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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Autumn Examinations 2007/
Module Title: Engineering Mathematics 101
Module Code: MATH 6005
School: School of Building & Civil Engineering School of Electrical & Electronic Engineering School of Mechanical & Process Engineering
Programme Title: Bachelor of Engineering(Honours) in Mechanical Engineering – Year 1 Bachelor of Engineering(Honours) in Electronic Engineering – Year 1 Bachelor of Engineering(Honours) in Chemical & Process Engineering – Year 1 Bachelor of Engineering(Honours) in Structural Engineering – Year 1
Programme Code: EMECH_8_Y EELXE_8_Y ECPEN_8_Y CSTRU_8_Y
External Examiner(s): Dr. P. Robinson Internal Examiner(s): Mr. G. O’Driscoll, 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: Autumn 2008
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.
1.(a) Given the matrices A =
and B =
find the products (i) AB , (ii) BA
(iii) ABT^ and (iv) AAT^. What do you notice in the case of (iv)? (7 marks)
(b) Find P−^1 where P =
b a b a b
For what values of a and b is P−^1 = P
(7 marks)
(c) A complex number z is represented by the following expression: 0 0 2 0 3
∠ × ∠ −
Evaluate the complex number expressing in both polar and cartesian form: Hence find z^4.
(6 marks)
(d) Define the functions cosh x and sinh x and indicate the shape of their graphs. Prove that cosh 2 x = 2 cosh 2 x − 1 (7 marks)
(e) A triangle has vertices P (3, − 2, 1), Q ( 4,1, −3) and R ( 2, 5, 1)−. Find (i) position vectors defining the triangle (ii) any one angle in the triangle (iii) the area of the triangle. (7 marks)
(f) Find the volume of the tetrahedron with vertices A (5, -3, 2) , B (1,4,3) , C (3,-1,0) and D (1, -5, 3). (6 marks)
4.(a) Define the scalar product and the vector product of two vectors a and b.
Are these products commutative? A force F of magnitude 20N acts in the direction of the line joining ( 2 , 1 ,− 3 )to (− 5 , 4 , 2 ). Find the unit force vector and hence find the work done by F in moving an object from ( 1 , 3 ,− 2 )to ( 2 , 3 , 4 ).Assume the unit of displacement is the metre.
(10 marks)
(b) Given a = 2 i − j + 3 k , b = i + j − 2 k and c = − + i 2 j − 5 k (i) show that the vectors form a triangle and find the length of any one of its medians. (ii) find a unit vector perpendicular to a and b. (iii) find the area of the parallelogram with sides a and b. (10 marks)
(c) Find the constant α such that the vectors a = 2 i − j + k , b = i + 2 j − 3 k and
c = 3 i + α j + 5 k are coplanar.
Find the vector triple products (i) ( a × b ) × c (ii) a × ( b × c )and comment on your result. (10 marks)
