Maths-2a 3ei4u

Complex Numbers 1. if Sol:

= 4 determine the locus of z. let z = x1 + iy1

(x1 - 3)2 + (y1 + 1)2 = 16 x12 - 6x1 + 9 + y12 + 2y1 + 1 - 16 = 0. x12 + y12 - 6x1 + 2y1 - 6 = 0 ... Required locus is x2 + y2 - 6x + 2y - 6 = 0 2. If z = 2 - 3i, then show that z2 - 4z + 13 = 0 Sol: Consider z = 2 - 3i => z - 2 = - 3i Squaring on both sides we get (z - 2)2 = (-3i)2 z2 - 4z + 4 = 9i2 z2 - 4z + 4 = -9 (... i2 = -1) z2 - 4z + 13 = 0. 3. Find the multiplicative inverse of 7 + 24i Sol: The multiplicative inverse of a + ib is

conjugate of z1 is z2 5. Find the square root of (3 + 4i) sol: Square root of a + ib

Comparing real parts we get x=

2x = 1

4x2 = 1 4x2 - 1 = 0

7. Express the complex number into modulus - amplitude form, z = - 1 - i Sol:

Given that z = - 1 - i Let z = x + iy Comparing we get x = - 1, y = -

We know that x = r cos , y = r sin

.. .

cos and sin are negative, the required angle lies in the third quadrant, so angle is negative.

The amplitude of a complex number is known as argument denoted by Arg (z) = Arg (x + iy) = tan-1 ( ) Arg ( ) = Arg (x - iy) Arg (z1. z2) = Arg z1 + Arg z2 + n , n {-1, 0, 1} Arg (

) = Arg z - Arg z + n 1 2

,n

{-1, 0, 1}

The sign of argument changes depending on the quadrants accordingly. is required argument.

8. If the Arg ( ) and Arg ( Sol:

) are

respectively, find (Arg z1 + Arg z2)

Let z1 = x1 - i y1, => = x1 + iy1 So the point lies in the IV quadrant

De Moiver's Theorem

3. Show that (−1 + i ) 3n + (−1 − i ) 3n = (−1)3n 23n+1 cos nπ . Sol. L.H.S = (−1 + i ) 3n + (−1 − i ) 3n

= [−1 (1 − i )] 3n + [−1 (1 + i )] 3n = (−1)3n [(1 − i ) 3n + (1 + i )3n] Let us find the Mod - amplitude form of 1 + i Let 1 + i = x + iy. x = 1 y =

.