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Cork Institute of Technology
Higher Certificate in Engineering in Electronic Engineering-Award
(National Certificate in Engineering in Electronic Engineering-Award)
Autumn 2005
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
=^1 -^13
A B C =
determine the following products (i) AB (ii) BA (iii) ABC Hence state AT^ BT (6 marks)
Q1b (i) Find the inverse of 1 3 4
A and confirm your answer.
(6 marks) (ii) Use the inverse the solve the equations 3 x + 5 y - 2 z = - 2 x - 3 z = 0
- x + 3 y + 4 z = - 1 (4 marks) (iii) Use Cramers Rule to confirm your answer for z (4 marks)
Q2a Find the sum of the first 20 terms of the series 3 + 8 + 13 + …… and determine the smallest number of terms for which the sum of the terms is greater than 3000. (6 marks)
Q2b Use the binomial theorem to find the first four terms of a series for (^) ( 1 +^1 x ) 2.
Hence obtain a series for (^) ( 1 − 12 x ) 2. State the values of x , for which each series is
valid. Use the series to evaluate (^0).^16 2. Compare this to the accurate value for (^0).^162 and comment. (7 marks)
Q2c State the Maclaurin series and use it to determine the first four terms of the series expansion for f ( x )= ex. Use this series to obtain series for (i) g ( x )= ex^ − e − x (ii) h ( x )= 1 − e − x as far as the term in x^3.
Use this series to evaluate ∫
0
(^1) dx x
e x
(7 marks)
Section B
Q3a Students were asked about how much money was spent on social activities per month. Students who spent less than €200 are detailed below Money (€) No. of Students 0 but less than 30 2 30 but less than 60 8 60 but less than 100 26 100 but less than 150 60 150 but less than 170 24 170 but less than 190 16 190 but less than 200 4
(i) Use an assumed mean of €160 to calculate the mean money spent, x , and
standard deviation, s (ii) Establish a cumulative frequency table and hence plot the cumulative frequency ogive.
(iii) Use your graph to estimate the number of students in the range x ± s
(14 marks) Q3b If two students are chosen at random determine the probability that (i) both spent less than € (ii) one spent less than €60 and one more than €
(iii) both spent within the range x ± s
(6 marks)
Q6a Determine each of the following integrals:
(i) (^) ∫ (^) ( 2 +^1 3 x ) dx (ii) (^) ∫ 5 xe −^0.^4 xdx (iii) (^) ∫ ( 1 − x ) 2 x − x^2 dx
(8 marks)
Q6b Find the mean value of (^9) +^10 4 x 2 in the interval [0, 1.5]
(5 marks)
Q6c Determine where the graphs of the functions y 1 = 3 - 2 x and y 2 = x^2 − 5 x − 7 intersect and hence the area between the graphs. (7 marks)
Q7a A graph of a function passes through the point (0, 8). Determine the equation of the function if the slope at any point ( x , y ) is given by (i) –2 x (ii) –2 y (7 marks)
Q7b An object is dropped from a point A and t s later, its acceleration is given by 10 e −^0.^5^ t ms-2^. Find an expression for its velocity v at this time t. Determine the velocity of the object and its distance from A when t = 10. (7 marks)
Q7c A tank of height 3m is initially full of liquid. It is then emptied and t s later
when the depth of liquid in the tank is h m it can be shown that h^2 dtdh =− 0. 125 h Determine the time taken to empty the tank i.e determine t when h = 0. (6 marks)
