Download (a) Solve . Express your solution in simplified surd form.

Wellington Girls’ College Mathematics Department
Achievement Standard 90638 (version 2)
Manipulate real and complex numbers, and solve equations
May 2010
QUESTION ONE
(a)
Solve 𝟐𝒙𝟐 – 𝟒𝒙 – 𝟕 = 𝟎. Express your solution in simplified surd form.
(b)
Solve the equation 𝑒 3𝑥−2 = 4
𝜋
4
(c)
Write (3𝑐𝑖𝑠 ( 6 )) as a complex number in the form a+bi.
(d)
Solve
(e)
𝑧 = 2𝑖 is a solution to x 3  x2  kx  4  0 , where k is a real number.
Find the value of k.
(f)
𝑧 = 𝑥 + 𝑖𝑦 is any non zero complex number. If 𝑧 + 𝑧 = 𝑘, with k real, prove that
either 𝑦 = 0 or 𝑥 2 + 𝑦 2 = 1.
x  3  3x .
1
QUESTION TWO
4
3
2
1
u
-3
-2
-1
0
1
2
3
4
Re
-1
-2
v
-3
(a)
Write u in polar form, rcis .
(b)
Write uv in rectangular form, a  bi .
(c)
Solve log x  2 27  3 .
(d)
Find all the solutions to 𝑤 3 = 8, where w is a complex number.
(e)
Hence or otherwise solve (𝑧 2 − 2)3 = 8, where z is a complex number.