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Cork Institute of Technology
Higher Certificate in Engineering in Electronic Engineering-
Award
Summer 2006
Mathematics
(Time: 3 Hours)
Instructions Answer FIVE questions, at least ONE question from each Section. All questions carry equal marks.
Examiners: Mr. J. Berry Dr. R O Dubhghaill Ms. M. Harley
Section A
Q1a Given that
M = N
Determine (i) M + N (ii) ( M + NT ) (iii) MN
Hence state (i) M T^ + N (ii) N T^ M T Do not evaluate these matrices
(6 marks)
Q1b Find the inverse of
A and confirm your answer. (8 marks)
Q1c Use Cramers Rule to solve for z (to 1 decimal place) in the equations: -2 x + y + 4 z = - x - y + z = - 2 x + y - z = -1 (6 marks)
Q2a (i) Write an expression for S (^) N , the sum of N terms of the series –16 – 12 – 8 -…. Hence find the value of N for which S (^) N > 8000 OR (ii) The 5th term of a geometric series is 162 and the 10th term is –39366. Determine the first term and find the sum of the series from 162 to – inclusive. (6 marks)
Q2b Use the binomial theorem to find the first three terms of a series for 1 + x.
and state the values of x for which each series is valid.
Use the series to estimate (i) 1. 4 and (ii) 98.
Comment on the relative accuracy of the two answers. (8 marks)
Q2c State the Maclaurin series and use it to determine the first three terms of the series
expansion for f ( x )= sin( x ). Use this series to obtain a series for
y ( t ) =
t
sin( 2 t ) to the term in t 2 and hence find a linear approximation for y ′( t )
(6 marks)
Section B
Q3a An insurance company carried out an examination of a random sample of 220 of its car insurance policies. The cost of insurance in each case recorded and the findings tabulated as follows: Insurance Cost (€’s) No. of Policies less than 200 6 200 but less than 300 17 300 but less than 400 47 400 but less than 450 62 450 but less than 500 58 500 but less than 600 21 600 but less than 800 9 (i) Using an assumed mean of 425 calculate the mean and standard deviation of the distribution. (ii) Establish a cumulative frequency table and hence plot the cumulative frequency curve. (iii) Use your graph to estimate the cost of 10% most expensive policies
and determine the number of insurance policies costing in the range x ± s
(15 marks) Q3b Two polices are chosen at random. Determine the probability that (i) both polices cost €500 or more (ii) one costs €500 or more and the other costs less than € (5 marks)
Q4a An electronics supplier claims that 95% of its catalogue items are in stock at any time. Electronics Dept of CIT sends in an order for 15 different components. What is the probability that (i) all of the ordered components will be in stock (ii) 2 items will be missing (iii) less than 13 items will be in stock (7 marks)
Q6c
(6 marks)
Q7a The current i in a circuit is given by i = dqdt = 5 e^ −^ 2 10×^4 t where q is the charge. If the charge is zero initially, find an expression for q at time t. (4 marks)
Q7b Solve the differential equations (i) (^) dtdy^ = 15 − 2 t (ii) dydt^ = 15 − 2 y given that y = 3 at t = 0 in each case. (8 marks)
Q7c The velocity of a particle is given by^ dx^ 3. 25 4 x^2 dt
= − where x ( t ) is the displacement of the particle at time t. Determine the displacement if the displacement is 1.25 initially. Determine the acceleration of the particle in terms of its displacement x. (8 marks)
The diagram shows the graph of y = ( x − 1)(9 − x^2 ) Determine the total area enclosed between the graph and the x -axis.
