = a + 5iiii) (a+b) (2 + i) = b + 1 + (10 + 2a)iiv) abi = 3a - b + 12iv)1/a+ib=3-2ivi) (a+ib) (1+i) =2+i​

Answers

Answered by Anonymous
4

i)

a + 2b + 2ai = 4

(a + 2b) + 2ai = 4 + 0i

  • Equating real and imaginary parts

a + 2b = 4 and 2ai = 0i

a + 2b = 4... .1 and 2a = 0

a = 0/2 = 0 ......2

  • Substituting eq 2 in 1

•°• 0 + 2b = 4

2b = 4

b = 4/2 = 2

Answer. a = 0 and b = 2

ii)

(a - b) + (a +b)i = a + 5i

  • Equating real and imaginary parts

•°• (a - b) = a and (a + b)i = 5i

a - b = a and a + b = 5 ....1

b = 0 ...2

  • substituting 2 in 1

a + 0 = 5

a = 5

Answer. a = 5 and b = 0

iii)

(a + b) (2 + i) = b + 1 + (10 + 2a)i

2(a + b) + i(a + b) = b + 1 + (10 + 2a)i

  • Equating real and imaginary parts

2(a + b) = b + 1 and (a + b)i = (10 + 2a)i

2a + 2b = b + 1 and a + b = 10 + 2a

2a + 2b - b = 1 and b = 10 + 2a - a

2a + b = 1 and a - b = -10

  • Solving both the Equations.
  • Adding the Equations

•°• 2a + b = 1

a - b = -10

3a = -9

a = -9/3 = -3

substituting a = -3 in a - b = -10

•°• -3 - b = -10

-3 + 10 = b

7 = b

Answer. a = -3 and b = 7

vi)

abi = 3a - b + 12i

0 + abi = 3a - b + 12i

  • Equating real and imaginary parts

0 = 3a - b and abi = 12i

3a = b and ab = 12

a = b/3 ....1 and ab = 12 ....2

substituting eq 1 in 2

•°• (b/3) × b = 12

b²/3 = 12

b² = 12 × 3 = 36

b = 6

substituting b = 6 in eq 1

•°• a = 6/3 = 2

a = 2

Answer. a = 2 and b = 6

v)

1/(a + bi) = 3 - 2i

1 = (3 - 2i) (a + bi)

1 = a (3 - 2i) + bi(3 - 2i)

1 = 3a - 2ai + 3bi - 2bi² ....{i² = -1}

1 = 3a - 2ai + 3bi + 2b

1 = 3a + 2b + (-2a + 3b)i

1 + 0i = 3a + 2b + (-2a + 3b)i

  • Equating real and imaginary parts

3a + 2b = 1 and (-2a + 3b)i = 0i

3a + 2b = 1 and -2a + 3b = 0

3a + 2b = 1 and 3b = 2a

3a + 2b = 1 ...1 and b = 2a/3 ......2

substituting eq 2 in 1

•°• 3a + 2 × 2a/3 = 1

3a + 4a/3 = 1

(9a + 4a)/3 = 1

13a = 3

a = 3/13

substituting a = 3/13 in eq 1

3/13 + 2b = 1

2b = 1 - 3/13

2b = 1 - 3/13

2b = (13 - 3)/13

2b = 10/13

b = 10/13 × 2 = 5/13

Answer. a = 3/13 and b = 5/13

vi)

(a + bi) (1 + i) = 2 + i

1(a + bi) + i(a + bi) = 2 + i

a + bi + ai + bi² = 2 + i

a + bi + ai - b = 2 + i

a - b + (a + b)i = 2 + i

  • Equating real and imaginary parts

a - b = 2 and (a + b)i = i

a - b = 2 and a + b = 1

Adding both the Equations

•°• a - b = 2

a + b = 1

2a = 3

a = 2/3

substituting a = 2/3 in Equation a - b = 2

•°• 2/3 - b = 2

2/3 - 2 = b

(2 - 6)/3 = b

b = -4/3

Answer. a = 2/3 and b = -4/3

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