Challenging +1.2 This is a Further Maths polar area question requiring the standard formula A = ½∫r²dθ with appropriate limits. While it involves integration of cos²(4θ) requiring double-angle formulas and determining the correct limits for a rose curve (k = 1/8 for one petal, or recognizing the full curve), these are standard techniques for FM students. The 'show detailed reasoning' requirement and multi-step nature elevate it above average, but it's a textbook application of polar area methods without novel insight.
3 In this question you must show detailed reasoning.
The diagram below shows the curve \(r = 2 \cos 4 \theta\) for \(- k \pi \leq \theta \leq k \pi\) where \(k\) is a constant to be determined.
Calculate the exact area enclosed by the curve.
Must show f(1(cid:83))(cid:16)f((cid:16)1(cid:83))
Answer
Marks
8 8
Must be seen
Must be seen
A0 for decimal answer
3
n
e
m
i
c
e
p
S
Question 3:
3 | DR
r(cid:32)0(cid:159)k (cid:32)1
8
1(cid:83) 1(cid:83)
A(cid:32) 1(cid:179)8 r2 d(cid:84)(cid:32) 1(cid:179)8 (cid:11)2cos4(cid:84)(cid:12)2dp(cid:84)
2 (cid:16)1(cid:83) 2 (cid:16)1(cid:83)
8 8
1(cid:83)
(cid:32)2(cid:179)8 cos24(cid:84)d(cid:84) S
(cid:16)1(cid:83)
8
1(cid:83)
1(cid:83) (cid:170) sin8(cid:84)(cid:186)8
(cid:32)(cid:179)8 (cid:11)1(cid:14)cos8(cid:84)(cid:12)d(cid:84)(cid:32) (cid:84)(cid:14)
(cid:171) (cid:187)
(cid:16)1(cid:83) (cid:172) 8 (cid:188)
8 (cid:16)1(cid:83)
8
(cid:167) sin(cid:83)(cid:183) (cid:167) sin((cid:16)(cid:83))(cid:183)
(cid:32) 1(cid:83)(cid:14) (cid:16) (cid:16)1(cid:83)(cid:14) (cid:32) 1(cid:83)
(cid:168) (cid:184) (cid:168) (cid:184)
(cid:169) 8 8 (cid:185) (cid:169) 8 8 (cid:185) 4 | eB1
M1
A1
A1
M1
A1
[6] | i
c
2.2a
3.1a
1.1
1.1
3.1a
1.1 | Limits not required
Correct form to be integrated
Correct indefinite integral
Using correct limits
Must show f(1(cid:83))(cid:16)f((cid:16)1(cid:83))
8 8 | Must be seen
Must be seen
A0 for decimal answer
3
n
e
m
i
c
e
p
S
3 In this question you must show detailed reasoning.
The diagram below shows the curve $r = 2 \cos 4 \theta$ for $- k \pi \leq \theta \leq k \pi$ where $k$ is a constant to be determined.
Calculate the exact area enclosed by the curve.
\hfill \mbox{\textit{OCR Further Pure Core 1 Q3 [6]}}