AQA Further Paper 1 2019 June — Question 9 9 marks

Exam BoardAQA
ModuleFurther Paper 1 (Further Paper 1)
Year2019
SessionJune
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex numbers 2
TypeDirect nth roots: roots with geometric or algebraic follow-up
DifficultyChallenging +1.8 This is a challenging Further Maths question requiring multiple sophisticated techniques: converting complex numbers to polar form with non-standard arguments, finding cube roots using De Moivre's theorem, then applying matrix transformations to complex numbers interpreted geometrically, and calculating areas under transformation using determinants. The combination of complex numbers and linear transformations, plus the need for exact trigonometric values, places this well above average difficulty.
Spec4.02r nth roots: of complex numbers4.03i Determinant: area scale factor and orientation
  1. Solve the equation \(z^3 = \sqrt{2} - \sqrt{6}i\), giving your answers in the form \(re^{i\theta}\) where \(r > 0\) and \(0 \leq \theta < 2\pi\) [5 marks]
  2. The transformation represented by the matrix \(\mathbf{M} = \begin{pmatrix} 5 & 1 \\ 1 & 3 \end{pmatrix}\) acts on the points on an Argand Diagram which represent the roots of the equation in part (a). Find the exact area of the shape formed by joining the transformed points. [4 marks]
Question 9:
AnswerMarks
9(a)Writes complex number in
Eulerian form or
equivalent.
AnswerMarks Guidance
PI correct r&θAO1.1b B1
z3 =2 2e 3
r = 2
−π
θ=
9
5π 11π 17π
θ= , ,
9 9 9
5πi 11πi 17πi
z = 2e 9 , 2e 9 , 2e 9
Obtains r by taking cube
root of their modulus of z3,
accept AWRT 1.41 or
( )1
3
AnswerMarks Guidance
2 2 OEAO1.1b B1F
Divides their argument by
AnswerMarks Guidance
3AO1.1a M1
Finds three correct angles
5π 11π −7π 17π −π
θ= , or  , or 
AnswerMarks Guidance
9 9  9  9  9 AO2.2a A1
Finds fully correct solution,
accept decimal equivalents
( )1
3
& 2 2 OE
Accept
5π 11π −7π 17π −π
θ= , or  , or 
AnswerMarks Guidance
9 9  9  9  9 AO2.2a A1
AnswerMarks
9(b)Finds the area of their
triangle from part (a) - ft
their r only
Or
Applies Matrix Mto the
AnswerMarks Guidance
three pointsAO3.1a M1
Area of ∆ =3× × 2× 2×sin 
2  3 
3 3
=
2
M =14
3 3
Required Area =14×
2
=21 3.
Finds correct area of
original triangle
Or
Finds three correct new
AnswerMarks Guidance
pointsAO1.1b A1
Finds correct M and
uses as area scale factor
with their area of original
triangle
Or
Works out area of new
AnswerMarks Guidance
triangleAO2.2a M1
Finds correct answer from
correct reasoning in exact
AnswerMarks Guidance
form onlyAO1.1b A1
Total9
QMarking Instructions AO 
Question 9:
--- 9(a) ---
9(a) | Writes complex number in
Eulerian form or
equivalent.
PI correct r&θ | AO1.1b | B1 | −πi
z3 =2 2e 3
r = 2
−π
θ=
9
5π 11π 17π
θ= , ,
9 9 9
5πi 11πi 17πi
z = 2e 9 , 2e 9 , 2e 9
Obtains r by taking cube
root of their modulus of z3,
accept AWRT 1.41 or
( )1
3
2 2 OE | AO1.1b | B1F
Divides their argument by
3 | AO1.1a | M1
Finds three correct angles
5π 11π −7π 17π −π
θ= , or  , or 
9 9  9  9  9  | AO2.2a | A1
Finds fully correct solution,
accept decimal equivalents
( )1
3
& 2 2 OE
Accept
5π 11π −7π 17π −π
θ= , or  , or 
9 9  9  9  9  | AO2.2a | A1
--- 9(b) ---
9(b) | Finds the area of their
triangle from part (a) - ft
their r only
Or
Applies Matrix Mto the
three points | AO3.1a | M1 | 1 2π
Area of ∆ =3× × 2× 2×sin 
2  3 
3 3
=
2
M =14
3 3
Required Area =14×
2
=21 3.
Finds correct area of
original triangle
Or
Finds three correct new
points | AO1.1b | A1
Finds correct M and
uses as area scale factor
with their area of original
triangle
Or
Works out area of new
triangle | AO2.2a | M1
Finds correct answer from
correct reasoning in exact
form only | AO1.1b | A1
Total | 9
Q | Marking Instructions | AO | Marks | Typical solution
\begin{enumerate}[label=(\alph*)]
\item Solve the equation $z^3 = \sqrt{2} - \sqrt{6}i$, giving your answers in the form $re^{i\theta}$ where $r > 0$ and $0 \leq \theta < 2\pi$
[5 marks]

\item The transformation represented by the matrix $\mathbf{M} = \begin{pmatrix} 5 & 1 \\ 1 & 3 \end{pmatrix}$ acts on the points on an Argand Diagram which represent the roots of the equation in part (a).

Find the exact area of the shape formed by joining the transformed points.
[4 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA Further Paper 1 2019 Q9 [9]}}
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