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Cork Institute of Technology
Bachelor of Engineering (Honours) in Mechanical Engineering – Stage 1
(Bachelor of Engineering in Mechanical Engineering – Stage 1)
(NFQ – Level 8)
Summer 2005
Mathematics
(Time: 3 Hours)
Answer FIVE questions. Answer TWO
questions from Section A and THREE questions
from Section B. All questions carry equal marks.
Examiners: Mr. G. O’Driscoll
Mr. J. Hegarty
Prof. J. Monaghan
Section A
- (a) Given the matrix
A
obtain
A.
Find the scalars α and β such that A A I
= α + β where I is the 3 x 3 identity matrix.
(10 marks)
(b) Using Gaussian elimination determine the values of p and q for which the system of
equations
1 2 3
1 2 3
1 2 3
x x px
x x x q
x x x
is inconsistent.
Solve the system when
(i) p = 1 and q =− 1
(ii) p =− 1 and q = 1
(10 marks)
- (a) Using properties of scalar and vector product simplify
(i) ( 2 A − B ) (. 2 A + B )
(ii) ( 2 A − B ) x( 2 A + B )
(iii) A .[( 2 A − B ) x( 2 A + B )]
(7 marks)
(b) A force F of magnitude 14 N acts at the point (2, 3, 5) in the direction of the line joining
(2, 1, 1) to (4, 7, -2).
Find
(i) the direction cosines of F
(ii) the moment of the force about the point (1, 2, 3)
(7 marks)
(c) Given the vectors a = 3 i − 2 j + 4 k , b = 2 i − 5 j − 4 k and c = i − 8 j − 12 k determine
whether or not
(i) a and b are perpendicular
(ii) a , b and c are coplanar.
Find a vector perpendicular to both a and b.
(6 marks)
- (a) If z 1 = 8 ∠ 30 ° and z (^) 2 = 2 ∠ 45 ° express the following in polar form
(i) z 1 (^) z 2 (ii)
1
z
(iii)
1
4 2
z
z (iv) z 1 (^) − z 2
(7 marks)
(b) Find the modulus and argument of the complex number
( )
j
j z − +
2
.
Use De Moivre’s theorem to find the cube roots of z and express the roots in Cartesian
form.
(8 marks)
(c) Find the locus of z if
(i) z − j = 2 z − 1
(ii) Re ( 2 z + 1 + 3 j ) =Im( z − 2 − 4 j )
(5 marks)
- (a) Find
(i)
4
0
sin 4 cos 2
π
x x dx
(ii)
dx x x
x
2
(iii)
2
1
2 10 9
dx
x x
(14 marks)
(b) Find the r.m.s. value of y = cos 2 t between t = 0 and 4
π t =.
(6 marks)
- (a) Find the general solution of the following differential equations:
(i) 2 ( 1 ) ( 3 )
2 2
dy yx
(ii) x xy
xy y
dx
dy
2
2
(iii) 12 36 0 2
2
− + y = dx
dy
dx
d y
(14 marks)
(b) Solve the differential equation
x y xe dx
dy (^) 3 4
−
given that y = 1 when x = 0.
(6 marks)
