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Complex Numbers sheet
1) Find the output of the following operations performed on complex numbers
2+๐
๐)
๐+1
d (
2
โ3โ ๐
g)
10
+
1
๐
10
)
(2 + ๐)(1 โ ๐)
3โ2๐
j) (โ2 โ 3 ๐) (โ3 + 4๐)
(1โ๐ )6
b) (1 โ โ3 ๐)
c)
e) (1 + ๐)5
f) (1 + โ3 ๐) (๐ โ 1)7
h)
1
๐(3+2๐)2
(โ2โ3 ๐)
k)
(โ3+4๐)
(๐+1)7
5
i)
l)
(โ3+โ2 ๐)3
(โ2โโ3๐)
1
(โ3+4๐)
2) Find the modulus ( |Z |) and the argument ( arg(Z) ) for next complex numbers
3)
๐) ๐ = (1 โ ๐)
b) ๐ = 11๐
c) ๐ = โ
d) ๐ = (โ2 + 2โ3๐)
e) ๐ = โ4๐
f) ๐ =
๐
4
1
2
Find the roots of complex numbers where
๐) ๐ 3 = โ8
d)
๐ 4 = (โ2 โ 2โ3๐)
b) ๐ 3 = 27๐
e)
๐ 3 = 4โ2๐ โ 4โ2
1
c) ๐ 2 = (โ1 + โ3 ๐)
f) ๐ 6 = 64๐
Complex Numbers sheet
g) ๐ 6 = โ64๐
j) ๐ 3 =
โ2
2
โ
4
h) ๐ 8 = โ16
โ2
2
1
2
i) ๐ = โ
โ3
๐
4) Given Z= cos(3) + sin(3) ๐ prove that 1 + ๐ง = (1 + ๐ง)๐ง
5) Calculate (cos(2) + sin(2)๐ + 1)๐
6) Given : n is a positive integer
Z is a complex number with modulus 1, such that ๐ง 2๐ โ โ1
show that
๐ง๐
1+๐ง 2๐
is a real number.
7) Let ๐ง the conjugate complex number of z. find z such that
๐ง 2 + (๐ง)2 = ๐ง๐๐๐
8) Find the value of k for the quotient
(2โ๐๐)
(๐โ๐)
if it is :
- A pure imaginary number
- A real number
9) The complex number, 2 + 2
is rotated 45ยฐ about the origin of its
coordinates in an anti-clockwise direction. Find the complex number
obtained after the turn.
10)
Determine the value of b for the quotient
, if it equals:
Note the following trigonometric Identities
a) cos 2 ๐ฅ + sin2 ๐ฅ = 1
b) cos(2๐ฅ) = cos 2 ๐ฅ โ sin2 ๐ฅ = 2 cos 2 ๐ฅ โ 1 = 1 โ 2 sin2 ๐ฅ
c) sin(2๐ฅ) = 2sin(๐ฅ)cos(๐ฅ)
2
2
๐
Maths sheet-1.pdf (PDF, 323.86 KB)
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