OCR MEI Further Pure Core 2019 June — Question 7 8 marks

Exam BoardOCR MEI
ModuleFurther Pure Core (Further Pure Core)
Year2019
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPolar coordinates
TypeShow polar curve has Cartesian form
DifficultyStandard +0.3 This is a straightforward Further Maths polar coordinates question with standard conversions (x²+y²=r², xy=r²sinθcosθ), routine sketching of a lemniscate, and direct application of the polar area formula ½∫r²dθ. While Further Maths content, it requires only mechanical substitution and standard techniques without novel insight.
Spec4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve
7 A curve has cartesian equation \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2 c ^ { 2 } x y\), where \(c\) is a positive constant.
  1. Show that the polar equation of the curve is \(r ^ { 2 } = c ^ { 2 } \sin 2 \theta\).
  2. Sketch the curves \(r = c \sqrt { \sin 2 \theta }\) and \(r = - c \sqrt { \sin 2 \theta }\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  3. Find the area of the region enclosed by one of the loops in part (b). Section B (110 marks)
    Answer all the questions.
Question 7:
AnswerMarks Guidance
7(a) (x2y2)22c2xy  (r2)2 2c2rcosrsin
 r2 2c2cossinc2sin2*M1
A1
AnswerMarks
[2]1.1b
2.2asubstituting for r2, x and y
NB AG
AnswerMarks Guidance
7(b) B1
B1*
B1dep
AnswerMarks
[3]1.1b
1.1b
AnswerMarks
2.5one loop shown
both shown (no extras)
AnswerMarks
ve r with broken lineallow sep diags
in correct quadrant
dep B1*
AnswerMarks Guidance
7(c)
1
A2 c2sin2d
0 2
 1 2
  c2cos2
 
 4 
0
1
 c2
AnswerMarks
2B1
B1
B1cao
AnswerMarks
[3]1.1a
1.1b
AnswerMarks
1.1bcondone missing d
1
sin2d cos2
AnswerMarks
2limits soi 
Question 7:
7 | (a) | (x2y2)22c2xy  (r2)2 2c2rcosrsin
 r2 2c2cossinc2sin2* | M1
A1
[2] | 1.1b
2.2a | substituting for r2, x and y
NB AG
7 | (b) | B1
B1*
B1dep
[3] | 1.1b
1.1b
2.5 | one loop shown
both shown (no extras)
ve r with broken line | allow sep diags
in correct quadrant
dep B1*
7 | (c) | 
1
A2 c2sin2d
0 2

 1 2
  c2cos2
 
 4 
0
1
 c2
2 | B1
B1
B1cao
[3] | 1.1a
1.1b
1.1b | condone missing d
1
sin2d cos2
2 | limits soi
7 A curve has cartesian equation $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2 c ^ { 2 } x y$, where $c$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Show that the polar equation of the curve is $r ^ { 2 } = c ^ { 2 } \sin 2 \theta$.
\item Sketch the curves $r = c \sqrt { \sin 2 \theta }$ and $r = - c \sqrt { \sin 2 \theta }$ for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.
\item Find the area of the region enclosed by one of the loops in part (b).

Section B (110 marks)\\
Answer all the questions.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Pure Core 2019 Q7 [8]}}
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