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.

ecorrect to four places of decimals. (10 Marks)

(b) Find the maximum and minimum values of y = lnx – x2 + 2x. (10 Marks)

Q3. Integrate

(a) dx

x

ex

2

1

5.0

1

−

∫ (4 Marks)

(b)

∫+2

16

1

x

dx (4 Marks)

(c)

()

xdxx .1

3

2

3

2

∫+ (4 Marks)

(d) xdxx ln

2

∫

(e)

()()()

dx

xxx

xx

∫−−−

−−

432

12

2

(4 Marks)

2

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) 2

1x

dx

dy −= (3 Marks)

(ii)

()

1

22 −= xy

dx

dy (3 Marks)

(iii) 3

2

2

x

dx

yd = (3 Marks)

(iv) 2

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

θ

θ

k

dt

d−= where k is a constant.

Find the solution of this equation given

θ

= 60 when t = 0. (8 Marks)

3

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)

x y

14 20.3

19 26.4

24 32.9

29 40.2

34 48.3

39 53.7

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.

(10 Marks)

4

Required formulae:

McLaurins expansion: f(x )= f(0) + f’(0).x +

(

)

!2

0" 2

xf + f’’’(0) !3

3

x

Least Squares: y = ao+ a1 x

where

ao =

()

()

()

(

)

()

()

2

2

2

∑∑

∑

∑

∑∑

−

−

xxN

xyxxy

where

a1 =

(

)

(

)

()

()

2

2

.

∑∑

∑

∑∑

−

−

xxN

yxxyN

Confidence limits:

x - zα. N

S

zx

N

S.

α

µ

+≤≤

##### Document information

Uploaded by:
dwar

Views: 1383

Downloads :
0

University:
Chhattisgarh Swami Vivekanand Technical University

Subject:
Mathematics

Upload date:
28/03/2013