(c) they did not play competitive sport, given that they had undertaken paid work? (1 mark) Question 3 (5 marks) The universal set is the set of integers from 1 to 20, is the set of numbers divisible by 3, is the set of square numbers and is the set of prime numbers. (a) Describe the set (1 mark) (b) Determine (2 marks) (c) Determine (1 mark) (d) A number is selected at random from the universal set, determine (1 mark) Question 4 (5 marks) Expand and simplify (3 marks)
END OF TEST QUESTIONS Summary table of syllabus content assessed by Task 7 (Test 3) Question Syllabus reference 1 1.3.8 1.3.9 1.3.13 use everyday occurrences to illustrate set descriptions and representations of events and set operations review probability as a measure of ‘the likelihood of occurrence’ of an event understand the notion of a conditional probability and recognise and use language that indicates conditionality 2 1.3.9 1.3.13 review probability as a measure of ‘the likelihood of occurrence’ of an event understand the notion of a conditional probability and recognise and use language that indicates conditionality 3 1.3.7 1.3.14 use set language and notation for events use the notation ... 4 1.3.3 1.3.5 expand for small positive integers use Pascal’s triangle and its properties 5 1.1.24 1.1.27 use function notation ... examine dilations and the graphs of and 6 1.1.4 1.2.3 1.2.8 recognise features of the graph of , including its linear nature, its intercepts and its slope or gradient examine the relationship between the angle of inclination of a line and the gradient of that line recognise the exact values of ... 7 1.3.4 recognise the numbers as binomial coefficients (as coefficients in the expansion of ) 8 1.3.1 1.3.2 understand the notion of a combination as a set of objects taken from a set of distinct objects use the notation and the formula for the number of combinations of objects taken from a set of distinct object 9 1.3.16 establish and use the formula for independence of events and , and recognise the symmetry of independence 10 1.3.12 1.3.14 use relative frequencies obtained from data as estimates of probabilities use the notation and the formula 11 1.3.14 use the notation and the formula
Tan θ = -1 θ = 135o Section Two: Calculator-assumed (25 marks) Question 7 (4 marks) Solution Marks Task (S/C) (a) 4860 (b) (a) (b) Behaviours: Selects the correct coefficient from the expanded expression on a calculator Recognises combinations as coefficients of expansion Allows for power of the coefficients in original expression 1 1 1 S S C Question 8 (5 marks) Solution Behaviours Marks Task (S/C) (a) Uses combination notation and calculates number of combinations 1 S (b) Identifies number of ways for girls Multiplies this by number of ways for boys 1 1 S S (c) Selects number of ways these two girls can be chosen Divides by the number of ways that eight girls can be chosen 1 1 C C Question 9 (7 marks) Solution Behaviours Marks Task (S/C) (a) 0.9 x 0.7 = 0.63 so 63% Uses the multiplication principle 1 S (b) 1 – (0.1 x 0.4) = 0.96 so 96% Identifies those failing both tests Uses the fact that probability sums to 1 1 1 S S
(c) Identifies proportion of the intersection of the two sets Identifies proportion of students not passing both tests. 1 1 C C (d) So events not independent Uses correct rule to test for independence Selects correct probabilities to substitute into the rule 1 1 C C Question 10 (3 marks) Solution Behaviours Marks Task (S/C) (a) = 1 – (0.05+0.16+0.44+0.23+0.07) = 0.05 Identifies probability sums to 1 1 S (b) =0.957 Applies the conditional probability formula Evaluates correctly 1 1 C S Question 11 (6 marks) Solution Marks Task (S/C) (a) If M and K are mutually exclusive then Behaviours Applies the formula for conditional probability Calculates Uses the fact that the intersection is empty if sets are mutually exclusive 1 1 1 C C S (b) If sets are independent, then Behaviours Applies rule for independence Expands and substitutes the value for Solves the equation for 1 1 1 S C C