##### Document information

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-*2*x + 4x* (4 Marks)

(b) *y = eSin x* (4 Marks)

(c) *x*3* – *y3* – x*3* y*3* = 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 – x*2 *+ 2x.* (10 Marks)

Q3. Integrate

(a) *dx
x
*

*e x *2
1

5.0

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)

2

Q4. (a) Find the area between the curve *y*2 = 4*x* 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

*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

θθ *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 = ( )( ) ( )( )

( ) ( )22 2

∑∑ ∑∑∑∑

−

−

*xxN
*

*xyxxy
*

where a1 = ( )( )

( ) ( )22 .

∑∑ ∑∑∑

−

−

*xxN
*

*yxxyN
*

Confidence limits: *x * - zα.
*N
Szx
*

*N
S *.αµ +≤≤