Polar Coordinates (Edexcel A Level Further Maths: Core Pure)

Exam Questions

41 mins4 questions
1
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9 marks

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The curve C shown in Figure 1 has polar equation

r space equals space 4 space plus space cos space 2 theta      0 less or equal than theta less or equal than straight pi over 2

At the point A on C, the value of r is 9 over 2

The point N lies on the initial line and A N is perpendicular to the initial line.

The finite region R comma shown shaded in Figure 1, is bounded by the curve C, the initial line and the line A N.

Find the exact area of the shaded region R, giving your answer in the form  p pi plus q square root of 3 where space p and q are rational numbers to be found.

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2a
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6 marks
(a)
(i)
Show on an Argand diagram the locus of points given by the values of z satisfying

vertical line z space minus space 4 space minus space 3 straight i vertical line space equals space 5

Taking the initial line as the positive real axis with the pole at the origin and given that


      theta element of open square brackets alpha comma space alpha plus straight pi close square brackets comma spacewhere alpha equals negative arctan space open parentheses 4 over 3 close parentheses comma

(ii)
show that this locus of points can be represented by the polar curve with equation

r space equals space 8 space cos theta space plus space 6 space sin theta

2b
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7 marks

The set of points A is defined by

A equals open curly brackets z space colon space 0 space less or equal than space arg space z space less or equal than straight pi over 3 close curly brackets intersection open curly brackets z space colon space open vertical bar z minus 4 minus 3 straight i close vertical bar less or equal than space 5 close curly brackets

(b)
(i)
Show, by shading on your Argand diagram, the set of points A.

(ii)
Find the exact area of the region defined by A, giving your answer in simplest form.

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3
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9 marks

qp3-9fm0_01-ms-june-2020

Figure 1 shows a sketch of two curves C subscript 1 and C subscript 2 with polar equations

C subscript 1 space colon space r space equals space left parenthesis 1 space plus space sin space theta right parenthesis space space space space space space 0 space less or equal than space theta space less than space 2 pi

C subscript 2 colon space r equals 3 left parenthesis 1 minus sin space theta right parenthesis space space space space space space space space 0 space less or equal than space theta space less than space 2 pi

The region R lies inside C subscript 1 and outside C subscript 2 and is shown shaded in Figure 1.

Show that the area of R is

space p square root of 3 space minus space q pi

wherespace p and q are integers to be determined.

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4a
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2 marks

qp3a

Figure 1 shows the design for a table top in the shape of a rectangle A B C D. The length of the table, A B, is 1.2 m. The area inside the closed curve is made of glass and the surrounding area, shown shaded in Figure 1, is made of wood.

The perimeter of the glass is modelled by the curve with polar equation

r space equals space 0.4 space plus space a space cos space 2 theta space space space space space 0 less or equal than theta less than 2 straight pi

where a is a constant.

a)
Show that a equals 0.2
4b
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8 marks

Hence, given that A D = 60 cm,

(b)
find the area of the wooden part of the table top, giving your answer in m2 to 3 significant figures.

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