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Cork Institute of Technology
Bachelor of Engineering (Honours) in Mechanical Engineering – Stage 1
(NFQ – Level 8)
Autumn 2007
Mathematics
(Time: 3 Hours)
Answer TWO questions from Section A and THREE questions from Section B. All questions carry equal marks.
Examiners: Mr. G. O’Driscoll Mr. P. Clarke Prof. M. Gilchrist
Section A
- (a) Given the matrix 1 3 3 A 3 1 3 3 3 1
= ^
find the scalars r and s such that A^2 + r A + s I = Owhere I is the identity matrix. (7 marks)
(b) Find the inverse of the matrix 1 3 2 3 6 2 1 6 3
and hence solve the simultaneous equations 3 2 9 3 6 2 1 6 3 13
x y z x y z x y z
(13 marks)
- (a) A vector A is of magnitude 5 and lies in the direction of the line joining (1, -1, 2)
to (2, 1, 0). A vector B is of magnitude 14 and lies in a direction parallel to the vector 2 i − 6 j + 3 k. Find the vectors A and B. Determine the angle between A and B. (6 marks)
(b) Given the vectors a = 2 i − 3 j + 4 k , b = i + 2 j + 5 k and c = 4 i + j − 2 k find (i) (^) a x b (ii) a unit vector perpendicular to a and b
(iii) the volume of the parallelepiped enclosed by a , b and c
(8 marks)
(c) Determine whether or not the vectors a = i + j + k , b = i − j and c = i + j − 2 k are mutually perpendicular.
Verify that a x ( b x c ) =( a. c ) b −( a. b ) c
(6 marks)
- (a) Find the complex numbers z which satisfy the equation
j
z z z z 13 4
(6 marks)
(b) If z (^) 1 = 1 + 3 j and z (^) 2 = 3 − 4 j find z if 1 2
z z z
Express z in polar form. Hence find z^5 and express the answer in the form x + jy. (8 marks)
(c) Find the locus of z if
(i) ( z − 2 − j )( z − 2 + j ) = 3
(ii) z = z + 5 (6 marks)
- (a) Evaluate the integrals
(i) ∫
2
0
sin 4 sin
π x x dx
(ii) ∫
3
2 3 2 2
(^1) dx x x
(iii)
dx xx
x x 1
1 2
2
(14 marks)
(b) The area bounded by the curve y = 2 x ln x , the x axis and the ordinates at x = 0 and x = 2 , is rotated through a complete revolution about the x axis. Find the volume generated. (6 marks)
- Find the general solution of the following differential equations:
(i) xe^2^ y^ dydx = x^2 + 1 (5 marks) (ii) x^2 dy xy y^2 dx
(6 marks) (iii) dy^ y x dx
(6 marks) (iv)
2 2 4 4 0
d y dy (^) y dx dx
(3 marks)
