OCR Further Pure Core 1 Specimen — Question 10 10 marks

Exam BoardOCR
ModuleFurther Pure Core 1 (Further Pure Core 1)
SessionSpecimen
Marks10
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Mark schemeDownload PDF ↗
TopicComplex numbers 2
TypeBasic roots of unity properties
DifficultyStandard +0.3 This is a structured question on basic roots of unity properties with guided parts requiring standard justifications (modulus equality, symmetry, conjugate pairs, sum of roots). Part (ii) involves finding midpoints and forming a polynomial, which is routine manipulation. All parts follow directly from fundamental complex number theory without requiring novel insight or extended problem-solving.
Spec4.02r nth roots: of complex numbers4.05a Roots and coefficients: symmetric functions
10 The Argand diagram below shows the origin \(O\) and pentagon \(A B C D E\), where \(A , B , C , D\) and \(E\) are the points that represent the complex numbers \(a , b , c , d\) and \(e\), and where \(a\) is a positive real number. You are given that these five complex numbers are the roots of the equation \(z ^ { 5 } - a ^ { 5 } = 0\). \includegraphics[max width=\textwidth, alt={}, center]{94ecfc6e-df52-45a0-8f7b-f33fda391b15-4_903_883_477_502}
  1. Justify each of the following statements.
    1. \(A , B , C , D\) and \(E\) lie on a circle with centre \(O\).
    2. \(A B C D E\) is a regular pentagon.
    3. \(b \times \mathrm { e } ^ { \frac { 2 \mathrm { i } \pi } { 5 } } = c\)
    4. \(b ^ { * } = e\)
    5. \(a + b + c + d + e = 0\)
    6. The midpoints of sides \(A B , B C , C D , D E\) and \(E A\) represent the complex numbers \(p , q , r , s\) and \(t\). Determine a polynomial equation, with real coefficients, that has roots \(p , q , r , s\) and \(t\).
Question 10:
AnswerMarks Guidance
10(i) (a)
E1
AnswerMarks
[1]c
2.2aClear justification that all points are
equidistant from O
AnswerMarks Guidance
10(i) (b)
S
All points are distance a from O
(cid:167)2k(cid:83)(cid:183)
Each root of the form a cis(cid:168) (cid:184) hence
(cid:169) 5 (cid:185)
2(cid:83)
spaced at angular intervals of around this
5
AnswerMarks
circleB1
E1
AnswerMarks Guidance
[2]1.1
2.4Using the answer to part (b)
10(i) (c)
2(cid:83)i 2(cid:83)i
arg b(cid:117)e5 (cid:32)arg(cid:11)b(cid:12)(cid:14)arg e5
(cid:32) 4(cid:83)(cid:32)arg(c)
AnswerMarks Guidance
5E1
[1]2.2a Multiplying any complex number by
2i(cid:83) 2(cid:83)
e 5 rotates it through an angle of ,
5
so b (cid:111) c
AnswerMarks Guidance
10(i) (d)
b* = cis(cid:168) (cid:184) = cis(cid:168) (cid:184) = cis(cid:168) (cid:184)= e
AnswerMarks Guidance
(cid:169) 5 (cid:185) (cid:169) 5 (cid:185) (cid:169) 5 (cid:185)E1
[1]2.4 n
eOr by noting that complex roots
of a real polynomial occur in
conjugate pairs (the pairing is
obvious)
AnswerMarks Guidance
10(i) (e)
0
so sum of a, b, c, d and e is (cid:16) (cid:32)0
AnswerMarks
1M1
E1
AnswerMarks Guidance
[2]2.1
i1.1m Accept any complete
justification
AnswerMarks Guidance
10(ii) All roots of equal magnitude a cos(cid:83)
5
2(cid:83)
All roots spaced at angles of p
5
r is negative real root (cid:159) z5 + (a cos(cid:83))5 = 0
AnswerMarks
5eM1
B1
A1
AnswerMarks
[3]c
3.1a
3.1a
AnswerMarks
2.2aFrom any sensible right-angled triangle
Must justify equally spread arguments
Allow correct eqn. from 1st M1 only
10(ii)
p
S
Question 10:
10 | (i) | (a) | All roots satisfy | z5 | = a5 (cid:159) | z | = a | e
E1
[1] | c
2.2a | Clear justification that all points are
equidistant from O
10 | (i) | (b) | p
S
All points are distance a from O
(cid:167)2k(cid:83)(cid:183)
Each root of the form a cis(cid:168) (cid:184) hence
(cid:169) 5 (cid:185)
2(cid:83)
spaced at angular intervals of around this
5
circle | B1
E1
[2] | 1.1
2.4 | Using the answer to part (b)
10 | (i) | (c) | (cid:11) (cid:12) (cid:11) (cid:12)
2(cid:83)i 2(cid:83)i
arg b(cid:117)e5 (cid:32)arg(cid:11)b(cid:12)(cid:14)arg e5
(cid:32) 4(cid:83)(cid:32)arg(c)
5 | E1
[1] | 2.2a | Multiplying any complex number by
2i(cid:83) 2(cid:83)
e 5 rotates it through an angle of ,
5
so b (cid:111) c
10 | (i) | (d) | (cid:167)2(cid:83)(cid:183)* (cid:167)(cid:16)2(cid:83)(cid:183) (cid:167)8(cid:83)(cid:183)
b* = cis(cid:168) (cid:184) = cis(cid:168) (cid:184) = cis(cid:168) (cid:184)= e
(cid:169) 5 (cid:185) (cid:169) 5 (cid:185) (cid:169) 5 (cid:185) | E1
[1] | 2.4 | n
e | Or by noting that complex roots
of a real polynomial occur in
conjugate pairs (the pairing is
obvious)
10 | (i) | (e) | a, b, c, d and e are the roots of z5(cid:16)a5 (cid:32)0
0
so sum of a, b, c, d and e is (cid:16) (cid:32)0
1 | M1
E1
[2] | 2.1
i1.1 | m | Accept any complete
justification
10 | (ii) | All roots of equal magnitude a cos(cid:83)
5
2(cid:83)
All roots spaced at angles of p
5
r is negative real root (cid:159) z5 + (a cos(cid:83))5 = 0
5 | eM1
B1
A1
[3] | c
3.1a
3.1a
2.2a | From any sensible right-angled triangle
Must justify equally spread arguments
Allow correct eqn. from 1st M1 only
--- 10(ii) ---
10(ii)
p
S
10 The Argand diagram below shows the origin $O$ and pentagon $A B C D E$, where $A , B , C , D$ and $E$ are the points that represent the complex numbers $a , b , c , d$ and $e$, and where $a$ is a positive real number. You are given that these five complex numbers are the roots of the equation $z ^ { 5 } - a ^ { 5 } = 0$.\\
\includegraphics[max width=\textwidth, alt={}, center]{94ecfc6e-df52-45a0-8f7b-f33fda391b15-4_903_883_477_502}\\
(i) Justify each of the following statements.
\begin{enumerate}[label=(\alph*)]
\item $A , B , C , D$ and $E$ lie on a circle with centre $O$.
\item $A B C D E$ is a regular pentagon.
\item $b \times \mathrm { e } ^ { \frac { 2 \mathrm { i } \pi } { 5 } } = c$
\item $b ^ { * } = e$
\item $a + b + c + d + e = 0$\\
(ii) The midpoints of sides $A B , B C , C D , D E$ and $E A$ represent the complex numbers $p , q , r , s$ and $t$. Determine a polynomial equation, with real coefficients, that has roots $p , q , r , s$ and $t$.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Pure Core 1  Q10 [10]}}
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