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Cork Institute of Technology
Bachelor of Engineering in Building Services Engineering -
Stage 1
(NFQ - Level 7)
Autumn 2007
Mathematics
(Time: 3 Hours)
Instructions Answer FIVE questions. All questions carry equal marks.
Examiners: Ms.H.Lordan Mr.M.Smyth Mr.D. Leonard
Q1 (a) Using the laws of indices:
(i) Solve for x : 3 x^ +^2 ร 91 โ^ x^ = 272 x โ^3.
(ii) Simplify:
2 1 5 2 2
ab c a b
โ โ.
(8 marks)
(b) Solve for x using the laws of logarithms: (i) log x + log( x + 2) = log 3. (ii) 3 x^ +^2 = 12 x โ^1. (6 marks)
(c) For the formula v = c 2 gh (i) Make h the subject. (ii) Find the value of h if v = 1.8 ร10 , 3 c = 1.3 ร 10 2 and g = 9.81.
Q2 (a) Solve for x and y.
x + y = 18 1 1 9 x y 40
(8 marks)
(b) Draw the graph of y = 5 e โ1.2 x for values of x from x = 0 to 2 using intervals of 0.4. From the graph or otherwise find the value of x for which y = 3. (6 marks)
(c) Resolve into its partial fractions
x x x
(6 marks)
Q3 (a) Write each of the following in linear form. In all cases a and b are constants. State clearly the slope and intercept. (i) b T
a S = +
(ii) R = aV^2 + bV
(iii) F โ b = a L.
(6 marks)
(b) It is believed that x and y are related by a law of the form y = aekx where a and k are constants. Values of x and y were measured and the results are as shown:
x 0.25 0.9 2.1 2.8 3.7 4. y 6.0 10.0 25.0 42.5 85.0 198. Show by plotting ln y against x that the law is true and find approximate values for a and k.
Q7 (a) Differentiate by rule:
(i)
5 3 2 4 6 3 10 3
x y = x โ + x โ.
(ii)
3
3 2 5
e^ x y x
(iii) y = (2 x + 3) sin x. (9 marks)
(b) Find the co-ordinates of the maximum value and minimum value of the curve y = x^3 โ 6 x^2 + 9 x + 2. (11 marks)
Q8 (a) Obtain
(i) โซ( x + 2)(2 x โ3) dx
(ii)
(^2 )
โซ 1 (^ x^ โ^3 x^ +3) dx
(iii)
3 2 2
dx
โซ x
(iv) โซ (2 x +3)^5 dx.
(12 marks)
(b) Find the roots of y = x^2 โ 9. Sketch the curve. Find the area bounded by the curve y = x^2 โ 9 and the x axis.
