Cambridge International Mathematics

666 Investigation and modelling questions (Chapter 34)

13 Nov 1997, Paper 4

A “Pythagorean triple” is a set of threewholenumbers that could be the lengths of the three sides of a

right-angled triangle.

a Show that f 5 , 12 , 13 g is a Pythagorean triple.

b

c The largest number in a Pythagorean triple isxand one of the other numbers isx¡ 2.

i If the third number isy, show that y=

p

4 x¡ 4 :

ii If x=50, find the other two numbers in the triple.

iii If x= 101, find the other two numbers in the triple.

iv

14

a Show that

i 212 £ 123 =2^12 +1^23 ii iii

b

c

d Write down a generalisation of what you have discovered and prove it algebraically.

15

Maria thinks of 3 possible savings schemes for her baby son.

Scheme A: save$10on his 1 st birthday,$20on his 2 nd birthday,$30on his 3 rd birthday,

$40on his 4 th birthday, ......

Scheme B: save$1on his 1 st birthday,$2on his 2 nd birthday,$4on his 3 rd birthday,

$8on his 4 th birthday, ......

Scheme C: save$1on his 1 st birthday,$4on his 2 nd birthday,$9on his 3 rd birthday,

$16on his 4 th birthday, ......

She puts these ideas in a table. Scheme/Birthday 1 st 2 nd 3 rd 4 th

A $10 $20 $30 $40

B $1 $2 $4 $8

C $1 $4 $9 $16

a Write down, for each of the Schemes A, B and C, the amount to be saved on

i his 7 th birthday ii hisnth birthday.

b The formulae for thetotalamount saved up to and including hisnth birthday are as follows.

6

i Foreachof the schemes A, B and C, find the total amount saved up to and including his 10 th

birthday.

ii Which scheme gives the smallest total amount of savings up to and including his 18 th birthday?

iii Find the birthday when the scheme you have selected inbiifirst gives the smallest total

amount of savings.

Two of the numbers in a Pythagorean triple are 24 and 25. Find the third number.

From your observations inb, find another statement like those inawhich is true.

Scheme A: total=$5n(n+1)

Scheme B: total= $(2n¡1)

Scheme C: total=$

n(n+ 1)(2n+1)

Find two other Pythagorean triples in the form fy,x¡ 2 ,xg, where x< 40.

Remember that all three numbers must be whole numbers.

134 £ 213 =1^34 +2^13215 £ 156 =2^15 +1^56

Write 212 and 123 in the form 1+xy and repeat for 134 and 213.

Adapted from June 1991, Paper 4

Adapted from June 1997, Paper 4

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Y:\HAESE\IGCSE01\IG01_34\666IGCSE01_34.CDR Friday, 14 November 2008 12:00:16 PM PETER