polar qns past papers

POLAR COORDINATES EXAM QUESTIONS FROM ‘OLD’ PAPERS

1. The curve C has polar equation r = 3a cos ,

π2

π2 .

The curve D has polar equation r = a(1 + cos ), < .

Given that a is a positive constant,

(a) sketch, on the same diagram, the graphs of C and D, indicating where each curve cuts the initial line.

(4)The graphs of C intersect at the pole O and at the points P and Q. (b) Find the polar coordinates of P and Q.

(3)(c) Use integration to find the exact value of the area enclosed by the curve D and the lines

= 0 and =

π3 .

(7)The region R contains all points which lie outside D and inside C.

Given that the value of the smaller area enclosed by the curve C and the line =

π3 is

3 a2

16 (2 33),(d) show that the area of R is a2.

(4)[P4 January 2002 Qn 8]

2. Figure 1

The curve C shown in Fig. 1 has polar equation

r = a(3 + 5 cos ), < .

(a) Find the polar coordinates of the points P and Q where the tangents to C are parallel to the initial line.

(6)

The curve C represents the perimeter of the surface of a swimming pool. The direct distance from P to Q is 20 m.

(b) Calculate the value of a. (3)

(c) Find the area of the surface of the pool. (6)

[P4 June 2002 Qn 8]

FP2 exam questions from past papers – Polar Coordinates

3. Figure 1

Figure 1 shows a sketch of the cardioid C with equation r = a(1 + cos ), < . Also shown are the tangents to C that are parallel and perpendicular to the initial line. These tangents form a rectangle WXYZ.

(a) Find the area of the finite region, shaded in Fig. 1, bounded by the curve C. (6)

(b) Find the polar coordinates of the points A and B where WZ touches the curve C. (5)

(c) Hence find the length of WX.(2)

Given that the length of WZ is

3√3a2 ,

(d) find the area of the rectangle WXYZ.(1)

A heart-shape is modelled by the cardioid C, where a = 10 cm. The heart shape is cut from the rectangular card WXYZ, shown in Fig. 1.

(e) Find a numerical value for the area of card wasted in making this heart shape. (2)

[P4 January 2003 Qn 8]

4. Figure 1

D

O B A Initial line

C

A logo is designed which consists of two overlapping closed curves.

The polar equations of these curves are

r = a(3 + 2cos ) and r = a(5 – 2 cos ), 0 < 2.

Figure 1 is a sketch (not to scale) of these two curves.


(a) Write down the polar coordinates of the points A and B where the curves meet the initial line.

(2)(b) Find the polar coordinates of the points C and D where the two curves meet.

(4)(c) Show that the area of the overlapping region, which is shaded in the figure, is

a2

3 (49 483).

(8)[P4 June 2003 Qn 7]

5. (a) Sketch the curve with polar equation

r = 3 cos 2, –

π4 <

π4 .

(2)

(b) Find the area of the smaller finite region enclosed between the curve and the half-line

=

π6 .

(6)(c) Find the exact distance between the two tangents which are parallel to the initial line.

(8)[P4 January 2004 Qn 7]

6. Figure 1

Figure 1 is a sketch of the two curves C1 and C2 with polar equations

C1 : r = 3a(1 cos ),

and C2 : r = a(1 + cos ), .

The curves meet at the pole O, and at the points A and B.

(a) Find, in terms of a, the polar coordinates of the points A and B.(4)

(b) Show that the length of the line AB is

3√ 32 a.

(2)


The region inside C2 and outside C1 is shown shaded in Fig. 1.

(c) Find, in terms of a, the area of this region.(7)

A badge is designed which has the shape of the shaded region.

Given that the length of the line AB is 4.5 cm,

(d) calculate the area of this badge, giving your answer to three significant figures.(3)

[P4 June 2004 Qn 8]

7. The curve C has polar equation r = 6 cos , –

π2 <

π2 ,

and the line D has polar equation r = 3 sec ( π

3−θ)

, –

π6 <

5 π6 .

(a) Find a cartesian equation of C and a cartesian equation of D.(5)

(b) Sketch on the same diagram the graphs of C and D, indicating where each cuts the initial line.

