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Cork Institute of Technology
Bachelor of Engineering (Honours) in Mechanical Engineering – Stage 1
(NFQ – Level 8)
Summer 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 matrices 2 1 2 A 2 4 1 1 1 1
= ^ −
and
B 1 0 1
= ^
find the matrix C such that AC = B + BT. (10 marks)
(b) Using Gaussian elimination find the values of k for which the system of equations
x y ( k ) z k
x y z
x y z
2
does not have a unique solution. For these values of k determine whether the resulting equations are consistent and, if so, find the solutions. (10 marks)
- (a) Define the scalar product a b. and the vector product a x b of vectors a and b.
Hence prove that
( a. b )^2 + a x b^2 = a^2 b^2
Show also that if a + b is perpendicular to a − b then a = b (7 marks)
(b) A force F of magnitude 22N acts through the point Q (2, 3, − 6 ) in the direction of the vector 6 i − 2 j + 9 k. Find the moment of the force about the point P (1, 1, − 1 ). (5 marks)
(c) If a = i + 3 j − 4 k , b = 2 i − 3 j + k and c = 3 i − 4 j + λ k find
(i) the area of the parallelogram with sides a and b (ii) a vector of magnitude 15 perpendicular to the plane containing a and b
(iii) the value of λ for which a , b and c are coplanar
(8 marks)
- (a) If z (^) 1 = x + jy and if z 2 (^) = ajz 1 where a is real , prove that
2 2 2 z 1 (^) + z 2 (^) = z 1 (^) − z 2 (5 marks)
(b) (i) Express 8 1 3
z j
in the form x + jy
(ii) Find the modulus and argument of z and hence express z in polar form (iii) Using De Moivre’s theorem find the cube roots of (^) z and express the answers in Cartesian form. (8 marks)
(c) Find the complex number z such that
z = z − 4 − 2 j and arg 2( )
z = π.
Find ez. (7 marks)
Section B
(i)
3 2 0
sin 3 2 cos 3
x (^) dx x
π ∫ +
(ii)
8 2 4 6 25
dx ∫ x − x +
(iii)
(^4 ) 2 3
x x (^) dx x x
∫ −
(14 marks)
(b) The work done by a variable force F in moving a body from x = a to x = b is given by b
a ∫ F dx. Find the work done by the force^^ F^ =^ xe^2 x in moving a body from^ x^ =^0 to^ x^ =^3. Find also the mean value of the force over this interval. (6 marks)
- (a) Find the general solution of the following differential equations:
(i) (^3)
2 1
x
x y dx
dy
(ii) xy dy y^2 x^2 dx
(iii)^ dy^ y tan x cos x dx
(14 marks)
(b) The deflection θ in a galvanometer satisfies the differential equation
2 2 2 0
d d dt dt
Given that θ = 0.3and^ d 0
dt
θ = at t = 0 , solve this equation for θ.
(6 marks)
