1. Find the real and imaginary parts of

    
Re(Z)   =  -1/5 and Im (Z)  = -7/5

2. If Z₁ = 2 + i, Z₂ = 3 - 2i and . Find the conjugate of Z₁Z₂.

                                  = (2-i) (3 + 2i)
                                  = 6 + 4i - 3i - 2i²
                                  = 6 + 4i - 3i + 2
                                  = 8 + i

3. If Z₁ = 2+i, Z₂ = 3 - 2i  and

4. Express in the standard form a+ib :

                 = (-3 + i) (-i)  [Since ¹/_i   =  -i]
                 = 3i  -  i²
                 = 3i + 1
                 = 1 + 3i

5. Express in the standard form a+ i b  :

6. Find the real values of x and y for which the equation is satisfied (1-i) x + (1 + i)y  = 1 - 3i.

(1-i) x + (1 + i) y  = x - ix + y + iy
                          = (x+y) + i (y - x )
                          = 1 - 3i (given).

So equating their RP and IP we get

                 x + y = 1 → (1)
y - x = -3 ⇒  x- y= 3 → (2)
x + y = 1  ;  x - y = 3
    2x = 4 ⇒     x  = 2

Substituting x = 2 in (1) we get y = 1
So  x = 2, y = -1

7. Find the modulus or the absolute value of

8. Find the modulus of - √2  +  i √2

Let √2 + i √2   = r (cos θ  + sin θ)

Equating the real and imaginary parts separately.

          r cos θ   = -√2         r sin θ  = √2
       r² cos² θ   = 2            r²sin²θ = 2
       r² (cos2θ +sin²θ)  = 4
                              r    = √4
                              r    = 2

9. Find the argument of -√2 +i√2

   
 Modulus r = 2,   argument

10. Find the square root of (-7 + 24i)

Squaring -7 + 24i   = (x² - y²) + 2i xy
Equating the real and imaginary parts
x² - y² = -7 and 2xy  = 24

Solving x² - y²  = -7 and x² + y² = 2 π we get x² = 9 ⇒ x = ± 3 and y2 = 16 ⇒ y = ± 4

Since xy is positive, x and y have the same sign

∴ (x = 3, y = 4) or  (x = -3, y = -4)