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Date May Specimen paper Marks available 3 Reference code SPM.3.AHL.TZ0.1
Level Additional Higher Level Paper Paper 3 Time zone Time zone 0
Command term Find Question number 1 Adapted from N/A

Question

This question asks you to investigate regular n -sided polygons inscribed and circumscribed in a circle, and the perimeter of these as n tends to infinity, to make an approximation for π .

Let P i ( n ) represent the perimeter of any n -sided regular polygon inscribed in a circle of radius 1 unit.

Consider an equilateral triangle ABC of side length, x units, circumscribed about a circle of radius 1 unit and centre O as shown in the following diagram.

Let P c ( n ) represent the perimeter of any n -sided regular polygon circumscribed about a circle of radius 1 unit.

Consider an equilateral triangle ABC of side length, x units, inscribed in a circle of radius 1 unit and centre O as shown in the following diagram.

The equilateral triangle ABC can be divided into three smaller isosceles triangles, each subtending an angle of  2 π 3  at O, as shown in the following diagram.

Using right-angled trigonometry or otherwise, show that the perimeter of the equilateral triangle ABC is equal to  3 3 units.

[3]
a.

Consider a square of side length, x units, inscribed in a circle of radius 1 unit. By dividing the inscribed square into four isosceles triangles, find the exact perimeter of the inscribed square.

 

[3]
b.

Find the perimeter of a regular hexagon, of side length, x units, inscribed in a circle of radius 1 unit.

 

[2]
c.

Show that  P i ( n ) = 2 n sin ( π n ) .

[3]
d.

Use an appropriate Maclaurin series expansion to find  lim n P i ( n ) and interpret this result geometrically.

[5]
e.

Show that  P c ( n ) = 2 n tan ( π n ) .

[4]
f.

By writing  P c ( n )  in the form  2 tan ( π n ) 1 n , find  lim n P c ( n ) .

[5]
g.

Use the results from part (d) and part (f) to determine an inequality for the value of π in terms of n .

[2]
h.

The inequality found in part (h) can be used to determine lower and upper bound approximations for the value of π .

Determine the least value for n such that the lower bound and upper bound approximations are both within 0.005 of π .

[3]
i.

Markscheme

METHOD 1

consider right-angled triangle OCX where CX  = x 2

sin π 3 = x 2 1        M1A1

x 2 = 3 2 x = 3       A1

P i = 3 × x = 3 3       AG

 

METHOD 2

eg  use of the cosine rule  x 2 = 1 2 + 1 2 2 ( 1 ) ( 1 ) cos 2 π 3           M1A1    

x = 3       A1

P i = 3 × x = 3 3       AG

Note: Accept use of sine rule.

 

[3 marks]

a.

sin π 4 = 1 x where x  = side of square      M1

x = 2        A1

P i = 4 2        A1

[3 marks]

b.

6 equilateral triangles ⇒ x = 1       A1

P i = 6       A1

[2 marks]

c.

in right-angled triangle  sin ( π n ) = x 2 1      M1

x = 2 sin ( π n )      A1

P i = n × x

P i = n × 2 sin ( π n )      M1

P i = 2 n sin ( π n )      AG

[3 marks]

d.

consider  lim n 2 n sin ( π n )

use of  sin x = x x 3 3 ! + x 5 5 !       M1

2 n sin ( π n ) = 2 n ( π n π 3 6 n 3 + π 5 120 n 5 )       (A1)

= 2 ( π π 3 6 n 2 + π 5 120 n 4 )       A1

lim n 2 n sin ( π n ) = 2 π      A1

as  n polygon becomes a circle of radius 1 and  P i = 2 π      R1

[5 marks]

e.

consider an n -sided polygon of side length x

2 n right-angled triangles with angle  2 π 2 n = π n  at centre       M1A1

opposite side  x 2 = tan ( π n ) x = 2 tan ( π n )        M1A1

Perimeter  P c = 2 n tan ( π n )        AG

[4 marks]

f.

consider  lim n 2 n tan ( π n ) = lim n ( 2 tan ( π n ) 1 n )

= lim n ( 2 tan ( π n ) 1 n ) = 0 0          R1

attempt to use L’Hopital’s rule        M1

= lim n ( 2 π n 2 se c 2 ( π n ) 1 n 2 )        A1A1

= 2 π        A1

[5 marks]

g.

P i < 2 π < P c

2 n sin ( π n ) < 2 π < 2 n tan ( π n )        M1

n sin ( π n ) < π < n tan ( π n )        A1

[2 marks]

h.

attempt to find the lower bound and upper bound approximations within 0.005 of π     (M1)

n = 46        A2

[3 marks]

i.

Examiners report

[N/A]
a.
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b.
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c.
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d.
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e.
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f.
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g.
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h.
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i.

Syllabus sections

Topic 5 —Calculus » AHL 5.13—Limits and L’Hopitals
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Topic 5 —Calculus » AHL 5.19—Maclaurin series
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