Download IB Problems File

1.
Consider the polynomial p(x) = x4 + ax3 + bx2 + cx + d, where a, b, c, d 
.
Given that 1 + i and 1 – 2i are zeros of p(x), find the values of a, b, c and d.
(Total 7 marks)
2.
(a)
Solve the equation z3 = –2 + 2i, giving your answers in modulus–argument form.
(6)
(b)
Hence show that one of the solutions is 1 + i when written in Cartesian form.
(1)
(Total 7 marks)
3.
Given that z = cosθ + i sin θ show that
(a)
1 

Im  z n  n   0, n 
z 

+
;
(2)
(b)
 z 1 
Re 
 = 0, z ≠ –1.
 z 1
(5)
(Total 7 marks)
4.
The complex numbers z1 = 2 – 2i and z2 = 1 – i 3 are represented by the points A and B
respectively on an Argand diagram. Given that O is the origin,
(a)
find AB, giving your answer in the form a b  3 , where a, b 
+
;
(3)
(b)
calculate AÔB in terms of π.
(3)
(Total 6 marks)
IB Questionbank Mathematics Higher Level 3rd edition
1
5.
Consider the complex number ω =
(a)
z i
, where z = x + iy and i =
z2
1 .
If ω = i, determine z in the form z = r cis θ.
(6)
(b)
Prove that ω =
( x 2  2 x  y 2  y)  i( x  2 y  2)
( x  2) 2  y 2
.
(3)
(c)
Hence show that when Re(ω) = 1 the points (x, y) lie on a straight line, l1, and write down
its gradient.
(4)
(d)
Given arg (z) = arg(ω) =
π
, find │z│.
4
(6)
(Total 19 marks)
6.
 2π 
 2π 
Consider ω = cos   i sin   .
 3 
 3 
(a)
Show that
(i)
ω3 = 1;
(ii)
1 + ω + ω2 = 0.
(5)
(b)
iθ
 2π 
i   
3 

(i)
Deduce that e + e
(ii)
Illustrate this result for θ =
e
 4π 
i   
3 

= 0.
π
on an Argand diagram.
2
(4)
(c)
(i)
Expand and simplify F(z) = (z – 1)(z – ω)(z – ω2) where z is a complex number.
(ii)
Solve F(z) = 7, giving your answers in terms of ω.
(7)
IB Questionbank Mathematics Higher Level 3rd edition
2
(Total 16 marks)
7.
(a)
Factorize z3 + 1 into a linear and quadratic factor.
(2)
Let γ =
(b)
1 i 3
.
2
(i)
Show that γ is one of the cube roots of –1.
(ii)
Show that γ2 = γ – 1.
(iii)
Hence find the value of (1 – γ)6.
(9)
(Total 11 marks)
8.
(a)
Write down the expansion of (cos θ + i sin θ)3 in the form a + ib, where a and b are in
terms of sin θ and cos θ.
(2)
(b)
Hence show that cos 3θ = 4 cos3 θ – 3 cos θ.
(3)
(c)
Similarly show that cos 5θ = 16 cos5 θ – 20 cos3 θ + 5 cos θ.
(3)
(d)
 π π
Hence solve the equation cos 5θ + cos 3θ + cos θ = 0, where θ   ,  .
 2 2
(6)
(e)
By considering the solutions of the equation cos 5θ = 0, show that
cos
7π
π
5 5
and state the value of cos
.

10
8
10
(8)
(Total 22 marks)
IB Questionbank Mathematics Higher Level 3rd edition
3