Download Advanced Higher Mathematics Unit 2

Advanced Higher Mathematics Unit 2
Complex Numbers 1
1.
2.
If a  1  i, b  2  i, c  2  3i, express in the form x  iy :
(i)
ab
(ii)
(v)
a2  b2
(vi)
(ix)
b
a
(x)
(iii)
ab
1
a
b
ac
(vii)
(xi)
a 2c
1
ab
ab
c2
(iv)
(viii)
(xii)
(a  b)c 2
a
b
(a  b) 2
Solve the following equations, giving the roots in the form x  iy (where either x or y may be
zero).
(i)
(iii)
x 2  4 x  20  0
x 2  4x  7  0
(ii)
(iv)
x 2  36  0
x 2  2ix  2  0
3.
If z  x  iy , write down the complex conjugate, z , and express
4.
If z  1  i, express z in the form x  iy , and hence evaluate zz .
5.
If x  iy 
3i
, find x and y.
2i
6.
If a  ib 
1
, show that (a 2  b 2 )( x 2  y 2 )  1.
x  iy
7.
If z  cos   i sin , find the value of
8.
If z  cos   i sin , find the values of z 
9.
Find the modulus and argument of:
(i)
1 i
3  4i
.
1 i
10.
Find
11.
 3i
.
Find Arg 

 1 i 
(ii)
z
in the form x  iy .
z
1
.
z
1
1
and z  .
z
z
3  5i
(iii)
 2  3i


12.
Find 1  i 
13.
If z  1, describe the locus of the point ( x, y ) , where z  x  iy.
14.
Evaluate
15.
Find the modulus and argument of
16.
 1 i 
Find Arg 
,
 3 i
(i)
(ii)
3i .
1 i
.
7i
1  3i
.
8i
By finding the arguments and subtracting.
1 i
By first writing
in the form a  ib.
3i
17.


 


Simplify  cos  i sin  cos  i sin  .

3
3 
6
6
18.


 
2
2 
Simplify  cos  i sin  cos
 i sin  .

5
5 
5
5 
19.



Simplify  cos  i sin  .

6
6
20.
Simplify
cos 2  i sin 2
.
cos   i sin 
21.
Simplify
cos 3  i sin 3
.
cos 2  i sin 2
22.
 1 i 3 
 .
Evaluate 

2


23.
Evaluate 1  i  .
2
3
3
6
24.
Expand cos 4 i sin 4 by De Moivre’s Theorem.
Use your result to express cos 4 as a polynomial in cos .
4
[Write cos 4 i sin 4   cos  i sin  . ]
25.
By writing cos 5 i sin 5   cos  i sin  , express
5
(i)
cos5 in terms of cos
(ii)
sin 5 in terms of sin .
3
26.
 3i
 .
Evaluate 

2


27.
(a)
If z  cos   i sin , show that
1
1
 2 cos  and z   2i sin .
z
z
1
1
z n  n  2 cos n and z n  n  2i sin n.
(ii)
z
z
7
 1
7
By writing  2 cos    z   , and expanding the R.H.S., express cos 7  in terms of

z
multiple angles.
Express sin 7  in terms of multiple angles.
(i)
(b)
(c)
z
28.
Solve the equation z 4  1 ( z  C).
29.
Find the three cube roots of i.
30.
Solve the equation z 3  27 ( z  C).
31.
(a)
Verify that x  2  3i is a root of the equation x 4  7 x 2  12 x  130  0.
(b)
Hence find all roots in the form a  ib (a, b  R ).
32.
Solve the equation z 5  1  0, giving the non-real roots in conjugate pairs.
33.
Show that z  1  i is a root of the equation z 4  4  0. Hence find all the roots.
34.
Show that z  3  i is a root of the equation z 4  3z 3  4 z 2  6 z  40  0. Hence find all the roots.
35.
Show that z  4  3i is a root of the equation z 3  21z 2  129 z  325  0. Hence find all roots.
36.
If z  x  iy , find the equations of the loci defined by
(i)
z4 3
(ii)
37.
Obtain the expansion of sin 7 in powers of sin .
38.
Obtain the expansion of cos7 in powers of cos .
arg( z  2) 

6
.