THE UNIVERSITY OF HONG KONG
BACHELOR OF ENGINEERING: LEVEL 11 EXAMINATION
DEPARTMENT OF CIVIL ENGINEERING
ENGINEERING MATHEMATICS Il CIVB2001
January 6, 2001 9:30am- 12:30 pm (3 hours)
Answer FIVE questions: ALL OF section A and Two from Section B.
All questions carry equal marks.
Use of Electronic Calculators:
Candidates taking examination that permits the use of calculators may use any calculator
which fulfils the following criteria:
(a) it should be self-contained, silent, battery-operated and pocket-sized;
(b) it should have numeral-display facilities only and should be used only for the purpose
of calculation;
(c) it should not have any printing device, alphanumeric keyboard, or graphic display; and
(d) it should not contain any recorded data or program.
It is the candidate1s responsibility to ensure that the calculator operates satisfactorily and the
candidate must record the name and type of the calculator on the front page of the
examination scripts. Lists of permitted/prohibited calculators will not be made available to
candidates for reference, and the onus will be on the candidate to ensure that the calculator
used will not be in violation of the criteria listed above.
Section A
Answer ALL questions of this section.
For all questions of this section, determine whether each statement IS true or false.
WARNING: Negative marks will be given to wrong answers.
1. (a) icos zl : :; 1 for all complex z.
(b)
(c) f(z)=z is differentiable at z=O .
(d) If !m z > 0 , then je'= > 1 . I
(e)
t ---;. (ez -1) dz = in, where C is the unit circle.
z
1
- e ·.dz = zn
. , w here C is the unit circle.
(f)
tc z
I
-
(g)
t z e = dz = in , where C is the unit circle.
(h) !z = 2: :zo c-1r cz -1 r ror a11 z such that lz -tl < 1 .
(i) res ( 1 ,
') = --i .
1
2 2
(z +1) 4
(j) A bilinear map can be found, which maps conformally a crescent onto another
crescent with different apical angles.
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2. (a) A subset of a linearly independent set of vectors is linearly independent.
(b) In a vector space of dimension n , a set of n linearly independent vectors is a basis.
(c) In an inner product space, an orthogonal set of vectors is a basis.
(d) Let A be n x n and X n x 1 . The equation AX =0 has non-trivial solution
~A is invertible.
(e) Let A be n x n , and X and Yn be n x 1 . The equation AX =Y, always has
a solution.
(f) Similar matrices have the same rank, the same determinant, the same trace, and the
same eigenvalues.
(g) A matrix coordinate of an isomorphism is invertible.
(h) If an eigenvalue of A is zero, then A is not invertible.
(i) G~) is diagonalizable over R.
[~
1
G) 0 is diagonalizable over R .
0 IJ
3. (a) res( coshz , zn(l
. + 2k) ) = -1 for any integer k .
ez +1
] l~ ~)
1 0
(b) 0 is similar to -1
[: 1 0
(c) L "" n= I
sinnt
--
n.I
= e cosr sm
• { •
smt ) .
(d) ltl = 1t2 -~ ""'
~
Jl' n=
0
cos(2n+1)t
(2n+ 1)
2 , -1t'5ot'5o1t .
1
(e) The Fourier transform j of f(x)=--2 is given by:
1+x
f(}. )=~e-f-tl, -oo<A<oo.
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Section 8
Choose TWO questions from this section.
4. (a) Compute [ ~ by the method of residues, using the contour as shown.
X +1
0
>x
(b) Given w = e;~ z- a , where fjJ is real and jaj < 1 .
1-az
i) Show that jzj == 1 <::::> jwj = 1 , and interprete this result geometrically.
ii) Show that a direction at a is rotated by the angle t/J under this map.
5. (a) Compute £~ dx by contour integration. Sketch the contour used, and explain
~
x 3 +1
why a certain integral approaches zero.
(b) i) Find A" where A =G ;) .
Pn+l = 3 Pn +4qn
ii) Let {
qn+l = Pn +3qn , n =0, 1,2, ...
Express Pn and qn in terms of Po and qo .
6. (a) Let B={e~>e 2 ,e 3 } be the standard basis of R
3
, and B={v1,v 2 ,vj where
v1 == e1 + e 2 + e3 , v2 = 2e2 + e 3 , v3 = 3 e 3 •
i) Find the matrix coordinates M%' (1) and M%. (1) of the identity map on R 3 •
How are they related?
ii) Are these matrices diagonalizable? Give supporting reasons for your answer.
(b) Let A be an m xn matrix with rank r, and k=dimker A. Consider the equation
AX = Yo . Discuss briefly the nature of the solutions of this euqation as they are
related to the integers m, n, r, and k .
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7. (a) Solve the following problem of heat flow in a uniform rod:
DE: c 2 u:u =u1 , 05,x5,L, t:?.O
BC: ux (O,t)=O, u(L,t)=O
{
JC: u(x,O)= f(x)
(b) Solve the corresponding problem with a source term.
I.e., solve the problem in which the DE is
c 2 uxx =u1 +,(x,t),
while the BC and IC remain the same as in (a). Express the solution as a sum of the
transient and the asymptotic parts, and explain briefly what each of these parts
represents.
END OF PAPER
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