Question 3 - A-Level Maths

OCR Further Pure Core 1 2022 June — Question 3 11 marks

Exam Board	OCR
Module	Further Pure Core 1 (Further Pure Core 1)
Year	2022
Session	June
Marks	11
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Complex Numbers Argand & Loci
Type	Region shading with multiple inequalities
Difficulty	Standard +0.3 This is a standard Further Maths question combining routine quadratic formula application with locus sketching and region identification. While it requires multiple techniques (finding complex roots, interpreting modulus loci as perpendicular bisector and circle, shading regions), each component is straightforward textbook material with no novel insight required. The verification in part (c) involves direct substitution rather than creative problem-solving.
Spec	4.02i Quadratic equations: with complex roots 4.02k Argand diagrams: geometric interpretation 4.02o Loci in Argand diagram: circles, half-lines

3 In this question you must show detailed reasoning.

Find the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$. The loci $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$ are given by $| z | = | z - 2 \mathrm { i } |$ and $| z - 2 | = \sqrt { 5 }$ respectively.
1. Sketch on a single Argand diagram the loci $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$, showing any intercepts with the imaginary axis.
2. Indicate, by shading on your Argand diagram, the region $$\{ z : | z | \leqslant | z - 2 i | \} \cap \{ z : | z - 2 | \leqslant \sqrt { 5 } \} .$$
1. Show that both of the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$ satisfy $| z - 2 | < \sqrt { 5 }$.
2. State, with a reason, which root of the equation $2 z ^ { 2 } - 2 z + 5 = 0$ satisfies $| z | < | z - 2 i |$.
On the same Argand diagram as part (b), indicate the positions of the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$.

Show mark scheme Show mark scheme source

Question 3:

Answer	Marks	Guidance
3	(a)	DR

2± −36 1 3

z = ⇒ z = ± i

4 2 2

or

( )  1 9

2 z2 −z =−5⇒2z2 −z+  =−

 4 2

2  1 9 1 3 1 3

z−  =− ⇒ z− =± i⇒ z = ± i

Answer	Marks
 2 4 2 2 2 2	M1

A1

Answer	Marks
[2]	1.1
1.1	Using correct quadratic formula with correct substitutions

Or completing the square (Root of negative number must be

seen)

Final answer

Answer	Marks	Guidance
(b)	(i)	B1

B1

Answer	Marks
[3]	1.1

1.1

Answer	Marks
2.2a	Correct line Im

(𝑧𝑧) = 1

Circle centre 2

Circle intersect on imaginary axis at ±i

Answer	Marks	Guidance
(ii)	Major segment (area below line and inside
circle) shaded.	B1
[1]	1.1	ft their circle and their horizontal line from (ii)
3	(c)	(i)

2 2

3 3  3  3

z−2 = − ± i =  −  +  ± 

2 2  2  2

9  10 

= < = 5

 

Answer	Marks
2  2 	M1

A1

Answer	Marks
[2]	2.1
2.2a	ft For calculating modulus of their for at least one of

their values of z.

Answer	Marks
𝑧𝑧−2

AG. Must see clear statement of inequality (eg as shown or

by values). Both roots

If shown that , must explain that

follows from 2 .

Answer	Marks	Guidance
𝑧𝑧−2	< 5	𝑧𝑧−2
(ii)	1 3 1 3  3

− ibecause Im − i =− <1

Answer	Marks	Guidance
2 2 2 2  2	B1
[1]	2.2a

1 3 5 1 3 5 2

Or − i = < − i−2i = ft their roots as long as

2 2 2 2 2 2

their roots are conjugate pairs with imaginary part of

magnitude greater than one.

Could be explained in terms of loci (ie “because (the point

representing) is closer to O than it is to (the point

1 3

representing) 2i”) but must be consistent with their diagram

2−2i

after (d).

Or “below the perpendicular bisector” or “below axis”

isw after acceptable answer

Answer	Marks
(d)	M1

A1

Answer	Marks
[2]	3.1a
1.1	For a complex conjugate pair.

Approximate positions inside their circle above and below the

line

Question 3:
3 | (a) | DR
2± −36 1 3
z = ⇒ z = ± i
4 2 2
or
( )  1 9
2 z2 −z =−5⇒2z2 −z+  =−
 4 2
2
 1 9 1 3 1 3
z−  =− ⇒ z− =± i⇒ z = ± i
 2 4 2 2 2 2 | M1
A1
[2] | 1.1
1.1 | Using correct quadratic formula with correct substitutions
Or completing the square (Root of negative number must be
seen)
Final answer
(b) | (i) | B1
B1
B1
[3] | 1.1
1.1
2.2a | Correct line Im
(𝑧𝑧) = 1
Circle centre 2
Circle intersect on imaginary axis at ±i
(ii) | Major segment (area below line and inside
circle) shaded. | B1
[1] | 1.1 | ft their circle and their horizontal line from (ii)
3 | (c) | (i) | DR
2 2
3 3  3  3
z−2 = − ± i =  −  +  ± 
2 2  2  2
9  10 
= < = 5
 
2  2  | M1
A1
[2] | 2.1
2.2a | ft For calculating modulus of their for at least one of
their values of z.
|𝑧𝑧−2|
AG. Must see clear statement of inequality (eg as shown or
by values). Both roots
If shown that , must explain that
follows from 2 .
|𝑧𝑧−2| < 5 |𝑧𝑧−2| < √5
(ii) | 1 3 1 3  3
− ibecause Im − i =− <1
2 2 2 2  2 | B1
[1] | 2.2a | |𝑧𝑧−2| ≥ 0
1 3 5 1 3 5 2
Or − i = < − i−2i = ft their roots as long as
2 2 2 2 2 2
their roots are conjugate pairs with imaginary part of
magnitude greater than one.
Could be explained in terms of loci (ie “because (the point
representing) is closer to O than it is to (the point
1 3
representing) 2i”) but must be consistent with their diagram
2−2i
after (d).
Or “below the perpendicular bisector” or “below axis”
isw after acceptable answer
(d) | M1
A1
[2] | 3.1a
1.1 | For a complex conjugate pair.
Approximate positions inside their circle above and below the
line

Show LaTeX source

3 In this question you must show detailed reasoning.
\begin{enumerate}[label=(\alph*)]
\item Find the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$.

The loci $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$ are given by $| z | = | z - 2 \mathrm { i } |$ and $| z - 2 | = \sqrt { 5 }$ respectively.
\item \begin{enumerate}[label=(\roman*)]
\item Sketch on a single Argand diagram the loci $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$, showing any intercepts with the imaginary axis.
\item Indicate, by shading on your Argand diagram, the region

$$\{ z : | z | \leqslant | z - 2 i | \} \cap \{ z : | z - 2 | \leqslant \sqrt { 5 } \} .$$
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Show that both of the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$ satisfy $| z - 2 | < \sqrt { 5 }$.
\item State, with a reason, which root of the equation $2 z ^ { 2 } - 2 z + 5 = 0$ satisfies $| z | < | z - 2 i |$.
\end{enumerate}\item On the same Argand diagram as part (b), indicate the positions of the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Pure Core 1 2022 Q3 [11]}}

This paper (9 questions)

View full paper

Q1 6 Q2 9 Q3 11 Q4 4 Q5 11 Q6 6 Q7 10 Q8 13 Q9 5