Question 8 - A-Level Maths

OCR MEI Further Pure Core 2022 June — Question 8 11 marks

Exam Board	OCR MEI
Module	Further Pure Core (Further Pure Core)
Year	2022
Session	June
Marks	11
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Complex Numbers Argand & Loci
Type	Intersection of two loci
Difficulty	Challenging +1.2 This is a standard Further Maths locus intersection problem requiring geometric interpretation and algebraic solution. Students must recognize that arg(z-10) = 3π/4 is a half-line from (10,0), use the imaginary axis condition to find k, then solve the system. While it requires multiple steps and careful reasoning, the techniques are well-practiced in Further Pure courses and the 'imaginary axis' hint significantly guides the solution path.
Spec	4.02k Argand diagrams: geometric interpretation 4.02o Loci in Argand diagram: circles, half-lines

8 Two sets of complex numbers are given by \(\left\{ z : \arg ( z - 10 ) = \frac { 3 } { 4 } \pi \right\}\) and \(\{ z : | z - 3 - 6 i | = k \}\), where \(k\) is a positive constant. In an Argand diagram, one of the points of intersection of the two loci representing these sets lies on the imaginary axis.

Sketch the loci on an Argand diagram.
In this question you must show detailed reasoning. Find the complex numbers represented by the points of intersection.

Show mark scheme Show mark scheme source

Question 8:

Answer	Marks	Guidance
8	(a)	M1

A1

B1

Answer	Marks
[4]	1.1

1.1

Answer	Marks
1.1	half line from 10

at 45 to Real axis implied by angle shown or meeting the

Imaginary axis at 10

circle centre 3+6𝑖

circle meeting half line on Imaginary axis

Answer	Marks	Guidance
8	(b)	DR

one point is 10i

𝑘2 = (3−0)2+(6−10)2

⇒ 𝑘 = 5

line 𝑥+𝑦 = 10

(𝑥−3)2+(10−𝑥−6)2 = 25

⇒ 2𝑥2−14𝑥 = 0

⇒ 𝑥 = 7, 𝑦 = 3

Answer	Marks
other point is 7+3𝑖	B1

M1*

A1

M1de

p*

M1

A1

Answer	Marks
A1	1.1

3.1a

1.1 3.1a

1.1

Answer	Marks
3.2a	soi

solving 𝑥+𝑦 = 10 and circle equation simultaneously

Rearranging into a quadratic = 0

Could see solutions as √58(cos0.405+𝑖sin0.405)or

√58𝑒0.405𝑖

Alternative solution

Answer	Marks	Guidance
one point is 10i	B1
eqn of perp from (3, 6) to chord is 𝑦 = 𝑥+3	eqn of perp from (3, 6) to chord is 𝑦 = 𝑥+3	M1

(3, 6)

Answer	Marks
solving with 𝑥+𝑦 = 10	M1
A1	or by inspection
oe	or by inspection

1 1

midpoint of chord is (3 , 6 )

Answer	Marks
2 2	oe

1 1

other end of chord is (2×3 −0, 2×6 −10)

Answer	Marks	Guidance
2 2	M1
 other point of intersection is (7, 3)	A1
this represents 7+3𝑖	this represents 7+3𝑖	A1

√58𝑒0.405𝑖

[7]

M1

A1

Question 8:
8 | (a) | M1
A1
B1
B1
[4] | 1.1
1.1
1.1
1.1 | half line from 10
at 45 to Real axis implied by angle shown or meeting the
Imaginary axis at 10
circle centre 3+6𝑖
circle meeting half line on Imaginary axis
8 | (b) | DR
one point is 10i
𝑘2 = (3−0)2+(6−10)2
⇒ 𝑘 = 5
line 𝑥+𝑦 = 10
(𝑥−3)2+(10−𝑥−6)2 = 25
⇒ 2𝑥2−14𝑥 = 0
⇒ 𝑥 = 7, 𝑦 = 3
other point is 7+3𝑖 | B1
M1*
A1
M1de
p*
M1
A1
A1 | 1.1
3.1a
1.1
3.1a
1.1
1.1
3.2a | soi
solving 𝑥+𝑦 = 10 and circle equation simultaneously
Rearranging into a quadratic = 0
Could see solutions as √58(cos0.405+𝑖sin0.405)or
√58𝑒0.405𝑖
Alternative solution
one point is 10i | B1
eqn of perp from (3, 6) to chord is 𝑦 = 𝑥+3 | eqn of perp from (3, 6) to chord is 𝑦 = 𝑥+3 | M1 | M1 | oe, e.g. midpoint of chord has equal x and y displacements from
(3, 6)
solving with 𝑥+𝑦 = 10 | M1
A1 | or by inspection
oe | or by inspection
1 1
midpoint of chord is (3 , 6 )
2 2 | oe
1 1
other end of chord is (2×3 −0, 2×6 −10)
2 2 | M1
 other point of intersection is (7, 3) | A1
this represents 7+3𝑖 | this represents 7+3𝑖 | A1 | A1 | Could see solutions as √58(cos0.405+𝑖sin0.405)or
√58𝑒0.405𝑖
[7]
M1
A1

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8 Two sets of complex numbers are given by $\left\{ z : \arg ( z - 10 ) = \frac { 3 } { 4 } \pi \right\}$ and $\{ z : | z - 3 - 6 i | = k \}$, where $k$ is a positive constant. In an Argand diagram, one of the points of intersection of the two loci representing these sets lies on the imaginary axis.
\begin{enumerate}[label=(\alph*)]
\item Sketch the loci on an Argand diagram.
\item In this question you must show detailed reasoning.

Find the complex numbers represented by the points of intersection.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Pure Core 2022 Q8 [11]}}

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