Question -
Answer -
(i) (x + iy) (2 – 3i) = 4+ i
Given:
(x + iy) (2 – 3i) = 4+ i
Let us simplify theexpression we get,
x(2 – 3i) + iy(2 – 3i)= 4 + i
2x – 3xi + 2yi – 3yi2 =4 + i
2x + (-3x+2y)i – 3y(-1) = 4 + i [since, i2 = -1]
2x + (-3x+2y)i + 3y =4 + i [since, i2 = -1]
(2x+3y) + i(-3x+2y) =4 + i
Equating Real andImaginary parts on both sides, we get
2x+3y = 4… (i)
And -3x+2y = 1… (ii)
Multiply (i) by 3 and(ii) by 2 and add
On solving we get,
6x – 6x – 9y + 4y = 12+ 2
13y = 14
y = 14/13
Substitute the valueof y in (i) we get,
2x+3y = 4
2x + 3(14/13) = 4
2x = 4 – (42/13)
= (52-42)/13
2x = 10/13
x = 5/13
x = 5/13, y = 14/13
∴ The real valuesof x and y are 5/13, 14/13
(ii) (3x – 2i y) (2 + i)2 =10(1 + i)
Given:
(3x – 2iy) (2+i)2 =10(1+i)
(3x – 2yi) (22+i2+2(2)(i))= 10+10i
(3x – 2yi) (4 +(-1)+4i) = 10+10i [since, i2 = -1]
(3x – 2yi) (3+4i) =10+10i
Let us divide with3+4i on both sides we get,
(3x – 2yi) =(10+10i)/(3+4i)
= Now multiply anddivide with (3-4i)
= [10(3-4i) +10i(3-4i)] / (32 – (4i)2)
= [30-40i+30i-40i2]/ (9 – 16i2)
= [30-10i-40(-1)] /(9-16(-1))
= [70-10i]/25
Now, equating Real andImaginary parts on both sides we get
3x = 70/25 and -2y =-10/25
x = 70/75 and y = 1/5
x = 14/15 and y = 1/5
∴ The real valuesof x and y are 14/15, 1/5
(4+2i) x-3i-3 +(9-7i)y = 10i
(4x+9y-3) + i(2x-7y-3)= 10i
Now, equating Real andImaginary parts on both sides we get,
4x+9y-3 = 0 … (i)
And 2x-7y-3 = 10
2x-7y = 13 … (ii)
Multiply (i) by 7 and(ii) by 9 and add
On solving theseequations we get
28x + 18x + 63y – 63y= 117 + 21
46x = 117 + 21
46x = 138
x = 138/46
= 3
Substitute the valueof x in (i) we get,
4x+9y-3 = 0
9y = -9
y = -9/9
= -1
x = 3 and y = -1
∴ The real valuesof x and y are 3 and -1
(iv) (1 + i) (x + iy) = 2 –5i
Given:
(1 + i) (x + iy) = 2 –5i
Divide with (1+i) onboth the sides we get,
(x + iy) = (2 –5i)/(1+i)
Multiply and divide by(1-i)
= (2 – 5i)/(1+i) ×(1-i)/(1-i)
= [2(1-i) – 5i (1-i)]/ (12 – i2)
= [2 – 7i + 5(-1)] / 2[since, i2 = -1]
= (-3-7i)/2
Now, equating Real andImaginary parts on both sides we get
x = -3/2 and y = -7/2
∴ Thee real valuesof x and y are -3/2, -7/2