(3)The graphs of C and D intersect at the points P and Q.

(c) Find the polar coordinates of P and Q. (5)

[FP1/P4 January 2005 Qn 7]

8. Figure 1

The curve C which passes through O has polar equation

r = 4a(1 + cos ), – < .

The line l has polar equation

r = 3a sec , –

π2 < <

π2 .

The line l cuts C at the points P and Q, as shown in Figure 1.(a) Prove that PQ = 63a.


C

lP

O = 0

R

initial lineO

C

m

(6)The region R, shown shaded in Figure 1, is bounded by l and C.

(b) Use calculus to find the exact area of R. (7)

[FP1/P4 June 2005 Qn8]

9. Figure 1

A curve C has polar equation r2 = a2 cos 2, 0

π4 . The line l is parallel to the initial

line, and l is the tangent to C at the point P, as shown in Figure 1.

(a) (i) Show that, for any point on C, r2 sin2 can be expressed in terms of sin and a only.

(1)

(ii) Hence, using differentiation, show that the polar coordinates of P are ( a√ 2

,π6 )

.(6)

The shaded region R, shown in Figure 1, is bounded by C, the line l and the half-line with

equation =

π2 .

(b) Show that the area of R is

a2

16 (33 – 4). (8)

[FP1/P4 January 2006 Qn 7]

10. Figure 1


=

π2

Figure 1 shows a curve C with polar equation r = 4a cos 2, 0

π4 , and a line m with

polar equation =

π8 . The shaded region, shown in Figure 1, is bounded by C and m. Use

calculus to show that the area of the shaded region is 12 a2( – 2).

(7)[FP1 June 2006 Qn 2]

11. Figure 2

Figure 2 shows a sketch of the curve C with polar equation

The tangent to C at the point P is perpendicular to the initial line.

(a) Show that P has polar coordinates (6)

The point Q on C has polar coordinates (√ 2 ,

π4 )

.

The shaded region R is bounded by OP, OQ and C, as shown in Figure 2.

(b) Show that the area of R is given by

∫π6

π4 (sin2 2θ cos2 θ+ 1

2−1

2cos4θ)

d.(3)

(c) Hence, or otherwise, find the area of R, giving your answer in the form , where a and b are rational numbers.

(4)[FP1 January 2007 Qn 8]


12. (a) Sketch the curve C with polar equationr = 5 + 3 cos , 0 < 2.

(2)

(b) Find the polar coordinates of the points where the tangents to C are parallel to the initial line = 0. Give your answers to 3 significant figures where appropriate.

(6)

(c) Using integration, find the area enclosed by the curve C, giving your answer in terms of .

(6)[FP1 June 2007 Qn 7]

13. Figure 1

Figure 1 shows the curve C1 which has polar equation r = a(3 + 2 cos ), 0 < 2, and

the circle C2 with equation r = 4a, 0 < 2, where a is a positive constant.

(a) Find, in terms of a, the polar coordinates of the points where the curve C1 meets the

circle C2 .(4)

The regions enclosed by the curves C1 and C2 overlap and this common region R is shaded in the figure.

(b) Find, in terms of a, an exact expression for the area of the shaded region R.(8)

(c) In a single diagram, copy the two curves in Figure 1 and also sketch the curve C3 with polar equation r = 2a cos , 0 < 2. Show clearly the coordinates of the points of

intersection of C1 , C2 and C3 with the initial line, = 0. (3)

[FP1 January 2008 Qn 8]


14.

Figure 1The curve C shown in Figure 1 has polar equation

r = 4(1 – cos θ), 0 θ

π2 .

At the point P on C, the tangent to C is parallel to the line θ =

π2 .

(a) Show that P has polar coordinates (2 ,

π3 )

.(5)

The curve C meets the line θ =

π2 at the point A. The tangent to C at P meets the initial line at

the point N. The finite region R, shown shaded in Figure 1, is bounded by the initial line, the

line θ =

π2 , the arc AP of C and the line PN.

(b) Calculate the exact area of R.(8)

[FP1 June 2008 Qn 8]


polar qns past papers

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