1.[3, 3 and 3 = 9 marks]
Consider the function
ii)Determine the domain for which the function is increasing.
iii)Given that determine the values of x for which the function is concave downwards.
2.[1, 3 and 4 = 8 marks]
Consider the functions and
ii)Determine and state its domain and range
iii)Find, algebraically, the values of x for which
3.(3, 2, 2 and 2 = 9 marks)
Consider the system of equations shown;
a)Reduce this system of equations algebraically or using an augmented matrix.
b)Determine the value(s) of ‘k’ for which the system of equations has
ii)An infinite number of solutions
c)Determine the unique solution if k = 4.
4.(3, 3, 1, 3 and 3 = 13 marks)
A company makes two types of bags, the shoulder bag and the backpack. Each day the company produces at least 60 bags, with a maximum of 50 backpacks. To keep up with demands the company must ensure that the number of backpacks produced is at least the same as or more than the number of shoulder bags produced. Let x be the number of shoulder bags produced each day.
Let y be the number of backpacks produced each day.
a)Assuming x0 and y0, list the other three constraints on x and y as stated in the problem.
b)The graph displays the constraints listed in a).
Each backpack requires one tag and each shoulder bag requires two tags. To avoid an oversupply the company must use at least 80 tags each day. Write down this constraint and draw it on the graph above.
c)Shade the feasible region.
d)Each shoulder bag costs the company $12 to produce and each backpack costs the company $10 to produce. Determine the number of each type of bag the company should produce each day in order to minimise costs.
e)Production costs of the backpack rises. How much does the cost of each backpack need to increase by so that there is a definite change in the number of backpacks and/or shoulder bags produced each day.
5.(4, 1 and 3 = 8 marks)
Dave is concerned about pollution and is testing for the presence of dirt particles in the air. It is known that 15% of the air samples about to be tested were ‘dangerously dirty’ as shown in the tree diagram below.
1 - x
1 - y
Data shows that a positive test result given a sample is ‘Not Dangerous’ is 10 times more likely than a negative test result given that a sample is deemed ‘Dangerously Dirty’. It is also known that 21.6% of all samples tested give a positive result.
a)Find the values of x and y from the tree diagram.
i)Find the probability (to 4 d. p.) that Dave tested dangerously dirty and showed a positive result to the test.
ii)Given a positive result to the test, what is the probability (to 4 d. p.) that Dave tested dangerously dirty air?
a)Y is a discrete random variable with the following probability distribution function:
Find the value of k.
7.(2, 2, 3 and 3 = 10 marks)
A second hand car sales manager realizes that 6% of the vehicles for sale in his yards are defective in some minor way. He is prepared to fix all defects but only if the customer returns with a problem. Assume all customers with a problem return for assistance and that X, representing the number of vehicles returned for repairs per month, is a binomial random variable. a) Determine the probability that if 21 cars are sold during the month: i) none will be returned.
ii) no more than three will be returned.