OCR Further Pure Core 1 2024 June — Question 2 10 marks

Exam BoardOCR
ModuleFurther Pure Core 1 (Further Pure Core 1)
Year2024
SessionJune
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex Numbers Argand & Loci
TypeRegion shading with multiple inequalities
DifficultyStandard +0.3 This is a straightforward Further Maths question on argand diagrams requiring students to shade a region defined by argument inequalities, check point membership by calculation, and write set notation for a circle. While it's Further Maths content (inherently harder), the tasks are routine: shading between two rays from a point, calculating an argument to compare with bounds, and applying the standard circle inequality |z-a|<r. No novel insight or complex multi-step reasoning required.
Spec4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines4.02p Set notation: for loci
2 The locus \(C _ { 1 }\) is defined by \(C _ { 1 } = \left\{ z : 0 \leqslant \arg ( z + i ) \leqslant \frac { 1 } { 4 } \pi \right\}\).
  1. Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing \(C _ { 1 }\).
  2. Determine whether the complex number \(1.2 + 0.8\) is is \(C _ { 1 }\). The locus \(C _ { 2 }\) is the set of complex numbers represented by the interior of the circle with radius 2 and centre 3 . The locus \(C _ { 2 }\) is illustrated on the Argand diagram below. \includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-2_698_920_1009_239}
  3. Use set notation to define \(C _ { 2 }\).
  4. Determine whether the complex number \(1.2 + 0.8\) is in \(C _ { 2 }\).
Question 2:
AnswerMarks Guidance
2(a) M1
A1
AnswerMarks
[2]1.1
1.1Half-line starting at one of ( − 1 , 0 ) , ( 1 , 0 )
( 0 , 1 ) , or ( 0 , − 1 ) at an angle of approx. 
4
to the positive horizontal. This mark can be
awarded if the line is shown dashed rather
than solid. This mark can be implied by
shading that begins/ends where this line is
meant to be even if the line is not explicitly
shown.
Solid half-line starting at ( 0 , − 1 ) passing
through(1,0)with region between half-line
and y=−1 shaded. Neither line needs to be
shown if the shading alone exactly defines
the correct region. Condone dashed line for
y=−1. If shading the outside, then must
label the inside as C .
1
AnswerMarks
(b) 1 .8 
a r g (1 .2 + 0 .8 i + i ) = a r c t a n
1 .2
0 .9 8 ... =  so no, 1 .2 + 0 .8 i is not in C .
AnswerMarks
4 1M1
A1
AnswerMarks
[2]1.1
2.2aCorrect calculation for arg(1.2 + 1.8i) –
allow tan=1.8 (oe) for M1.
1.2
Correct justification and conclusion.
Evaluation of arctan must be given to at
least 2 d.p. rot (0.982793…). For reference:
0 . 7 8 ( 5 3 9 8 . . . ) .  = ‘No’ is sufficient as a
4
conclusion. Award A1 for both values
(0.98… and 0.78…) together with correct
conclusion but if not evaluated must see
4
comparison with 0.98… e.g. 0.98...
4
Alternative method
AnswerMarks Guidance
Cartesian equation of half-line is y = x − 1 so when x = 1 .2 , y = ...M1 Substitutes x=1.2 into y=x−1
0.20.8 so no, 1 .2 + 0 .8 i is not in C .
AnswerMarks Guidance
1A1 Correct justification and conclusion. Must
see comparison of 0.2 and 0.8 for this mark.
SC B2 For stating that 1.8 > 1.2 or 1.5 > 1
and ‘no’ cwo.
[2]
AnswerMarks
(c) 
z: z−3 2M1
A1
AnswerMarks Guidance
[2]1.1
2.5For 𝑧−3
z. Allow use of m o d ( z − 3 ) for both marks.
Set notation and inequality must be correct.
Allow any letter for z.
 
