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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 1 Examinations 2008/
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
Programme Code: EMECH_8_Y EBIOM_8_Y ECPEN_8_Y CSTRU_8_Y
External Examiner(s): Dr. P. Robinson Internal Examiner(s): Mr. G. O’Driscoll, Ms. 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 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.
- (a) Find adj A where
2 1 3 A 3 2 1 1 0 4
= ^
and show that A(adj A) = A I
(7 marks)
(b) Find the values of x for which the matrix
P 1 1
x x x
= ^
is singular.
(6 marks)
(c) Prove that cosh^2 x − sinh^2 x = 1 Hence solve the equation 2 cosh 2 x − 3sinh x = 1 (7 marks)
(d) Express
z i
in the form x + iy and find the polar form of z.
Hence find z^5 and convert the answer to Cartesian form. (7 marks)
(e) If a = 2 i + 4 j − k and b = 4 i − 2 j + 3 k find
(i) the area of the parallelogram with sides a and b (ii) a vector of magnitude 5 perpendicular to both a and b
(6 marks)
(f) Find the value of λ for which the points A(1, 2, 5), B(2, -1, 2), C(4, λ ,1) and
D(5, -2, -3) are coplanar. (7 marks)
- (a) A force F 1 of magnitude 18 lies in the direction of the line joining A(4, 9, -7)
to B(12, 5, -8). A force F 2 is of magnitude 60 and acts in the direction of the vector10 i − 2 j + 11 k.
Find (i) the resultant F of the two forces (ii) the work done by F in moving a particle from P(-1, 2, 4) to Q(1, -5, 6) (10 marks)
(b) Given the points P(1, 4, -3), Q(2, 6, -1) and R(3, 1, 3) show that the vector W =PQPR+PRPQ
makes equal angles with PR and PQ. (10 marks)
(c) Given the vectors a = − + i j − 2 k , b = i + 3 j + 4 k , c = 2 i + 7 j − 5 k and d = j − 13 k
find (i) the volume of the tetrahedron determined by a , b and c
(ii) b. ( c x d )and interpret the result geometrically
Verify that (^ )^
2 2 2 2 a x b + a. b = a b (10 marks)
