Standard +0.3 This is a straightforward system of two linear equations in complex variables requiring conjugate manipulation and algebraic substitution. While it involves complex conjugates, the solution method is mechanical: take conjugates of both equations to get four equations total, then solve by elimination. The algebra is routine for Further Maths students and requires no geometric insight or novel problem-solving approach.
Question 7:
7 | Writes
z = x+iy z*= x–iy
w=u+iv w*=u–iv
OE PI
or
Obtains the conjugate of
one of the equations
Eg z * + w = 5 | 1.1a | M1 | z = x + i y z * = x – i y
Let
w = u + i v w * = u – i v
where x , y , u , v
x + iy + u – iv = 5
Re: x + u = 5…(1)
Im: y – v = 0 y = v…(2)
3(x – iy) – (u + iv) = 6 + 4i
Re: 3x – u = 6…(3)
Im: –3y – v = 4…(4)
(2), (4) y = –1, v = –1
11 9
(1), (3) x= ,u =
4 4
1 1
z = – i
4
9
w = – i
4
Forms two of
x + u = 5
y – v = 0
OE
3 x – u = 6
– 3 y – v = 4
or
eliminates one complex
unknown | 1.1a | M1
Obtains at least two
correct values of
x, y, u or v
or
11
z* = +i
4
obtains one of
9
w* = +i
4 | 1.1a | M1
Obtains the values
1 1 9
, – 1 , , – 1
4 4 | 1.1b | A1
1 1
z = – i
4
Obtains
9
w = – i
4 | 1.1b | A1
Question total | 5
Q | Marking instructions | AO | Marks | Typical solution