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Cork Institute of Technology
Higher Certificate in Engineering in Electronic Engineering-Award
(NFQ – Level 6)
Autumn 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
= 0 1 = -2^0
3 2 -1^3
A B
− ^
^
− ^
determine (i) AB (ii) ( A + B ) T (iii) A + BT Hence state B AT^ T. Do not evaluate this matrix. (7 marks)
Q1b Find the inverse of
A and confirm your answer.
(7 marks)
Q1c Use Cramers Rule to solve for b in the equations: 2 a – 3 b - 4 c = 12 -3 a + b + 2 c = - 5 a – 4 b - c = 1 (6 marks)
Q2a A company lays cable 180m long. Estimate the cost of laying the cable if the cost is €50 for the first meter with an increase of €0.75 per meter thereafter (4 marks) Q2b A drilling machine is to have 4 speeds ranging from 100 rpm to 800rpm. Determine the speeds if they form a geometric progression (2 marks)
Q2c Write out the first three terms of the Binomial Series expansion of (^3)
+ x
and
state the values of x for which the series is valid. Use this series to (i) evaluate (^3)
1 (ii) express
( 1 )^3
x
x
− as a power series as far as the term in x 3
(7 marks)
Q2d State the Maclaurin series and use it to determine the first three terms of the series
expansion for f ( x )= cos( x ). Use this series to obtain a series for
g ( x ) = x cos( x ) to the term in x^3. Hence evaluate
0
∫ x^ cos( ) x dx.^ (7 marks)
Section B
Q3a The resistance in kΩ of a sample of 140 resistors was measured and the results are shown below Resistance (kΩ) No. of Resistors 88 but less than 92 6 92 but less than 96 19 94 but less than 98 21 98 but less than 100 29 100 but less than 102 25 102 but less than 106 24 106 but less than 112 16
(i) Using an assumed mean of 99 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 median and the number of resistors in the range
x ± s
(15 marks) Q3b Two resistors are chosen at random. Determine the probability that (i) both are ≥ 100 kΩ (ii) one is ≤ 96 kΩ and the other ≥ 102 kΩ (5 marks)
Q6a Determine each of the following integrals:
(i) (^14) (3 2 )
dx
∫ − x
(ii) ∫ 10 xe −^4 x^2 dx (iii) ∫3 sin(4 ) x x dx
(8 marks)
Q6b Find the mean and rms value of (^12) 1 4
y t
in the interval [0, 0.25]
(6 marks)
Q6c The diagram shows the graph of y = ( x − 4)( x^2 −4)
Determine the total area enclosed between the graph and the x -axis.
(6 marks)
Q7a Solve the differential equation ( ) 4 25 10^6
t x t e −
′ −^ ×
= given that x (0) = 0.
(6 marks)
Q7b Solve the differential equations
(i) dvdt^ = 1 − 4 t (ii) v
dt
dv = 1 − 4 given that v = 0 at t = 0 in each case.
(8 marks)
Q7c An object moves such that its distance s (metres) from a fixed point is given by
2
+ s =
dt
d s
Solve this equation subject to the initial conditions s = 4m and = 15
dt
ds ms- 1 when t = 0.
Evaluate the displacement and the velocity when t =^ π 5
(6 marks)
