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 -sided polygons inscribed and circumscribed in a circle, and the perimeter of these as tends to infinity, to make an approximation for .
Let represent the perimeter of any -sided regular polygon inscribed in a circle of radius 1 unit.
Consider an equilateral triangle ABC of side length, units, circumscribed about a circle of radius 1 unit and centre O as shown in the following diagram.
Let represent the perimeter of any -sided regular polygon circumscribed about a circle of radius 1 unit.
Consider an equilateral triangle ABC of side length, 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 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 units.
Consider a square of side length, 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.
Find the perimeter of a regular hexagon, of side length, units, inscribed in a circle of radius 1 unit.
Show that .
Use an appropriate Maclaurin series expansion to find and interpret this result geometrically.
Show that .
By writing in the form , find .
Use the results from part (d) and part (f) to determine an inequality for the value of in terms of .
The inequality found in part (h) can be used to determine lower and upper bound approximations for the value of .
Determine the least value for such that the lower bound and upper bound approximations are both within 0.005 of .
Markscheme
METHOD 1
consider right-angled triangle OCX where CX
M1A1
A1
AG
METHOD 2
eg use of the cosine rule M1A1
A1
AG
Note: Accept use of sine rule.
[3 marks]
where = side of square M1
A1
A1
[3 marks]
6 equilateral triangles ⇒ = 1 A1
A1
[2 marks]
in right-angled triangle M1
A1
M1
AG
[3 marks]
consider
use of M1
(A1)
A1
A1
as polygon becomes a circle of radius 1 and R1
[5 marks]
consider an -sided polygon of side length
2 right-angled triangles with angle at centre M1A1
opposite side M1A1
Perimeter AG
[4 marks]
consider
R1
attempt to use L’Hopital’s rule M1
A1A1
A1
[5 marks]
M1
A1
[2 marks]
attempt to find the lower bound and upper bound approximations within 0.005 of (M1)
= 46 A2
[3 marks]