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Cork Institute of Technology
Bachelor of Science (Honours) in Software Development & Computer Networking - Stage 1
(NFQ – Level 8)
Summer 2006
Mathematics
(Time: 3 Hours)
Instructions Answer FIVE questions, at least ONE question from each Section. All questions carry equal marks.
Examiners: Dr. J. Buckley Dr. A. Kinsella Ms. M. Harley
Section A
Q1a Express the following symbolically where the universe is Z the set of integers. (i) There is no even prime number. (ii) Some square numbers are odd (iii) Not all the odd factors of sixty are prime. (5 Marks) Q1b Express the following in English and state its truth-value ∃ x [( even x ( ) ∧ m 5( ) x ∧ square x ( )]where m 5( x ) represents x is a multiple of 5 Negate ∃ x [( even x ( ) ∧ m 5( ) x ∧ square x ( )] Simplify your answer, express it in English and state its truth-value. (5 Marks) Q1c Determine the validity of the argument a → b a ∨ c ¬ b ∴ c using (i) a truth table (ii) the laws of logic (10 Marks)
Q2a 60 students may take up to 3 extra options for extra credits. 32 take option A, 40 option B, 32 option C 28 take A and B, 23 take A and C, 25 take B and C The number of students taking none of the options is the same as that taking only one of them. Illustrate this data on a Venn diagram and use your diagram to determine x, the number of students taking all three options. How many students elect to take no options.
(10 Marks)
Q2b Explain what is meant by the following terms for any general sets (^) A and (^) B. (i) P ( A ), the power set of A (ii) the set A × B (iii) a relation from A to B (iv) a function from A to B For the cases where | A | = 3 and | B | = 5, determine | P ( A ) |, | A × B | State the number of binary relations from A to B and the number of functions from A to B. State how many of these functions are injective and how many are surjective. (10 Marks)
Q3a T = {( a, b ), ( b, a ), ( b , c )} and V = {( a , b ), ( a, c ), ( b , d ), ( c , b )} are relations on the set { a , b ,c, d }. List the elements of TV , VT , V^2 , V^3. Draw the digraph s of V, s ( V ) , ts ( V ) , t ( V ), st ( V ). State whether ts ( V ) = st ( V ) (8 Marks) Q3b R = {( x , y )| x - y is divisible by four} is a relation on Z , the set of integers Show that R is an equivalence relation. S = {( x , y )| x - y is divisible by four} is a relation on the set A= {1, 2, 3…9, 10} List S and state | S |. Determine the equivalence class E (2) and the other equivalence classes of A given by the relation S. Draw the digraph of S which illustrates how S partitions A. (12 Marks)
5a Given the functions V 1 (^) ( ) t = 15 e −^25 t and V 2 (^) ( ) t = 20(1 − e −^25 t ),
State the time constant for both. Evaluate V 2 (^) (0.15) Determine when V 1 (^) ( ) t = V 2 (^) ( ) t On the same axes sketch the functions labelling the diagram clearly. Illustrate your answers on it. (8 Marks)
5b i 1 (^) = 20sin(100 π t + π 6 )and i 2 (^) = 25sin(100 π t −^ π 4 )
State the amplitude and frequency of i 2 and the frequency of i 1 (^) + i 2 Draw a vector diagram to illustrate i 1 , i 2 , i 1 (^) + i 2 Represent i 1 , i 2 as complex numbers in polar and Cartesian form. Determine i 1 (^) + i 2 and hence express i 1 (^) + i 2 in the form A sin( n π t ±α) (8 Marks) 5c Determine the equations of the waveforms v 1 , v 2 below.
(4 Marks)
v 1
v 2
Section C
Q6a
(6 Marks)
Q6b Determine the derivatives of the following functions. Simplify your answers.
(i) 212 3 x − 4 x
(ii) 2 x^3^ e −^5 x (iii) e cos(5 ) x (8 marks)
Q6c Determine the instantaneous rate of change of the function y = − x^2 + 4 x + 8 at x = -1 and at x = 3
Find the average rate of change of the function y = − x^2 + 4 x + 8 between x = -1 and x = 3_._ Illustrate the three answers on a rough sketch. Do not plot the function accurately. (6 Marks)
Q7a The shape of a poster is a rectangle surmounted by a semicircle of radius r. If the perimeter
of the poster is 5m, show that the area is 2 2 (^5 2 ) A = r − r − π r. Hence find the maximum area of the poster. (7 Marks)
Q7b Liquid is pumped into a spherical tank of diameter 1m at the rate of 0.15m s^3 −^1.
The volume of liquid is given by V = π 3 (1.5 h^2 − h^3 ) m 3 when the depth of liquid present in the tank is h m. Find the rate at which the water level in the tank is rising when the tank is half full. (6 Marks)
For the function y = f ( x ) graphed across state the value(s) of x for which f ( x ) is not defined. Determine the following limits
(i) 1 lim ( ) x − f^ x →
(ii) 1 lim ( ) x
(iii) (^) lim x → 1 f ( ) x
(iv) (^) x lim →−∞ f ( ) x (vi) (^) x lim →+∞ f ( ) x Use the graph to estimate f (0) and the solution of f ( x ) = 0 For what value(s) of x is f ′( )^ x not defined? Use the graph below to estimate f ′(10)^.
