Download Ex.B

W.35 Pure Mathematics (Complex Number)
Ex.A
1.
(a) (i)
Express cos 4 sin 3 as a sum of sines of multiples of  .
(ii) Express cos3 sin 4 as a sum of cosines of multiples of  .
2.
(b) Hence, find

(a) Show that
cos 7  64 cos7 - 112 cos5  56 cos3 - 7 cos .
cos3 sin 3 cos   sin   d .
(b) Using (a), or otherwise, show that the roots of the equation 64 x 6 - 112 x 4  56 x 2 - 7  0
are
 cos
π
3π
5π
,  cos
and  cos
.
14
14
14
(c) Hence, deduce that cos
3.
i 

By expanding  1 

3 

n
(a)

k0
4.
(a) Show that
π
3π
5π
7
 cos
 cos

.
14
14
14
8
2n
, find the values of
 1
C
2n 2k  - 3 
n -1
k

(b)
k0
sin (2n  1)  sin 2n 1
n

k0
where n is a positive integer and 0 <  

2
2n C
- 1k 2n 1C2k 1
 1
2k 1  3 
k
.
cot2 n - k
.
(b) Hence, show that
n

k 1
5.
n (2n - 1)
 k 
cot 2 
 
3
 2n  1 
n
and

k 1
2n (n  1)
 k 
csc 2 
.
 
3
 2n  1 
(a) Let n be a positive integer.
Prove that
 1  i tan 

 1 - i tan 
Hence, show that



n

1  i tan n
.
1 - i tan n
C1 cot n -1 - C3 cot n -3  C5 cot n -5 - ...
n
n
n
tan n 
.
n
n
2
cot  - C2 cot   C4 cot n -4 - ...
n
n
(b) By considering the roots of the equation
xn 
C x n -1 - C2 x n -2 - C3 x n -3  C4 x n -4  C5 x n -5 - ...  0 ,
n 1
n
n
n
n
n
evaluate

k 1
cot
(4k - 1)
.
4n
Ex.B
1.
Let n be a positive integer. Show that
n -1
Hence, deduce that
1 x 
2n


1-n
cos 
  2

 2n 
k 1
k
 z2n - zn cos n  1 
-
 1- x 
2n
n -1
 4nx

 2
2  k 
 x  tan  2n  .


k 1 
n .
2.
Factorize
into real quadratic factors .
3.
(a) Find all the complex numbers z satisfying the equation
z  z n -1 .
(b) Find the sum 1  2w  3w 2  ...  nw n -1 where w is a root of the equation w n = 1.
4.
Solve for z where
5.
(a) Show that
 1  z 2n 1

 1 - z 2n 1
and show that
tan 2
π
2π
 tan 2
 5 .
5
5
2
4



 1   x 2 - 2x cos
 1
x 6  x5  x 4  x3  x 2  x  1   x 2 - 2x cos
7
7



6
 2

 1
 x - 2x cos
7


and deduce that sin
π
2π
3π
7
 sin
 sin

.
7
7
7
8
(b) Show that

3



 1
x 6 - x5  x 4 - x3  x 2 - x  1   x 2 - 2x cos  1   x 2 - 2x cos
7
7



5
 2

 1
 x - 2x cos
7


π
3π
5π
1
 sin
 sin
 .
and deduce that sin
14
14
14
8
6.
If  and  are the roots of the equation
(a)  - 2i and  - 2i
with the roots
7.
(1  i) z 2 - 2i z  (2 - 3i)  0 , construct the equation
(b)  2 and  2 .
(a) Write down all the 2n th roots of 1.
(b) Hence, factorize z 2n - 1 into real quadratic factors.
   z2 - 2z cos rn

n -1
(Answer : z 2n - 1 = z2 - 1
r 1
(c) Deduce from (b) that
(i)
sin n
 2n -1
sin 
(ii)
sin
n -1

 cos 


r 1
- cos
r 

n 
π
2π
(n - 1)
n
 sin
   sin

.
n
n
n
2n -1

 1

)
8.
If  is a complex cube root of 1, show that
(a)
(b)
(c)
9.
1     2 1 -    2   4
,
1   -  2 3 - 1 -    2 3  0 ,
 2  5  2 2 6   2  2  5 2 6  729 .
If  is a complex fifth root of 1, show that
1  
10. By considering the equation
1   2 2 1  3 3 1   4 4  -1 .
 z  i 8
x 4 - 28x 3  70x 2 - 28x  1  0

 z - i 8
are tan 2
 0 , prove that the roots of the equation
π
3π
5π
7π
, tan 2
, tan 2
and tan 2
.
16
16
16
16
Ex.C
1.
Let Tn (x) = cos
 n cos-1x  .
(82) (a) Prove that Tn+1(x) = 2x Tn(x) - Tn-1(x) and hence show that Tn(x) is a polynomial in x of
degree n with leading coefficient 2n-1 where n = 1, 2 , 3, … .
(b) By using De Moivre’s Theorem, or otherwise, find ak (k = 0 , 1 , 2, … , n) in
cos n 
n
=

k0
a
k
cos (n - 2k)  .
(c) By using (b), or otherwise, show that
xn =
1
2n 1
n -1
2

k0
n Ck Tn -2k (x)
for - 1  x  1
where n is any odd positive integer .
2.
(a) State De Moivre’s Theorem for a positive integral value.
(76) (b) Find a polynomial f (x) such that cos 5  cos  f  cos  .
Hence, find the value of  ( 0     ) such that f  cos  = 0 .
(c) By using (b), or otherwise, show that
 
3

 1 - cos
  1 - cos
10  
10

   3 

Hence, show that  sin
  sin
 
10 
 10  
7  
9 
1

.
  1 - cos
  1 - cos
 
10  
10  16

1
.
4
3.
(a) Show that if P (x) is a polynomial with real coefficients and r is a complex number for which
 
P (r) = 0, then P r  0 .
(b) If z is a complex number whose imaginary part is positive, show that for any complex number r,
 r - z  r - z


r - z  r - z


if Im r  0
if Im r  0
.
When does the equality hold?
(c) Let z1 , z2 , … , zn be n complex numbers whose imaginary parts are all strictly positive and put
 z - z   z n   a  i b  z n -1  ...   a  i b 
n
n

1
1
j

j1
n
j
j
j
where a1 , a2 , … , an and
b1 , b2 , … , bn are real .
Show that if r is a root of the equation x n  a1 x n -1  ...  a n  0 , then
n

j1
n
r - z
j
j


j1
r - z
j
.
j
(d) Hence, deduce from (b) and (c) that every root of the equation x n  a1 x n -1  ...  a n  0
is real .
4.
(a) By using De Moivre’s Theorem, show that
sin (2n  1)  sin 2n 1
n

r0
where n is a positive integer and 0 <  

2
- 1r 2n 1C2r 1
cot2 n - r
.
(b) By using (a) and the relations between the coefficients and the roots of the equation
n

r0
- 1r
n - r  0 , show that
2n 1C2r 1 x
n
and deduce that

k 1
n

k 1
n (2n  2)
 k 
csc 2 
.
 
3
 2n  1 
n (2n - 1)
 k 
cot 2 
 
3
 2n  1 