OCR Further Pure Core 2 2022 June — Question 2 5 marks

Exam BoardOCR
ModuleFurther Pure Core 2 (Further Pure Core 2)
Year2022
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPolar coordinates
TypePolar curve intersection points
DifficultyStandard +0.3 This is a straightforward Further Maths polar coordinates question requiring basic substitution to find intersections (setting θ=0 and equating the two curves) and sketching a simple spiral. While it's a Further Maths topic, the actual mathematical operations are routine with no problem-solving insight needed, making it slightly easier than average overall but typical for introductory polar work.
Spec4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)
2 Two polar curves, \(C _ { 1 }\) and \(C _ { 2 }\), are defined by \(C _ { 1 } : r = 2 \theta\) and \(C _ { 2 } : r = \theta + 1\) where \(0 \leqslant \theta \leqslant 2 \pi\). \(C _ { 1 }\) intersects the initial line at two points, the pole and the point \(A\).
  1. Write down the polar coordinates of \(A\).
  2. Determine the polar coordinates of the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\). The diagram below shows a sketch of \(C _ { 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{007f07ee-cb29-4a97-93d9-2328079c4aea-2_681_1353_1318_244}
  3. On the copy of this sketch in the Printed Answer Booklet, sketch \(C _ { 2 }\).
Question 2:
AnswerMarks Guidance
2(a) At A  = 2 so A[..., 2]
r = 2 = 22 = 4 so A[4, ...]B1
B1
AnswerMarks
[2]2.2a
1.1or just  = 2 . ISW
or just r = 4 . ISW
AnswerMarks
(b)At PoI 2 =  + 1
=> = 1 so [2, 1]M1
A1
AnswerMarks
[2]1.1
1.1Correct condition for PoI
or r = 2 and  = 1. ISW
AnswerMarks Guidance
(c)B1
[1]1.1 C drawn as a smooth curve
2
spiralling out from [1, 0] outside
C until a single point in the 1st
1
quadrant and then inside C .
1
AnswerMarks
Must stop on initial line.Start at [1, 0].
Intersection in 1st quadrant (by eye).
r increasing (by eye).
Reaches initial line and stops.
Ignore labels.
AnswerMarks Guidance
Alternative method:B1 Can be implied by 2nd B1.
u=−x
3
u = − − x
AnswerMarks
2B1
3
x = − − u
AnswerMarks Guidance
2M1 Rearrange and substitute into
original equation
3 2
 3   3   3 
4 − − u + 6 − − u − 3 − − u + 9 = 0
2 2 2
2 7 2 7 2 7
− − 2 7 u − 1 8 u 2 − 4 u 3 + + 1 8 u + 6 u 2 + + 3 u = 0
2 2 2
AnswerMarks
8u3+24u2+12u−27=02 7 2 7 2 7
− − 2 7 u − 1 8 u 2 − 4 u 3 + + 1 8 u + 6 u 2 + + 3 u = 0
AnswerMarks
2 2 2Award A1 only for coefficients of u 3
and one other correct in any form
Award A2 only for coefficients of u 3
and two others correct in any form
A3 for fully correct equation in the
AnswerMarks Guidance
8u3+24u2+12u−27=0A3 correct form.
[6]
B1
Can be implied by 2nd B1.
M1
Rearrange and substitute into
original equation
Award A1 only for coefficients of u 3
and one other correct in any form
Award A2 only for coefficients of u 3
and two others correct in any form
Question 2:
2 | (a) | At A  = 2 so A[..., 2]
r = 2 = 22 = 4 so A[4, ...] | B1
B1
[2] | 2.2a
1.1 | or just  = 2 . ISW
or just r = 4 . ISW
(b) | At PoI 2 =  + 1
=> = 1 so [2, 1] | M1
A1
[2] | 1.1
1.1 | Correct condition for PoI
or r = 2 and  = 1. ISW
(c) | B1
[1] | 1.1 | C drawn as a smooth curve
2
spiralling out from [1, 0] outside
C until a single point in the 1st
1
quadrant and then inside C .
1
Must stop on initial line. | Start at [1, 0].
Intersection in 1st quadrant (by eye).
r increasing (by eye).
Reaches initial line and stops.
Ignore labels.
Alternative method: | B1 | Can be implied by 2nd B1.
u=−x
3
u = − − x
2 | B1
3
x = − − u
2 | M1 | Rearrange and substitute into
original equation
3 2
 3   3   3 
4 − − u + 6 − − u − 3 − − u + 9 = 0
2 2 2
2 7 2 7 2 7
− − 2 7 u − 1 8 u 2 − 4 u 3 + + 1 8 u + 6 u 2 + + 3 u = 0
2 2 2
8u3+24u2+12u−27=0 | 2 7 2 7 2 7
− − 2 7 u − 1 8 u 2 − 4 u 3 + + 1 8 u + 6 u 2 + + 3 u = 0
2 2 2 | Award A1 only for coefficients of u 3
and one other correct in any form
Award A2 only for coefficients of u 3
and two others correct in any form
A3 for fully correct equation in the
8u3+24u2+12u−27=0 | A3 | correct form.
[6]
B1
Can be implied by 2nd B1.
M1
Rearrange and substitute into
original equation
Award A1 only for coefficients of u 3
and one other correct in any form
Award A2 only for coefficients of u 3
and two others correct in any form
2 Two polar curves, $C _ { 1 }$ and $C _ { 2 }$, are defined by $C _ { 1 } : r = 2 \theta$ and $C _ { 2 } : r = \theta + 1$ where $0 \leqslant \theta \leqslant 2 \pi$. $C _ { 1 }$ intersects the initial line at two points, the pole and the point $A$.
\begin{enumerate}[label=(\alph*)]
\item Write down the polar coordinates of $A$.
\item Determine the polar coordinates of the point of intersection of $C _ { 1 }$ and $C _ { 2 }$.

The diagram below shows a sketch of $C _ { 1 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{007f07ee-cb29-4a97-93d9-2328079c4aea-2_681_1353_1318_244}
\item On the copy of this sketch in the Printed Answer Booklet, sketch $C _ { 2 }$.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Pure Core 2 2022 Q2 [5]}}
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