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)

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University:
Chhattisgarh Swami Vivekanand Technical University

Subject:
Mathematics

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28/03/2013