Cork Institute of Technology Higher Certificate in Engineering in Civil Engineering – Award
(National Certificate in Engineering in Civil Engineering – Award) (NFQ – Level 6)
Summer 2005 Mathematics
(Time: 3 Hours) Instructions Answer FIVE questions; a minimum of TWO questions in each of Sections A and B should be attempted. All questions carry equal marks.
Examiners: Mr. L. O Hanlon Mr. P. Anthony Mr. J. Murphy
Section A Q1. Differentiate with respect to x:
(a) y = e-2x + 4x (4 Marks)
(b) y = eSin x (4 Marks)
(c) x3 – y3 – x3 y3 = 0 (4 Marks)
(d) x = a (Cost + bSint)
y = c(Sint + dCost) a,b,c,d are constants (4 Marks)
(e) y = xSinx (4 Marks)
Q2. (a) Write down the first four terms of the McLaurin expansion of f(x) = ex.
Hence calculate 1.e correct to four places of decimals. (10 Marks)
(b) Find the maximum and minimum values of y = lnx – x2 + 2x. (10 Marks)
(a) dx x
e x 2 1
1 −∫ (4 Marks)
(b) ∫ + 216 1
x dx (4 Marks)
(c) ( ) xdxx .13 2
32∫ + (4 Marks)
(d) xdxx ln2∫
(e) ( )( )( )dxxxx xx
∫ −−− −−
432 122 (4 Marks)
Q4. (a) Find the area between the curve y2 = 4x and the line y = x. (5 Marks)
(b) If the velocity v in m s-1 of a body, is given by 1+= tv , find the average value of v
from t = 1 to t = 4 s. (5 Marks)
(c) Find the root mean square value of
y = 1 + 2 Sin x between x = 0 and x = 2π. (5 Marks)
(d) The mass per unit length of a rod at xm from one rod is (1 + .04x) kg m-1. If the rod is 3
m long, find its mass. (5 Marks)
Section B Q5. (a) Solve the differential equations
(i) 21 x dx dy
−= (3 Marks)
(ii) ( )122 −= xy dx dy (3 Marks)
(iii) 32 2
yd = (3 Marks)
when3r and rSin d dr Π
===+ θθ θ
0 (3 Marks)
(b) For a body at temperature θ above its surroundings the rate of all of temperature is given
by the differential equation
θθ k dt d
−= where k is a constant.
Find the solution of this equation given θ = 60 when t = 0. (8 Marks)
Q6. For the variables in the given table:
(a) Plot the data. (4 Marks)
(b) Calculate the least square line of y or x. (8 Marks)
(c) Plot, on the same graph sheet, scales and axes, the least square line. (4 Marks)
(d) Estimate the value of y when x = 26.
(i) From the graph
(ii) From the least square line. (4 Marks)
Q7. (a) If the heights of 500 male students at CIT were NORMALLY DISTRIBUTED and had a
mean of 5’9” and a standard deviation of 3”, find how many students would be expected
to have a height between 5’5” and 6’0”. Assume measurements were taken to the nearest
inch. (10 Marks)
(b) If 1.5% of concrete blocks manufactured in a process were defective, find the probability
that out of 200 blocks chosen at random,
(i) Exactly two were defective.
(ii) More than two were defective.
(Use Poisson). (10 Marks)
Q8. (a) Find the probability of getting exactly two fours in five tosses of a fair dice.
(Use biomial). (10 Marks)
(b) The mean length of 60 planks is 1.7m with a standard deviation of .2m.
Find (i) 95% (ii) 99% confidence limits for the mean length of all the planks.
McLaurins expansion: f(x )= f(0) + f’(0).x + ( ) !2
0" 2xf + f’’’(0) !3
Least Squares: y = ao+ a1 x
where ao = ( )( ) ( )( )
( ) ( )22 2
where a1 = ( )( )
( ) ( )22 .
Confidence limits: x - zα. N Szx
N S .αµ +≤≤