Download Building Services Engineering Exam: Math & Computing, Summer 2006, BEng and more Exams Mathematics for Computing in PDF only on Docsity!
Cork Institute of Technology
Bachelor of Engineering in Building Services Engineering -
Award
(NFQ Level 7)
Summer 2006
Mathematics & Computing
(Time: 3 Hours)
Instructions Answer five questions. All questions carry equal marks.
Examiners: Dr. N. Hewitt Mr. D. Leonard Ms. M. Harley
Q1a Find the Laplace transforms of the following, expressing your answer as a single fraction. (i) t^4 − 4 t^2 − 12 (ii) e − t sin(100 π t )
(4 Marks) Q1b Deduce the Inverse Laplace Transform of the expressions
(i) (^) s^3 (^2) (^ − s^11 + 3)^ s (iii) 22 19 4 29
s s s
(8 Marks) Q1c Use Laplace Transforms to solve the differential equation
4 V ′( )^ t + V t ( ) = 5 e −0.5 t V (0) = 0
(8 Marks)
Q2 The current i in a circuit is given by Li ′′( ) t + Ri t ′( ) + (^) C^1 i t ( ) = 0 where
L = 0.2 and C = 80 × 10 −^6. Solve this differential equation in the cases where
(i) R = 60 (ii) R = 100 (iii) R = 125
given that i (0) = 0 and i ′(0) = 750 Sketch i ( t ) for one of these cases, to illustrate the particular features of the solution. Label axes clearly
(20 Marks)
Q3a The acceleration of an object at time t (s) is given by^ dvdt^ =^20020 − v
Find expressions for the velocity v and the displacement x at any time t if the velocity and displacement are zero initially. (10 Marks)
Q3b If h m. is the depth of liquid present in a spherical tank of diameter 4 m. then the volume
of liquid present is given by V = π 3 (6 h^2 − h^3 )m^3 .The tank is emptied through a valve of
c.s.a. 4 × 10 −^3 m^2 ., situated at the lowest point of the tank. If the velocity of emerging liquid is given by 2 gh ms -1^ , show that
(4 h h^2^ )^ dh 5.64 103 h dt
− = − × −.
Hence find the time taken to empty the tank if it was full initially. ( g = 9.81ms-2^ ) (10 Marks)
Q5a Given that
(^2 3) -2 0 1 = 0 1 = (^) -1 3 0 3 2
A B
− ^
^
(6 Marks)
Q5b Find the inverse of the matrix
2 3 1 3 1 2 1 2 5
−
−
and confirm your answer.
Hence solve the set of simultaneous equations below using matrix inversion.
1 2 3 1 2 3 1 2 3
2 3 2 3 2 7. 2 5 3.
V V V V V V V V V
Q5c Given the matrix M = 1 1 2
2 3 2
0 1 1
−
− , write out the matrix N = M + kI
where k is a constant and I is the identity matrix.
Show that N can be written in the form ( k + 1) − − −
− − −
1 0 0
Hence determine all values of k for which N is singular (5 Marks)
determine (i) AB (ii) ( A + B ) T (iii) A + BT Hence state B AT^ T
Q6a The relationship between y and x is of the form y = a 0 + a 1 x + a 2 x^2.
For a set of 4 data points, ∑ x =-2, ∑ y =9, ∑ x^2 =6,
∑^ x^^3 =-8,^ ∑^ x^^4 =18,^ ∑ xy^ =12,^ ∑ x^^2 y =-
Determine the constants a 0 (^) , a 1 (^) , a 2 using Cramers rule to solve the normal equations. (11 Marks) Q6b Solve the set of equations using Gaussian Elimination
Check your answer. (9 Marks)
Q7a The relationship between p and V for the data below is pV k = C
where C, and k are constants. Show that ln( P ) = − k ln( V ) +ln( C )
Use the Least Squares Method (working to 3 decimal places ) to estimate C and k. Hence estimate V when p = 2.0 (11 Marks)
Q7b The following table shows points and goals against for the top 6 teams in last years Premiership.
Goals against 9 17 14 31 26 27 Points 95 83 77 61 58 55
Evaluate the linear correlation coefficient and the coefficient of determination and comment on your answers. (9 Marks)
p 8.9 3.4 0.7 0.5 0. V 5 10 30 40 70
x + 2 y + 3 z + w = 7 2 x^ +^ y^ +^ z^ +^ w^ = 4 x + 2 y + z = 3 y + z + 2 w = -
The Coefficient of linear correlation
∑ ( )^ ∑ (^ )
x^2 Nx^2.^ y^2 N y^2
r xy Nxy
