Question 5 - A-Level Maths

AQA Further Paper 1 2022 June — Question 5 6 marks

Exam Board	AQA
Module	Further Paper 1 (Further Paper 1)
Year	2022
Session	June
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Roots of polynomials
Type	Complex roots with real coefficients
Difficulty	Standard +0.8 This is a Further Maths complex roots question requiring knowledge that complex roots come in conjugate pairs for real polynomials, finding k by substitution, factorizing the quartic, and solving an inequality. While systematic, it demands multiple techniques (complex arithmetic, polynomial division, inequality solving) and careful algebraic manipulation across 6 marks total, placing it moderately above average difficulty.
Spec	4.02g Conjugate pairs: real coefficient polynomials 4.02j Cubic/quartic equations: conjugate pairs and factor theorem

It is given that $z = -\frac{3}{2} + i\frac{\sqrt{11}}{2}$ is a root of the equation $$z^4 - 3z^3 - 5z^2 + kz + 40 = 0$$ where $k$ is a real number.

Find the other three roots. [5 marks]
Given that $x \in \mathbb{R}$, solve $$x^4 - 3x^3 - 5x^2 + kx + 40 < 0$$ [1 mark]

Show mark scheme Show mark scheme source

Question 5:

Answer	Marks	Guidance
5(a)	States complex conjugate	1.1b

3 com√1 1

Sum o f p lex roots = –3

𝑧𝑧 = −2−i 2

Product of complex roots = 5

So is a factor of

2 𝑧𝑧 + 3𝑧𝑧+5

4 3 2

T𝑧𝑧he− o3th𝑧𝑧er− qu5a𝑧𝑧dr+at𝑘𝑘ic𝑧𝑧 f+ac4to0r is

2 𝑧𝑧 −6𝑧𝑧+8

T=h(e𝑧𝑧 o−th2e)r( 𝑧𝑧ro−ot4s) are

3 √11

2,4,−2−i 2

Obtains a quadratic factor

or

Obtains sum and product of

their two complex roots

or

Substitutes given root or their

Answer	Marks	Guidance
conjugate into the quartic	1.1a	M1

Obtains

or 2

Answer	Marks	Guidance
Obtains 𝑧𝑧 +3𝑧𝑧 +5	1.1b	A1

𝑘𝑘 = −6

Forms a second quadratic

or

Solves the quartic equation with

Answer	Marks	Guidance
their value of	1.1a	M1

𝑘𝑘

Answer	Marks	Guidance
Obtains 2 and 4	1.1b	A1
Total	5
Q	Marking instructions	AO

Answer	Marks
5(b)	Deduces correct solution

(ft from their exactly two distinct

Answer	Marks	Guidance
real roots)	2.2a	B1F
Total	1
Question total	6
Q	Marking instructions	AO

Question 5:
--- 5(a) ---
5(a) | States complex conjugate | 1.1b | B1 | is another root
3 com√1 1
Sum o f p lex roots = –3
𝑧𝑧 = −2−i 2
Product of complex roots = 5
So is a factor of
2
𝑧𝑧 + 3𝑧𝑧+5
4 3 2
T𝑧𝑧he− o3th𝑧𝑧er− qu5a𝑧𝑧dr+at𝑘𝑘ic𝑧𝑧 f+ac4to0r is
2
𝑧𝑧 −6𝑧𝑧+8
T=h(e𝑧𝑧 o−th2e)r( 𝑧𝑧ro−ot4s) are
3 √11
2,4,−2−i 2
Obtains a quadratic factor
or
Obtains sum and product of
their two complex roots
or
Substitutes given root or their
conjugate into the quartic | 1.1a | M1
Obtains
or 2
Obtains 𝑧𝑧 +3𝑧𝑧 +5 | 1.1b | A1
𝑘𝑘 = −6
Forms a second quadratic
or
Solves the quartic equation with
their value of | 1.1a | M1
𝑘𝑘
Obtains 2 and 4 | 1.1b | A1
Total | 5
Q | Marking instructions | AO | Marks | Typical solution
--- 5(b) ---
5(b) | Deduces correct solution
(ft from their exactly two distinct
real roots) | 2.2a | B1F | 2 < x < 4
Total | 1
Question total | 6
Q | Marking instructions | AO | Marks | Typical solution

Show LaTeX source

It is given that $z = -\frac{3}{2} + i\frac{\sqrt{11}}{2}$ is a root of the equation
$$z^4 - 3z^3 - 5z^2 + kz + 40 = 0$$
where $k$ is a real number.

\begin{enumerate}[label=(\alph*)]
\item Find the other three roots. [5 marks]

\item Given that $x \in \mathbb{R}$, solve
$$x^4 - 3x^3 - 5x^2 + kx + 40 < 0$$ [1 mark]
\end{enumerate}

\hfill \mbox{\textit{AQA Further Paper 1 2022 Q5 [6]}}

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Q1 1 Q2 1 Q3 1 Q4 1 Q5 6 Q6 8 Q7 9 Q8 11 Q9 14 Q10 12 Q11 19 Q12 17