1/2
Q1.
Convert
(i + 1)2/(3 - i)
into the form of a + ib .
(a) - 1/5
+
3/5
i
(b) 2 + 3i
(c) 1/5
-
3/5
i
(d) 4 - 9i
Solution. a
(1 + i)2/(3 - i)
=
1 + i2 + 2i/(3 - i)
=
2i/(3 - i)
=
2i(3 + i)/(3 - i)(3 + i)
=
2i(3 + i)/(9 + 1)
=
i(3 + i)/5
=
3i + i2/5
=
-
1/5
+
3/5
i
Solution
Q2.
Find the multiplicative inverse of 4 + 3i
(a) 4 + 3i
(b)
4/25
-
3/25
i
(c) 4 - 3i
(d) 4/25
+
3/25
i
Solution.b
let z = 4 + 3i
multiplicative inverse is
1/z
1/4 + 3i
=
4 - 3i/(4 + 3i)(4 - 3i)
=
4 - 3i/16 - 9i2
=
4/25
-
3/25
i
or
shortcut
1/2
=
z̅/lzl2
Solution
Q3.
Find the value of
(
1 + i/1 - i
)8
(a) 1
(b) 2
(c) 3
(d) 4
Solution. a
(
1 + i/1 + i
)
(
1 + i/1 + i
) =
1 + 2i + i2/1 - (i2)
=
1 + 2i - 1/1 + 1
= i
= i
8 = (i
4)
2 = (1)
2 = 1
Solution
Q4.
zz̅ + (3 - i)z + (3 + i)z̅ + 1 = 0 represents a circle with
(a) centre (-3,-1) and radius 3
(b) centre (-3,1) and radius 3
(c) centre (-3,-1) and radius 4
(d) centre (-3,1) and radius 4
Solution. a
Given that z.z̅ + (3 - i) z +(3 + i)z̅ + 1 = 0
Put z = x + iy and z̅ = x - iy ,
we get (x + iy)(x - iy) + (3 - i)(x + iy) + (3 + i) (x - iy) + 1 = 0
⇒ x2 + y2 + 3x + 3iy - ix + y + 3x - 3iy + ix + y + 1 = 0
⇒ x2 + y2 + 6x + 2y + 1 = 0
∴ centre = (-g, -f) = (-3,-1)
and radius = √( g2 + f2 - c)
= √( 9 + 1 - 1) = √9 = 3
Solution
Q5.
Find the value of 1 + w27 + w30
(a) 1
(b) 3
(c) 2
(d) 4
Solution. b
1 + (w3)9 + (w3)10
∴ w3 = 1
= 1 + 1 + 1 = 3
Note : 1 + wr + w2r = 3 , if r is multiple of 3 .
Solution
