S1 Practice Paper “The Lower School”
One of a series of S1 practice papers written by DJS in 1995 with an active class (names included in questions). Sophie scored zero in coursework and still managed an A. Duncan went to Oxford to do International Law. Dan has a namesake in the department (no relation).
1. Masses of one of the classes in Year 7 were collected to use in data presentation. Here is a version of the data:
Class (kg) 20 ≤ x <40 40 ≤ x <45 45 ≤ x <50 50 ≤ x <60 60 ≤ x <70 70 ≤ x <80
Frequency 6 3 5 6 2 1
(i) Find the median mass [1]
(ii) By assuming that the data is evenly spread across each interval, calculate the inter-quartile range [3]
(iii) Draw a histogram for the data [4]
(iv) The class calculates the skew, midrange and outliers. Comment on their use within this data. [3]
2. Tim’s Gameboy has been confiscated and he wants it back as fast as possible, even to the point where he is contemplating cracking the combination to the lock on the relevant cupboard. The lock uses a numeric keypad with 14 keys, 0-9, X,Y,Z and C. He has observed just six key depressions when watching the lock be operated and is keen to establish better information.
i) How many different selections are possible from the information so far? [2]
ii) In practice, all users of this type of lock start with C, the cancel key, so in effect only 5 depressions matter. How many of the previous answer can be discarded? [2]
iii) Furthermore, the machinery inside is much simpler than you might expect and cannot tolerate repeated characters. What is the current number of different selections? [3]
iv) Having digested all of this, Tim overhears an arts-subject teacher commenting that they should have used more letters in the combination. How many ways now? [3]
v) Amazingly enough, the order of characters is immaterial with these locks. Rework the last calculation with this information. [2]
vi) Tim wonders whether his best strategy is to suggest that the code is already known so that it will be changed. If he assumes that when the combination is changed, all the letters will be used, suggest his next course of action [3]
3. The Year 8s and 9s, 120 pupils in all, have lost the habit of bringing maths equipment to lessons [weapons of maths construction]. I reckon that, on a typical day in 1995 two thirds will have brought calculators, half compasses, half protractors. Use tables to find the probability that a class of twenty has fewer than 10 pairs of compasses between them. [3]
Three quarters of those with protractors will also have compasses, probably because they’re in the geometry set. Typically only three out of twenty have all three items of kit and no-one ever has just a protractor. I notice that the numbers of pupils with exactly two out of the three items forms an arithmetic series. Find the probability that a child picked at random on a typical day has with them:
(i) no calculator (ii) only a calculator (iii) calculator and protractor [1,2,2]
(iv) no equipment (v) a protractor, given that they have a calculator [2,2]
4. In the boarding house, the boys are playing with dice, trying out a version of ‘craps”. Two dice are thrown and the scores added. A turn is ended by throwing either a pair of ones or an eleven.
i) Show that the probability of ending a turn at any throw is 1/12 [2]
ii) What is the most likely score on any one throw? [2]
iii) What is the probability that Daniel’s turn will be exactly ten throws long? [2]
iv) Show, by using the result 1+ x + x2 + x3 + x4….. = (1-x)-1 that a turn of ten throws or longer will have a probability of 0.457 [4]
v) Chris thinks Daniel’s turn is too long, that the dice are not behaving properly. How long would Dan’s turn have to be before you decided that the dice were somehow biased?
[Hint: logs] [4]
[60]
