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Cork Institute of Technology
Bachelor of Engineering in Civil Engineering – Stage 1
(NFQ Level 7)
Higher Certificate in Engineering in Civil Engineering (ACCS) – Stage 1
(NFQ Level 6)
Autumn 2007
Mathematics
(Time: 3 Hours)
Answer 5 Questions. Answer FIVE questions, choosing at least ONE question from Section B. All questions carry equal marks.
Examiners: Mr. D. Cremin Ms H. Lordan Mr. J. Lapthorne Mr. J. Kindregan
Section A
- (a) Simplify using the laws of indices:
( 4 a b −^ c ) × ab ÷ abc − (6 marks)
(b) (i) Solve the equation: 4 3 x = 26 x +^2 (ii) Express the following equation without logarithms:
10 R^ log 10 H^4 log 10^ S 4
log = 2 +^1 + (8 marks)
(c) Transpose the following equation to make λ the subject:
A
c z n 5
(^23) λ 72 4 ρ
μ = (6 marks)
- (a) Solve for x , y z , :
x y z
x y z
x y z (6 marks)
(b) Use the Remainder Theorem to find the factors of the equation: 4 x^3 − 39 x + 35 = 0 (7 marks)
(c) The roots of the quadratic equation x^2 + px + q = 0 are α, β where
x ∈ R and x ≠ 1. Show that the quadratic equation whose roots are α 2 β and
αβ 2 is x^2 + pqx + q^3 = 0 (7 marks)
- (a) Show that (cos x +sin x )^2 +(cos x −sin x )^2 simplifies to a constant. (6 marks)
(b) Solve the trigonometrical equation: 7 sin 3 x =− 4. 38 ( 0 ≤ x ≤ 3600 ) (6 marks)
(c) A quadrilateral ABCD has the following dimensions: AB=65m., BC=140m., CD=155m., DA=128m., and angle BAD= 67.^0 Calculate the length of BD and the angle BCD. ( marks)
- The compressive strength of 60 concrete samples, tested after seven days, was recorded and the results tabulated as follows:
(a) Taking the class mark of each class, calculate the mean x and the standard deviation
σ from the mean.
(8 marks) (b) Represent this information on a histogram. (4 marks) (c) From the histogram, read of the median and the mode. (4 marks)
(d) Estimate the number of samples that lie in the range x ± σ.
(4 marks)
Section B
- (a) Differentiate from first principles: f ( ) x = x^2 − 3 x
(5 marks) (b) Differentiate each of the following by rule: (i) 5 7 2 sin 2^3 3 4
y x e^ x x x
(ii) y = x^2^ + 10 x (5 x^2^ −7 ) x^4
(iii) s(3 ) 2 1
y co^ x x
(9 marks)
(c) A closed cylindrical can has a volume of 1000 cm^3. Show that its total surface area is given by A 2 r^2 r
= π + where r is the radius. Hence find the dimensions which require the least surface area. (6 marks)
Compressive Strength NM / m^2
Number of samples 8 10 19 15 6 2
- (a) Evaluate any three of the following:
(i) ∫( x^2 − 3)(2 x +7) dx
(ii)
(^3 ) 1
2 x 5 x dx x
(iii)
1 0
∫(cos 4^ x^ +sin 3 ) x dx
(iv)
(^14) 0
∫(3^ x^ −1) dx
(12 marks)
(b) Find the area between the curve y = 2 x^2 + 3 , the x - axis and the ordinates x = − 2 and x = 4. Sketch the graph. (8 marks)
