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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 1 Examinations 2009/
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 2009
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) If
A
find adj (adj A)
(6 marks)
(b) Show that 2 2 2
x x y y z z
= ( y − x )( z − x )( z − y )
(7 marks)
(c) Find
(^12) z if
1 1 2 2
z = + i
(6 marks)
(d) Solve the equation z^2^ + z + 1 = 0 (7 marks)
(e) For what values of a are the vectors 2 i + a j + 3 k and ai − a j + k perpendicular?
(7 marks)
(f) If a = (3, −2, 1), b = −( 1, 3, 4) and c = (2, 1, − 3)confirm that a × ( b × c ) = ( a c b ⋅ ) − ( a b c ⋅ )
(7 marks)
3.(a) Evaluate each of the following, expressing each result in the form r ∠ θ and in the form
x + iy :
(i)
0 0 2 0 3
∠ × ∠ −
(ii) ( 1− + i )^12 (iii)
i i i
(12 marks)
(b) Use De Moivre’s Theorem to find the fourth roots of z = − 5 + 12 i. Express the roots both in polar and Cartesian form. Represent the roots on an Argand Diagram. (8 marks)
(c) Find the locus of z if (i) z − 3 i = 3 z + i (ii) Re(2 z − 3 − i ) = Im( z + 2 −5 ) i (10 marks)
4.(a) Define the scalar product and the vector product of two vectors a and b. Given a = 2 i − 3 j − k , b = − 5 i + 3 j + 2 k and c = 3 i − k (i) Find the angle between a and b. (ii) Show that the vectors form a triangle. (iii) Find the area of the triangle. (iv) Find ( a – b ) and ( a + b ) and hence the vector product ( a – b ) × ( a + b ). Confirm your answer using the properties of vector product. (13 marks)
(b) A force F of magnitude 18N acts at the point A (3, − 1, 5)in the direction of the vector 6 i – 3 j + 2 k. (i) Find the moment of the force about the point B (2, − 4, 7). The unit of distance is the meter. (ii) Find the magnitude of the moment of the force. (7 marks)
(c) Determine the value of m such that the vectors a = 4 i + mj – 5 k , b = 2 i – j + 4 k and c = 2 i – 3 j + 6 k are coplanar. Find a unit vector perpendicular to the plane of a , b and c. If d = 3 i – j + 2 k find the volume of the tetrahedron determined by b , c and d. (10 marks)
