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Cork Institute of Technology
Bachelor of Science (Honours) in Software Development and Computer Networking - Stage 1
(Bachelor of Science in Software Development and Computer Networking - Stage 1)
(NFQ – Level 8)
Summer 2005
Mathematics
(Time: 3 Hours)
Instructions Answer FIVE questions, at least TWO questions from each Section. All questions carry equal marks.
Examiners: Dr. D. Chambers Mr. T. Parslow Mr. P. O Connor Ms. M. Harley
Section A
Q1a Express the following symbolically and state its truth-value. (i) There is no square multiple of 5. (ii) A square number cannot be a prime number Assume universe is the set of integers.
(4 Marks) Q1b Express ∀ x [( prime ( x )∧( 10 < x < 20 ))→¬ even ( x )]in English and state its truth-value
Negate the proposition, simplify it and express it in English and state its truth-value. (5 Marks) Q1c a b c f 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 (4 Marks) Q1d Explain what is meant by the terms tautology, contradiction. Use the laws of logic to classify [( a → c )∧(¬ b → a )∧¬ c ]→ b as one of these or neither. (7 Marks)
Find an expression for f in this truth table.
Q2a A survey of the efficiency of lights, brakes and steering of 106 motor vehicles reported the following results: Defective lights = 35 defective brakes = 40 defective steering = 41 Defective brakes and lights = 8 Defective lights and steering = 7 Defective brakes and steering = 6 5 vehicles were deemed non-defective. Use a Venn diagram to determine how many vehicles (i) had defective lights, brakes and steering (ii) had one defect only. (6 Marks)
Q2b R = {(1, 2) (1, 3), (2, 4) (4, 4)} and T = {(1, 1), (1, 2) (1, 3), (2, 3) (3, 2)} are relations defined on A = {1, 2, 3, 4}.
Illustrate each of the composite relations TR and T^2 in separate digraphs.
(6 Marks)
Q2c Explain what is meant by the terms partial order and equivalence relation. R = {( x , y )| x + y is even} and S (^) = {( x , y )| x is a multiple of y } are relations on Z , the set of integers. Investigate whether each relation is a partial order, an equivalence relation or neither. (8 Marks)
Q3a Determine the inverse of the real valued functions
(i) f ( x ) = 2 + 5 x (ii) f ( x ) = (^) x^2 + 3 (iii) f ( x ) = 9 − x^2 (6 Marks) Q3b Define the terms surjective, injectective , bijective as applied to functions. Each definition should include an example of a real-valued function, with domain and codomain stated, and also a sketch of the graph of the function (8 Marks) Q3c Determine whether the following functions are even, odd, neither. Justify your answer in each case (i) x^2 − 5 x + 6 (ii) sin( x ) (iii) ( e^2 x^ + e −^2 x ) (6 Marks)
Q4a Given the functions y (^) 1 ( t )= 20 e −^2 t and y (^) 2 ( t )= 15 ( 1 − e −^2^ t ), solve the equation y 1 (^) ( t )= y 2 ( t ) On the same axes sketch the functions labelling the diagram clearly. Illustrate on it your answer to y 1 (^) ( t )= y 2 ( t ) (6 Marks)
Section B
Q5a Determine the derivatives of the following functions. Simplify your answers.
(i) (^42 )
2 −
x
x x (ii) ) ln(sin( (^46)
θ + π (iii) 10 t^2 e −^5 t
(8 Marks) Q5b Find the values of a and b so that the function
( )^2
x^ x
ax b x
x x f x is continuous for all x.
Determine f ′(^ x ).
For what value(s) of x is f ′( x )not defined (6 Marks)
Q5c x = 2 cos( θ )+cos( 2 θ), y = 2 sin( θ )−sin( 2 θ)
are the parametric equations of a curve. Find dxdy^ at the point where θ =^ π 4
Hence determine the equation of the tangent to the curve at this point. (6 Marks)
Q6a If y ( t )= e −^2 t^ − e −^5 t , show that y ′′ ( t )+ 7 y ′( t )+ 10 y ( t )= 0
Thus determine the nature of any critical points. Evaluate y (0). Sketch y ( t ) for t > 0. (8 Marks)
Q6b The volume of a sphere is increasing at the rate of 2.5cm^3 s- Find the rate of change of the surface area of the sphere at the instant when the radius is 3m.
( volume and surface area of a sphere are 34 π r 3 and 4 π r 2 respectively)
(6 Marks)
Q6c Show that the equation ln( x ) = 2 x − 3 has a root in the interval [1, 2]
Determine this root correct to 3 decimal places using the Newton-Raphson approximation with three iterations. (6 Marks)
Q7a Determine the following integrals:
(i) (^) ∫
dx 2 x 3
(^1) (ii) ∫ (^) −
1 0 3
2 2 x 1 dx
x (^) (iii) dx ∫ (^25) − 4 x 2
(8 Marks)
Q7b Evaluate the mean value of y = 5 x sin( 2 x ) in the interval [0,^ π 2 ]
(5 Marks)
Q7c The diagram shows the graphs of the functions y = 2 x and y = x^3 − 2 x.
Determine the shaded area between the graphs. (7 Marks)
