Questions - Answers
1. if = 4 determine the locus of z.
Sol: 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 = 1/2 ⇒ 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 cosθ, y = r sinθ
... cosθ and sinθ 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 (y/x)
Arg = Arg (x - iy)
Arg (z1. z2) = Arg z1 + Arg z2 + nπ, n ∈ {-1, 0, 1}
Arg (Z1/Z2) = Arg z1 - Arg z2 + nπ, n ∈ {-1, 0, 1}
The sign of argument changes depending on the quadrants accordingly. θ is required argument.
8. If the Arg and Arg are respectively, find (Arg z1 + Arg z2)
Sol: Let z1 = x1 - iy1, => = x1 + iy1
So the point lies in the IV quadrant