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D P S INTERNATIONAL, SAKET
CLASS A1 A, B ENRICHEMENT COURCE
GENERAL INSTRUCTIONS
EACH PART OF THE HOME MUST BE DONE AS PRESENTATION (not necessarily a power
point presentation)
THE HOME WILL BE TAKEN AS THE PART OF CONTINUOS ASSESMENT.
THE WORK MUST BE SUBMITTED WITH IN THE FIRST WEEK OF REOPENING OF
SCHOOL.
1) GENETIC ENGEENERING
Genetic Fingerprinting was developed by Professor Alec Jeffreys at the University of
Leicester in 1984.
The technique is based on the fact that each of us has a unique genetic make-up,
contained in the molecule DNA, which is inherited from our natural parents, half from
our mother and half from our father.
DNA can be extracted from cells and body fluids and analysed to produce a characteristic
pattern of bands or genetic 'fingerprint'.
The sketch below shows how genetic fingerprinting can be used to identify a child's
father.
It is usual to compare between 10 and 20 bands. Experimental evidence has shown that in
unrelated people the probability of one band
matching is one in four. (0.25)
So for example, the probability of two bands
matching
= (0.25)2
= 0.0625 or a 1 in 16 chance.
Problem 1
Find the probability of 10 bands matching. Express your answer in the
form "1 in ? chance"
Problem 2
matching.
Repeat Problem 1, but using 0.5 as the probability of any single band
You will have noticed that the answer to Problems 1 and 2 change quite dramatically if
the underlying probability changes. In fact, the value of 0.25 has been the subject of some
speculation recently in a number of criminal trials.
Problem 3
Copy and complete the table below. Comment on the values found and
suggest the number of bands which should be compared, to be confident of a match not
happening be chance, when the probability is 0.25.
Probability
Number of bands
(p)
5
0.2
15
10
20
1 in 3125 ?
?
1 in 9.5 million million
0.25
?
?
?
?
0.5
?
?
?
?
2) DIPSTICK PROBLEM
Petrol stations very rarely run out of petrol. This is due partly to efficient deliveries but
also to precise stick control.
Each type of petrol (4 star, unleaded, diesel) is stored in an underground tank and the
amount left in each tank is carefully monitored using some form of dipstick.
It is easy to measure the height, say h, left in the tank. However, the volume will be
proportional to the cross-sectional area - not the height. Suppose the cross-section is
a circle (it is in fact elliptical, but a circle is a good approximation). We will find the
relationship between area, A, and height, h, and so provide a ready reckoner to convert
height to area.
For simplicity, we will take r = 1m. For values of h from 0 to 1, we will find the angle ø
and the area of oil.
Problem 1 Show that cos ø = 1 - h.
Problem 2 Show that the area of the sector OAB is given by
Problem 3 Show that the area of the triangle OAB is
(1 - h) sin ø.
Problem 4 Deduce the area of the cross-section of oil and express this as a fraction, A´,
of the complete cross-sectional area of the tank.
Problem 5 a) Using the equation in Problem 1, find the value of ø for each value of h
in the table below.
b) Use the formula deduced in Problem 4 to find the area fractions.
h
ø°
Area fraction
0
0
0
0.1
25.84 0.019
0.2
...
...
...
...
...
...
...
...
1.0
90
0.500
Problem 6 Plot a graph of A´ (vertical axis) against height h (horizontal axis).
Problem 7 Use your graph to estimate the height that corresponds to an area fraction of
a) 0.05
b) 0.10.
3) BAR CODES
Most grocery products include an identifying Bar Code on their wrappers and many
supermarkets now use these bar codes for totalling sales at the checkout, using a light pen
to read the code.
Problem 1
What advantages are there for the grocery trade in
using
bar
code
technology?
Are there disadvantages?
The UPC (Universal Product Code) was introduced in America in 1973 and adapted to
form EAN (European Article Code) in 1974. There are two versions of EAN - 13 digit
and 8 digit, but we will deal with the 8 digit version. An example is shown below.
This version is used by stores such as Sainsbury or Boots to
code their own label products.
The Number is divided into three parts
00
retailers'
code
34600
product
code
9
check
digit
The check digit is chosen so that
3 x (1st + 3rd + 5th+ 7th number) + (2nd + 4th + 6th + 8th number)
is exactly divisble by 10.
Problem 2
Do the following 8 digit EAN codes have the correct
check digit?
a) 00034548
Problem 3
b) 00396349
c) 50168622
Find the check digit, x, for the following 8 digit EAN
codes
a) 0008639x
b) 5021421x
c) 0042655x
Another 8 digit EAN is shown
opposite. It has left and right hand
guard bars and centre bars. In
between there are 8 bars of varying
thickness.
Each
number
is
represented by a unique set of 2
bars and 2 spaces. As can be seen
in the magnified version of 5, each
number code is made up of 7
modules.
We write 5 as 0110001 to indicate
whether a module is blank (0) or
black (1).
All left hand numbers start with 0 and end with 1, and only use a total of 3 or 5 black
modules. Right hand numbers are the complement of the corresponding left hand code
e.g. right hand 5 = 1001110.
Design all possible left hand codes using these rules
and use the examples on this worksheet to identify
the code for each number.
________________________________________________________________________
Problem 4
4) Make a power point presentation on Music and Mathematics.
Guidelines:Music has many mathematical elements in it: rhythm, pitch, scale, frequency, interval,
and ratio.
Find out about the notes of music.
Take different examples.
Find the sequence in the notes.
Relate to mathematics.
(Your class has already been divided into groups. You can use the same groups and do
the presentation. This presentation will be taken as part of your math activity. Do not
change the topic)
Finally work from the curriculum.
Finish miscellaneous exercises
of all the chapters done in the class
for P1 and S1.
Submit within in the first week
after school reopens
GET ASSESSED
ADVICE
DIVIDE YOUR WORK EVENLY THROUGHTOUT THE HOLIDAYS SO
THAT YOU ENJOY THE DAYS AS WELL.