Cambridge International Mathematics

664 Investigation and modelling questions (Chapter 34)

7

a Copy and complete the

following two sets of

calculations.

1=

1+2 =

1+2+3 =

1+2+3+4 =

13 =

13 +2^3 =

13 +2^3 +3^3 =

13 +2^3 +3^3 +4^3 =

b How are the two sets of results related?

c Find the value of 13 +2^3 +3^3 +4^3 +5^3 +6^3 +7^3 +8^3 +9^3.

d Given that the sum of the first 25 numbers, 1+2+3+::::::+25,is 325 , find the value of

13 +2^3 +3^3 +::::::+25^3.

e 1+2+3+4+5+::::::+n=an^2 +bn.

f Find the value of 13 +2^3 +3^3 +4^3 +::::::+ 250^3 :

8 June 1988, Paper 6

a Find a formula for each of the following:

i the number of triangles in thenth row

ii the total number of triangles in the firstnrows

iii the total number of white triangles in the firstnrows

iv the total number of black triangles in the firstnrows.

b Show algebraically that your answer toaiiis the sum of your answers toa iiiandiv.

9 Nov 2002, Paper 4

Sarah investigates cylindrical plant pots. The standard pot has base radiusrcm and heighthcm. Pot

A has radius 3 rand heighth. Pot B has radiusrand height 3 h. PotChas radius 3 rand height 3 h.

aiWrite down the volumes of pots A, B and C in terms of¼,randh.

ii Find in its lowest terms the ratio of the volumes of A : B : C.

iii Which one of the pots A, B or C is mathematically similar to the standard pot? Explain your

answer.

iv The surface area of the standard pot isScm^2. Write down in terms ofSthe surface area of

the similar pot.

b Sarah buys a cylindrical plant pot with radius 15 cm and height 20 cm. She wants to paint its

outside surface (base and curved surface area).

i Calculate the area she wants to paint.

ii Sarah buys a tin of paint which will cover 30 m^2. How many plant pots of this size could be

completelypainted on their outside surfaces using this tin of paint?

The diagram shows the first eight rows of a

continuing pattern of black and white

triangles.

row 1

row 2

row 3

row 4

row 5

row 6

row 7

row 8

h

r

h

3 r r

3 h

3 r

3 h

standard A

B C

Find the values ofaband ,and test your answers.

Adapted from June 1989, Paper 6

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Y:\HAESE\IGCSE01\IG01_34\664IGCSE01_34.CDR Friday, 14 November 2008 11:59:40 AM PETER