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Cork Institute of Technology
Higher Certificate in Engineering in Electronic Engineering-Award
(NFQ-Level 6)
Summer 2007
Mathematics
(Time: 3 Hours)
Instructions Answer FIVE questions, at least ONE question from each Section. All questions carry equal marks.
Examiners: Mr. D. Denieffe Dr. R O Dubhghaill Ms. M. Harley
Section A
Q1a Given that
A = 4 5
and B = (^32) - -^12 -^01 ļ£·ļ£· ļ£ø
determine (i) AB (ii) (3 A + 2 BT^ ) T (5 marks)
Q1b Rewrite the determinant 8 1 2
in the form
k
and hence evaluate the determinant (4 marks)
Q1c Find the inverse of 1 2 1
1 1 3
2 3 2
ļ£·
ļ£·
ļ£· ļ£ø
ļ£¶ ļ£¬
ļ£¬
ļ£¬ ļ£
ļ£«
ā
ā A and confirm your answer.
Use the inverse to solve the equations
2 1
3 8
2 3 2 7
1 2 3
1 2 3
1 2 3
ā + = ā
v v v
v v v
v v v
Use Cramers Rule to confirm the answer for v 2 (11 marks) Q2a 3 numbers form an arithmetic progression. Determine the numbers given that their sum is 42 and their product is 2618 (5 marks) Q2b The 6th term of a geometric series is 8 times the 3rd^. If the sum of the 7th^ and 8 th^ terms is 192, determine (i) the common ratio (ii) the 1st^ term (iii) the sum of the 5 th^ to 11th^ terms inclusive. (5 marks)
Q2c Use the binomial theorem to find the first three terms of a series for
(i) 1 + x
(^1) (ii) 1 2
1
(iii) (^2) 1 x
x
and state the values of x for which each series is valid. Use the series to evaluate
- 5
(^1) and0. 0 1 2
x
ā« + x
Comment on the relative accuracy of these answers compared to the accurate values of 0.8165 and 0. (10 marks)
Section B
Q3a The table gives the frequency distribution of anticipated annual salaries in thousands of ā¬ of a sample of 72 electronics students Anticipated Salary (ā¬1000) No. of students 10 but less than 20 2 20 but less than 22.5 8 22.5 but less than 25 26 25 but less than 30 17 30 but less than 35 15 35 but less than 50 4
(i) Using an assumed mean of 27.5 calculate the mean x and standard deviation s of the sample. (ii) Establish a cumulative frequency table and hence plot the cumulative frequency curve. (iii) Use your graph to estimate the number of students with expected salaries in
the range x Ā± s.
(16 marks) Q3b Two students are chosen at random. Determine the probability that (i) both would expect a salary less than ā¬25, (ii) one would expect less than ā¬25,000 and the other would expect at least ā¬35,
(4 marks)
Q4a The average percentage of employees absent from a firm each day is 3% A section in the firm has 8 employees. Determine the probability that for this section (i) all employees will be at work (ii) at most two employees will be absent on any given day (iii) 50 % of the section will be at work (6 marks)
Q6a Determine each of the following integrals: (i) 4 x^^3 dx x
ā« (ii)^2
x (^) dx x x
(iii) 5
ā« 1 ā 4 xdx
(10 marks)
Q6b Determine the area enclosed between the graph of y = 12 te ā0.5 t
and the ordinates t = 0 and t = 2. (5 marks)
Q6c Determine the mean of the function (^12) 16 + 9 x
in the interval [0, 13 ]
(5 marks)
Q7a The current i A in a circuit is given by i = 0.25^20 dq t dt =^ e^ ā where^ q^ is the charge at time
t Ī¼s State the time constant. Derive an expression for the charge given that the initial
charge is zero. Draw a rough sketch to show how q varies with time. Label axes appropriately (6 marks)
Q7b Solve the differential equations (i) y ā²( t )= 25 + 4 t^2 (ii) y ā²( t )= 25 ā 4 y^2 given that y (0) = 1.25 in each case. (8 marks)
Q7c The rate of cooling of an object is given by^ ddt Īø^ = ā0.15 Īø where ĪøĀ°C is the temperature at time t s. If the temperature is 100Ā°C initially, determine when the temperature reaches 50 Ā°C (6 marks)