Condone z : z − 3 2  2 2
   
SC1 for z : z + 3  2 or z : z − 3  4
AnswerMarks
(d)( )
1 .2 + 0 .8 i − 3 = ( − 1 .8 ) 2 + 0 .8 2 = 3 .2 4 + 0 .6 4 = 3 .8 8
3.88 2,so yes, 1 .2 + 0 .8 i is in C .
AnswerMarks
2M1
A1
AnswerMarks Guidance
[2]1.1
2.2aCalculating 1.2 + 0.8i ± 3
32 correctly for their 3 (which must be real)
from part (c). Or for calculating
 ( 3 − ( 1 . 2 + 0 . 8 i ) ) .
Comparing modulus with 2 (or modulus
squared with 4) and concluding that 1.2 +
0.8i is contained in C . Allow just √3.88 <
2
2 as a comparison. For reference
3 .8 8 = 1 .9 6 ( 9 7 7 ...) . Allow 3 . 8 8 2 .
‘Yes’ is sufficient as a conclusion. Allow
other correct surds for comparison e.g. 97.
5
Question 2:
2 | (a) | M1
A1
[2] | 1.1
1.1 | Half-line starting at one of ( − 1 , 0 ) , ( 1 , 0 )
( 0 , 1 ) , or ( 0 , − 1 ) at an angle of approx. 
4
to the positive horizontal. This mark can be
awarded if the line is shown dashed rather
than solid. This mark can be implied by
shading that begins/ends where this line is
meant to be even if the line is not explicitly
shown.
Solid half-line starting at ( 0 , − 1 ) passing
through(1,0)with region between half-line
and y=−1 shaded. Neither line needs to be
shown if the shading alone exactly defines
the correct region. Condone dashed line for
y=−1. If shading the outside, then must
label the inside as C .
1
(b) |  1 .8 
a r g (1 .2 + 0 .8 i + i ) = a r c t a n
1 .2

0 .9 8 ... =  so no, 1 .2 + 0 .8 i is not in C .
4 1 | M1
A1
[2] | 1.1
2.2a | Correct calculation for arg(1.2 + 1.8i) –
allow tan=1.8 (oe) for M1.
1.2
Correct justification and conclusion.
Evaluation of arctan must be given to at
least 2 d.p. rot (0.982793…). For reference:
0 . 7 8 ( 5 3 9 8 . . . ) .  = ‘No’ is sufficient as a
4
conclusion. Award A1 for both values
(0.98… and 0.78…) together with correct
conclusion but if not evaluated must see
4

comparison with 0.98… e.g. 0.98...
4
Alternative method
Cartesian equation of half-line is y = x − 1 so when x = 1 .2 , y = ... | M1 | Substitutes x=1.2 into y=x−1
0.20.8 so no, 1 .2 + 0 .8 i is not in C .
1 | A1 | Correct justification and conclusion. Must
see comparison of 0.2 and 0.8 for this mark.
SC B2 For stating that 1.8 > 1.2 or 1.5 > 1
and ‘no’ cwo.
[2]
(c) |  
z: z−3 2 | M1
A1
[2] | 1.1
2.5 | For|𝑧−3| and 2 seen. Allow any letter for
z. Allow use of m o d ( z − 3 ) for both marks.
Set notation and inequality must be correct.
Allow any letter for z.
 
Condone z : z − 3 2  2 2
   
SC1 for z : z + 3  2 or z : z − 3  4
(d) | ( )
1 .2 + 0 .8 i − 3 = ( − 1 .8 ) 2 + 0 .8 2 = 3 .2 4 + 0 .6 4 = 3 .8 8
3.88 2,so yes, 1 .2 + 0 .8 i is in C .
2 | M1
A1
[2] | 1.1
2.2a | Calculating |1.2 + 0.8i ± 3| or |1.2 +0.8i ±
3|2 correctly for their 3 (which must be real)
from part (c). Or for calculating
 ( 3 − ( 1 . 2 + 0 . 8 i ) ) .
Comparing modulus with 2 (or modulus
squared with 4) and concluding that 1.2 +
0.8i is contained in C . Allow just √3.88 <
2
2 as a comparison. For reference
3 .8 8 = 1 .9 6 ( 9 7 7 ...) . Allow 3 . 8 8 2 .
‘Yes’ is sufficient as a conclusion. Allow
other correct surds for comparison e.g. 97.
5
2 The locus $C _ { 1 }$ is defined by $C _ { 1 } = \left\{ z : 0 \leqslant \arg ( z + i ) \leqslant \frac { 1 } { 4 } \pi \right\}$.
\begin{enumerate}[label=(\alph*)]
\item Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing $C _ { 1 }$.
\item Determine whether the complex number $1.2 + 0.8$ is is $C _ { 1 }$.

The locus $C _ { 2 }$ is the set of complex numbers represented by the interior of the circle with radius 2 and centre 3 . The locus $C _ { 2 }$ is illustrated on the Argand diagram below.\\
\includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-2_698_920_1009_239}
\item Use set notation to define $C _ { 2 }$.
\item Determine whether the complex number $1.2 + 0.8$ is in $C _ { 2 }$.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Pure Core 1 2024 Q2 [10]}}
This paper (12 questions)
View full paper
Q1 3 Q2 10 Q3 8 Q4 3 Q5 5 Q6 4 Q7 9 Q8 5 Q9 6 Q10 10 Q11 7 Q12 7