Cambridge International Mathematics

h

H

S

T

O

R

r

ABC

656 Multi-Topic Questions (Chapter 33)

ii thetotalsurface area ofonepencil, giving your answer correct to 3 significant figures.

(The curved surface area,A, of a cone of radiusrandslantheightlis given by A=¼rl.)

18

An equilateral 16 -sided figure APA^0 QB..... is formed

when the square ABCD is rotated 45 oclockwise about

its centre to position A^0 B^0 C^0 D^0 .AB=12cm and

AP=xcm.

aiUse triangle PA^0 Q to explain why

2 x^2 = (12¡ 2 x)^2.

ii Show that this simplifies to

x^2 ¡ 24 x+72=0:

iii Solvex^2 ¡ 24 x+72 = 0. Give your answers

correct to 2 decimal places.

biCalculate the perimeter of the 16 -sided figure.

ii Calculate the area of the 16 -sided figure.

19

A cone, ATB, and a section of a sphere, ASB, share

the same circular base, centre C, radiusr. The height,

TC, of the cone ishand STOC is a straight line. The

radius, OB, of the sphere isRand the height, CS, of

the section of the sphere isH.

a r=6cm, h=14cm and R=10cm.

i Calculate the volume of the cone ABT.

(The volume of a cone with base radiusr

and heighthis^13 ¼r^2 h.)

ii Show that the height, SC, of the section of the sphere is 18 cm.

iii Calculate the volume of the section of the sphere ASB.

(The volume of a section of a sphere, radiusR, heightHis^13 ¼H^2 (3R¡H):)

iv Find the percentage of the volume of the section of the spherenotoccupied by the cone.

b In a different sphere section, R=3cm, h=2rcm and TS=1cm.

i Write down the height, SC, in terms ofrand show that OC=(2r¡2)cm.

ii Use Pythagoras’ theorem in triangle OCB to find OC^2 in terms ofr.

iii Use your answers to partsbiandbiito show that 5 r^2 ¡ 8 r¡5=0

iv Solve the equation 5 r^2 ¡ 8 r¡5=0.

v Write down the height of the cone.

AB

DC

A'

C'

D' B'

PQ

xcm

12 cm

.

Adapted from May 2002, Paper 4

Adapted from May 2004, Paper 4

Give your answers correct to 2 decimal places.

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Y:\HAESE\IGCSE01\IG01_33\656IGCSE01_33.CDR Friday, 14 November 2008 10:31:35 AM PETER