THIS PAPER IS WORTH 60% OF THE TOTAL SUBJECT MARK
Questions: 7 Students must attempt no more than 6 questions Marks per question: 10 Exam weighting: 60%.
Instructions to Students:
- Complete all of the details required on the front page of the examination booklet. - Make sure that you note the SIX(6) questions attempted on the front of your examination booklet - If Seven questions are attempted, the first SIX(6) will be marked. - You may use a non-programmable calculator - Graph paper will be provided. - Answers are to be written in ink. Pencils are permitted for graphing purposes. - The candidate may retain this paper.
a) A loan is to be repaid by a student. The student has debts of $10,000 to be paid at the end of the ﬁrst year, $5,000 to be paid in 18 months and $3,000 to be paid in the 24th month. The student would prefer to pay the debts as follows. $1,000 now, followed by payments at the end of the 6th, 20th and 30th month. The payment at the end of the 6th month is half the size of the payment at the end of the 20th and 30th months. Find the value of the ﬁnal repayment (using a focal date of the 30th month) if interest compounds monthly at 8%. (5 marks) b) i) For a car loan being repaid with fortnightly installments of $75, ﬁnd the original loan size if the term of the loan is 5 years, interest is calculated daily at 9%. ii) If the holder of the loan wished to pay out the loan at the end of the 3rd year, how much would be outstanding? iii) In this case (loan paid out in 3 years) what is the total ﬁnancial fee for this loan? (5 marks)
a) A ﬁrm has three categories of employees: juniors, seniors and supervisors. They earn $12, $18 and $24 per hour respectively. They have an output of 10, 16 and 22 units of product per hour respectively. The 250 employees of the plant have a total wage bill of $4,248 per hour and an hourly output of 3748 units. i) Place this into matrix form (Ax = b) to ﬁnd the number of each worker employed by the ﬁrm. ii) Solve using the adjoint method for the number of each type of worker employed by the ﬁrm. (6 marks) b) A ﬁrm uses three machines to process its work. Machine A is the most heavily used machine, machine B is used twice as often as machine C and half as often as machine A. If machine A produces a faultless product on 80% of occasions, machine B produces a faultless product on 70% of occasions and machine C loses 10% of its products to early faults, and of the product that remains, 90% is faultless. Find the probability that a faulty product has come from machine B. (4 marks)
a) A, B, C, and X are invertible, 3 × 3 Matrices such that:
C −1 AX −1 B −1 = AB −1 Find an expression for X. All steps must be shown. (1 mark) b) (i) Is this linear system solvable? (Use Matrices to say why(not))
2x + 4y + 8z = 34 2x + 7y + 4z = 12 x + 2y + 4z = 17
(1) (2) (3)
(ii) What are two features of a system of equations that would give it a determinant of zero. (4 marks) (c) Solve this system of linear equations using the adjoint method: −3x + 2y + z = 1 4x − 2z = 6 5x + 3y + 6z = 25 (5 marks)
(4) (5) (6)
The transport department promotes the use of buses for attending an event at a new showground. It allows the transport department to raise revenue. A maximum of 170,000 people are expected to arrive at the showground on a given day. At least 36,000 will use private transport. The number of buses that can arrive in an hour is 60, and each can carry up to 70 persons. The number of trains that can arrive in an hour is 20, each carrying up to 500 persons. Buses and trains arrive at the showgrounds for 10 hours per day. The number arriving on the bus is at least 25% of those arriving by train. Bus tickets cost $6 and train tickets cost $4. The transport...