Cambridge International Mathematics

Investigation and modelling questions (Chapter 34) 669

1 June 1990, Paper 4

A gardener has 357 tulip bulbs to plant.

a If she planted a rectangle of 15 rows, with 23 bulbs in each row, how many bulbs would be left

over?

b How many bulbs would there be in the largest square that she could plant?

ciIf she plantsxrows, withybulbs in each row, write down a formula for the number of bulbs

left over.

ii If 10 <x< 20 and y> 20 , find the value ofxand the value ofysuch that no bulbs are

left over.

2

t midnight noon 1300 1400

d 5 5 7 8 : 46

a

b Check that the formula is correct by substituting the fourth piece of information.

c Find the depth of water at 10 am.

d What is the greatest depth of water in the harbour?

e At what times of day is the depth of water greatest?

f What is the least depth of water in the harbour?

3

a In a chemical reaction, the massMgrams of a chemical is given by the formula M= 160a¡t

whereais a constant integer andtis the time (in minutes) after the start. A table of values fort

andMis given below.

t 0 1 2 3 4 5 6 7

M 160 80 40 20 q 5 r 1 : 25

i Find the value ofa.

ii Find the values ofqandr.

iii Sketch the graph ofMagainstt.

iv Draw an accurate graph and add to it a tangent at t=2. Estimate the rate of change in the

mass after 2 minutes.

b The other chemical in the same reaction has massmgrams where m= 160¡M:

i On the same graph as ina iii, sketch the graph ofmagainstt.

ii For what value oftdo the chemicals have equal mass?

iii State a single transformation which would give the graph formfrom the graph forM.

B MODELLING QUESTIONS

Using the first three pieces of information in the table, deduce the values of ,aband .c

The depth of water (dmetres) in a harbour is given by the formula d=a+bsin(ct)o wherea,band

care constants, andtis the time in hours after midnight. It is known that bothbandcare non-zero

and 20 <c< 35.

The following table gives depths at particular times:

Adapted from Nov 1994, Paper 4

IGCSE01

cyan magenta yellow black

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100
Y:\HAESE\IGCSE01\IG01_34\669IGCSE01_34.CDR Friday, 14 November 2008 12:01:18 PM PETER