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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)
Autumn 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 that
A
find the values of α and β such that A A I O
2
+α +β = where I is the 3 x 3 identity
matrix.
Using this equation, or otherwise, find
A.
(14 marks)
(b) Express the system of equations
1 2 3
1 2 3
1 2 3
x x x
x x x
x x x
in matrix form.
Solve the system using the inverse matrix obtained in (a).
(6 marks)
- (a) If A = 5 i − 3 j + 2 k and B = 4 i + 2 j + k find
(i) the direction cosines of A
(ii) the angle between A and B
(iii) the projection of A onto B
(iv) two vectors perpendicular to A and B
(9 marks)
(b) A force of magnitude 8 N acts at the point (3, 1, -4) in the direction of the line joining
(2, -1, 3) to (1, 0, 2). Find the moment of the force about the point (1, 0, -2).
(6 marks)
(c) If u = i + j , v = 2 i − 3 j + k and w = 2 j − k verify the identity
u x ( v x w ) =( u. w ) v −( u. v ) w
Hence show that for arbitrary vectors a , b and c
[ a x ( b x c )] x c =( a. c )( b x c )
(5 marks)
- (a) (i) Convert z (^) 1 = 1. 3 − 0. 7 j , z (^) 2 = 4 j and z (^) 3 = − 2. 1 + 1. 4 j to polar form
(ii) Hence evaluate (^3) 3
2
- 2
z
z z in polar form
(iii) Express the answer in (ii) in Cartesian form
(9 marks)
(b) Use De Moivre’s theorem to find the cube roots of z = 1 + j 3
(6 marks)
(c) Find the locus of z if
(i) z − 2 j = 5
(ii) (^) Re ( (^2) z − 1 ) (^) =Im( (^) z + 3 + 5 j )
(5 marks)
(i) dx x
x
1
0
2 4
(ii) ( )
dx x x
x
4
2
2 1
(iii) ∫ xe dx
2 x 3
(14 marks)
(b) Find the mean value of y sin 2 t
2
= between t = 0 and t = π.
(6 marks)
- Find the general solution of the following differential equations:
(i) (^2) 1 2
x
xy
dx
dy
(ii)
3 2 y x dx
dy x − =
(iii) x y
y
dx
dy
−
(iv) 2 2 9 0
2
dx
dt
d x
(20 marks)
