Question 1 :
If z1 = 2 − i and z2 = −4 + 3i , find the inverse of z1 z2 and z1/z2
Solution :
(i)
z1 z2 = (2 - i)(-4 + 3i)
= -8 + 6i + 4i - 3i2
= -8 + 10i - 3(-1)
= -8 + 10i + 3
= -5 + 10i
z1 z2 = -5(1 - 2i)
Inverse of z1 z2 = 1/z1 z2
= (-1/5)(1/(1 - 2i))
= (-1/5)(1/(1 - 2i))((1 + 2i)/(1 + 2i))
= (-1/5)(1 + 2i)/(1 - 4(-1))
= (-1/5)(1 + 2i)/(1 + 4)
= (-1/25)(1 + 2i)
= (-1 - 2i)/25
(ii) Inverse of z1/z2 = z2/z1
= [(-4 + 3i)/(2 - i)][(2 + i)/(2 + i)]
= (-4 + 3i)(2 + i) / (2 - i)(2 +i)
= (-8 - 4i + 6i + 3i2)/(4 -(-1))
= (-8 + 2i - 3)/(4 + 1)
= (-11 + 2i)/5
= (1/5) (-11 + 2i)
Question 2 :
The complex numbers u,v , and w are related by (1/u) = (1/v) + (1/w)
if v = 3 - 4i and w = 4 + 3i, find u in rectangular form.
Solution :
Given that
(1/u) = (1/v) + (1/w)
1/v = [1/(3 - 4i)][(3 + 4i)/(3 + 4i)]
= (3 + 4i)/(9 - 16(-1))
= (3 + 4i)/(9 + 16)
= (3 + 4i)/25
1/w = [1/(4 + 3i)][(4 - 3i)/(4 - 3i)]
= (4 - 3i)/(16 - 9(-1))
= (4 - 3i)/(16 + 9)
= (4 - 3i)/25
1/u = [(3 + 4i)/25] + [(4 - 3i)/25]
(1/u) = (7 + i)/25
By finding the inverse of 1/u, we get u
= [25/(7 + i)][(7 - i)/(7 - i)]
= 25(7 - i)/(49 + 1)
= 25(7 - i)/50
= (7 - i)/2
Hence the value of u in rectangular form is (1/2)(7 - i).
Question 4 :
Prove the following properties:
(i) z is real if and only if z = z bar
Solution :
Let us consider the complex number 4 + 3i
Here 4 is real part and 3 is the imaginary part.
z bar = 4 - 3i
There is no changes in real parts of the above complex numbers.
Hence z is real if and only if z = z bar.
(ii) Re(z) = (z + z bar)/2
Solution :
z = 4 + 3i
z bar = 4 - 3i
z + z bar = (4 + 3i) + (4 - 3i)
= (4 + 4) + (3i - 3i)
= 8
(z + z bar) / 2 = 8/2 = 4 ----(1)
Re(z) = 4 ----(2)
(1) = (2)
Hence proved.
(iii) im(z) = (z - z bar) / 2
z = 4 + 3i
z bar = 4 - 3i
z - z bar = (4 + 3i) - (4 - 3i)
= (4 - 4) + (3i + 3i)
= 6i
(z + z bar) / 2 = 6i/2 = 3 ----(1)
im(z) = 3 ----(2)
(1) = (2)
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