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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.