Further Mathematics · OCR H245 · Topic 4.09

Polar coordinates

Polar coordinates, sketching curves, area.

  • 3 sub-topics
  • Taught in Year 13
  • 39 practice questions
  • 3 animated lessons
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What the specification says

Every sub-topic OCR can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

  1. 4.09aPolar coordinatesY13
  2. 4.09bSketching curvesY13
  3. 4.09cAreaY13

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 39 questions in this topic, free, no account needed.

Polar coordinates

Not needed for Year 12Easy

Polar to Cartesian · Year 13 · 2 marks

  1. Find the Cartesian equation of r=6cos⁡θr = 6\cos\theta.
    (2)
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  1. Use x=rcos⁡θx = r \cos \theta, y=rsin⁡θy = r \sin \theta.Multiply by rr: r2=2r^{2} = 2a rcos⁡θr \cos \theta.
  2. (x−3)2+y2=9(x - 3)^2 + y^2 = 9

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Polar coordinates

Not needed for Year 12Medium

Sketching polar curves · Year 13 · 3 marks

r=6+5cos⁡θr = 6 + 5\cos\theta
  1. Describe the shape and give the greatest and least values of rr.
    (3)
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  1. Compare the constant with the cosine coefficient.It is a dimpled curve.
  2. rmax⁡r_{\max}=11 at θ{}= 11 \text{ at } \theta=0{}= 0
  3. rmin⁡r_{\min}=1 at θ{}= 1 \text{ at } \theta=π{}= \pi

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Polar coordinates

Not needed for Year 12Hard

Area enclosed by a polar curve · Year 13 · 4 marks

r=3(1+cos⁡θ)r = 3(1 + \cos\theta)
  1. Find the exact area enclosed by the curve.
    (4)
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  1. A = ½∫r² dθ.12∫02π9(1+cos⁡θ)2 dθ\tfrac{1}{2}\int_0^{2\pi} 9(1 + \cos\theta)^2\,\mathrm{d}\theta
  2. Double angle for cos2⁡\cos^{2}.(1+cos⁡θ)2(1 + \cos\theta)^2=1{}= 1+2cos⁡θ{}+ 2\cos\theta+12(1+cos⁡2θ){}+ \tfrac{1}{2}(1 + \cos 2\theta)
  3. 92[32θ+2sin⁡θ+14sin⁡2θ]02π\tfrac{9}{2}\left[\tfrac{3}{2}\theta + 2\sin\theta + \tfrac{1}{4}\sin 2\theta\right]_0^{2\pi}
  4. 272π\tfrac{27}{2}\pi

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