AQA Further Paper 1 Specimen — Question 5 6 marks

Exam BoardAQA
ModuleFurther Paper 1 (Further Paper 1)
SessionSpecimen
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex Numbers Arithmetic
TypeParametric polynomials with root conditions
DifficultyStandard +0.8 This is a Further Maths question requiring knowledge that complex roots come in conjugate pairs, polynomial division or factor theorem application, and solving quadratics. While systematic, it demands multiple techniques (finding conjugate root, forming quadratic factor, polynomial division, solving resulting quadratic) and careful algebraic manipulation across 6 marks total. More demanding than standard A-level but follows established procedures without requiring novel insight.
Spec4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots
\(p(z) = z^4 + 3z^2 + az + b\), \(a \in \mathbb{R}\), \(b \in \mathbb{R}\) \(2 - 3i\) is a root of the equation \(p(z) = 0\)
  1. Express \(p(z)\) as a product of quadratic factors with real coefficients. [5 marks]
  2. Solve the equation \(p(z) = 0\). [1 mark]
Question 5:
AnswerMarks Guidance
5(a)Makes a correct deduction about
another root (PI)AO2.2a B1
∴
p(z)=(z2−4z+13)(z2+cz+d)
(z2−4z+13)(z2+cz+d)≡z4+3z2+az+b
c−4=0
13−4c+d =3
⇒c=4, d =6
∴
( )( )
p(z)= z2 −4z+13 z2 +4z+6
Finds quadratic factor by expanding
brackets or using sum and product of
AnswerMarks Guidance
rootsAO1.1a M1
Finds a correct quadratic factorAO1.1b A1
Compares coefficients with quartic
AnswerMarks Guidance
z4 +3z2 +az+bAO1.1a M1
States the correct product of
AnswerMarks Guidance
quadratic factorsAO1.1b A1
(z−(2−3i))(z−(2+3i))=z2−4z+13
∴
p(z)=(z2−4z+13)(z2+cz+d)
(z2−4z+13)(z2+cz+d)≡z4+3z2+az+b
AnswerMarks Guidance
ALTALT Makes a correct deduction about
another root
AnswerMarks Guidance
Finds quadratic factor by expandingAO1.1a AO1.1a
brackets or using sum and product of
roots
α+β+γ+δ=0 ⇒ γ+δ= −4
AnswerMarks Guidance
Obtains a correct quadratic factorAO1.1b A1
(∑α )2 =∑α2 +2 ∑αβ
AnswerMarks Guidance
Uses coefficients/roots to set upAO1.1a AO1.1a
equations and find required∑α2+2×3
0=
coefficients
0= ( 2−3i)2 +( 2+3i)2 +γ2 +δ2 +6
States the correct product of
AnswerMarks Guidance
quadratic factorsStates the correct product of ∴γ2 +δ2 =4
quadratic factors
2γδ=(γ+δ)2 −γ2 +δ2
AnswerMarks
AO1.1bA1
16−4
γδ=
2
∴d =6
( ) ( )
p ( z )= z2 −4z+13 z2 +4z+6
AnswerMarks Guidance
QMarking Instructions AO
(b)States all four correct solutions
FT ‘their’ two quadratic factors from
part (a) provided both M1 marks
AnswerMarks Guidance
have been awardedAO1.1b B1F
Total6
QMarking Instructions AO 
Question 5:
--- 5(a) ---
5(a) | Makes a correct deduction about
another root (PI) | AO2.2a | B1 | (z−(2−3i))(z−(2+3i))=z2−4z+13
∴
p(z)=(z2−4z+13)(z2+cz+d)
(z2−4z+13)(z2+cz+d)≡z4+3z2+az+b
c−4=0
13−4c+d =3
⇒c=4, d =6
∴
( )( )
p(z)= z2 −4z+13 z2 +4z+6
Finds quadratic factor by expanding
brackets or using sum and product of
roots | AO1.1a | M1
Finds a correct quadratic factor | AO1.1b | A1
Compares coefficients with quartic
z4 +3z2 +az+b | AO1.1a | M1
States the correct product of
quadratic factors | AO1.1b | A1
(z−(2−3i))(z−(2+3i))=z2−4z+13
∴
p(z)=(z2−4z+13)(z2+cz+d)
(z2−4z+13)(z2+cz+d)≡z4+3z2+az+b
ALT | ALT | Makes a correct deduction about | AO2.2a | AO2.2a | B1 | B1
another root
Finds quadratic factor by expanding | AO1.1a | AO1.1a | M1 | M1
brackets or using sum and product of
roots
α+β+γ+δ=0 ⇒ γ+δ= −4
Obtains a correct quadratic factor | AO1.1b | A1 | ∴c=4
(∑α )2 =∑α2 +2 ∑αβ
Uses coefficients/roots to set up | AO1.1a | AO1.1a | M1 | M1
equations and find required | ∑α2+2×3
0=
coefficients
0= ( 2−3i)2 +( 2+3i)2 +γ2 +δ2 +6
States the correct product of
quadratic factors | States the correct product of | ∴γ2 +δ2 =4
quadratic factors
2γδ=(γ+δ)2 −γ2 +δ2
AO1.1b | A1
16−4
γδ=
2
∴d =6
( ) ( )
p ( z )= z2 −4z+13 z2 +4z+6
Q | Marking Instructions | AO | Marks | Typical Solution
(b) | States all four correct solutions
FT ‘their’ two quadratic factors from
part (a) provided both M1 marks
have been awarded | AO1.1b | B1F | z =2±3i,−2± 2i
Total | 6
Q | Marking Instructions | AO | Marks | Typical Solution
$p(z) = z^4 + 3z^2 + az + b$, $a \in \mathbb{R}$, $b \in \mathbb{R}$

$2 - 3i$ is a root of the equation $p(z) = 0$

\begin{enumerate}[label=(\alph*)]
\item Express $p(z)$ as a product of quadratic factors with real coefficients.
[5 marks]

\item Solve the equation $p(z) = 0$.
[1 mark]
\end{enumerate}

\hfill \mbox{\textit{AQA Further Paper 1  Q5 [6]}}
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Q1 1 Q2 2 Q3 6 Q4 2 Q5 6 Q6 7 Q7 11 Q8 5 Q9 13 Q10 10 Q11 6 Q12 3 Q13 5 Q14 12 Q15 11