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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 1 Examinations 2010/
Module Title: Engineering Mathematics 101
Module Code: MATH 6005
School: School of Building & Civil Engineering School of Mechanical & Process Engineering
Programme Title: Bachelor of Engineering(Honours) in Mechanical Engineering – Year 1 Bachelor of Engineering(Honours) in Biomedical Engineering – Year 1 Bachelor of Engineering(Honours) in Chemical & Process Engineering – Year 1 Bachelor of Engineering(Honours) in Structural Engineering – Year 1 Bachelor of Engineering(Honours) Common Entry – Year 1
Programme Code: EMECH_8_Y EBIOM_8_Y ECPEN_8_Y CSTRU_8_Y EOMNI_8_Y
External Examiner(s): Dr. P. Robinson Internal Examiner(s): Dr. V. Morari, Ms. F. Wood
Instructions: Answer QUESTION 1 (worth 40 points) and TWO other questions (worth 30 points each)
Duration: 2 HOURS
Sitting: Winter 2010
Requirements for this examination: Mathematics Tables
Note to Candidates: Please check the Programme Title and the Module Title to ensure that you are attempting the correct examination. If in doubt please contact an Invigilator.
- (a) Use Cramer’s Rule to find values for x and y in the following system of equations:
x y z x z x y z
Hence find a value for z. (6 marks)
(b) Given the matrices
A =
1 2 3 2 1 4 3 0 2
(^)
and M = 1 2 2
find the matrix P such that PA = M.
(7 marks)
(c) Given z 1 (^) 1 i and z 2 (^) 3 2 i (i) Express z 1 and z 2 in polar form. (ii) Find
2 2 1
z z and give your answer in Cartesian form.
(7 marks)
(d) Given that z 2 i is a root of the equation z^3 5 z^2 9 z 5 0 show that z 2 i is also a root. Hence find the remaining root. (7 marks)
(c) Use row and column operations to solve for t in the following equation:
t
t
t = 0.
(10 marks)
3.(a) Given z 1 (^) 8(cos30^0 i sin 30 )^0 and z 2 (^) 2(cos 45^0 i sin 45 )^0
(i) find z z 1 2 (ii) find
4 2 1
z z (6 marks)
(b) Find the modulus and argument of
(1 )^2 1
z i i
Use de Moivre’s Theorem to find the cube roots of z. (9 marks)
(c) Prove each of the following:
(i) z 1 (^) z 2 (^) z 1 (^) z 2
(ii) z z 1 2 (^) z 1 (^). z 2 (6 marks)
(d) Find the locus of z if z i 2 z 1 (9 marks)
4.(a) Given the vectors a 2 i 3 j , b j k and c 3 i 2 k verify that
a ( b c ) ( a c b ) ( a b c ) (8 marks)
(b) If 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. (8 marks)
(c) Find the volume of the parallelepiped with edges represented by a i j k , b i j 2 k and c 3 i 4 j k. (6 marks)
(d) A force F , of magnitude 14 N acts at a point ( 2, 3, 5) in the direction of the line joining ( 2, 1, 1) to ( 4, 7, – 2). (i) Find the moment of the force F about the point (1, 2, 3) (ii) Find the directional cosines of F. (8 marks)
