THE UNIVERSITY OF HONG KONG
BACHELOR OF ENGINEERING: PART I EXAMINATION
DEPARTMENT OF CIVll.., ENGINEERING
MATHEMATICS CNB 1002
DATE: December 29, 1999 TTIVUE: 9:30 am - 12:30 pm
Use of Electronic Calculators:
The Calculator: (i) should be silent and battery-operated;
(ii) should not have any printing device, alphanumeric display;
(iii) should not contain any recorded data or program.
It is the candidate's responsibility to ensure that his calculator operates satisfactorily. Candidates must
record the name and type of their calculators on the front page of their examination scripts.
Read the following carefully before starting to work:
( 1) Answer ALL 7 questions from Section A .
(2) Answer any 4 questions from Sections B and C, with at least 1 from Section C.
(3) Use a separate answer book for each section.
Section A
Ql. A straight line L is defined parametrically by the vector equation
r(t) = ro +td
where d is the direction vector and t ( + 1-) is any scalar quantity.
(a) Find an expression for the value t 0 of the parameter t such that the vector r(t 0 ) is
perpendicular to the line L .
(b) If ro is the line joining the ongm to the point ( .fj' 0, 0) and J is given by
1/ .J3 (i +] + k), find t 0 and the closest distance from the line to the origin.
2
Q2. (a) Find a solution of d : + x = e _2, sin 3t , by using the corresponding complex equation.
dt
(b) Explain why the above approach is preferred to the alternative method of substituting
x(t) = pe-21 cos3t+qe-2' sin3t.
s+l
Q3. Invert the Laplace transform 2
•
s(s +4)
Q4. Find the equation of the tangent plane at the point Q: (2, 1,-2) on the sphere x 2 + y 2 + z 2 = 9.
QS. Find the Fourier series of
f(t)=t 2 ,
· l val ues of (")
Then deduce the numenca ~-
1 L.J
1 and (1·1·) ~ (- 1) "-'
2L.J 2
n=l n n=l n
Q6. Solve the differential equation
yy' + al-bt = 0, t>O
where y(O) = 0, a and b are positive constants and y > 0 for t > 0.
2
Hint: d(u ) = 2 u du .
dx dx
Q7. A company finds that on average there is a claim for damages which it must pay seven times in
every ten years. It has expensive insurance to cover this situation. The premium has just been
increased, and the firm is considering letting the insurance lapse for 12 months as it can afford
to meet a single claim. Assuming an appropriate probability distribution, what is the
probability that there will be at least two claims during the year?
Section B
Q8. The following equation could represent the damped vertical n:otion of a mass supported by a
spring and subjected to an external periodic force:
x" + x' + 36x = lOsin mt, fort> 0,
the system being in equilibrium under no force for t ~ 0.
(a) Find the period of the free (damped) oscillations. Show that any free oscillation
stimulated at startup is reduced by a factor of about 14 after five periods of oscillation.
(b) Obtain expressions in terms of m (radian/sec) for the amplitude and phase of the forced
oscillation.
(c) Find the condition for resonance.
(d) Plot curves of amplitude and phase against m for a range 4 ~m~ 8.
[TURN OVER]
A system is described by the equation d ~ + 2 dx + 4x = 1. It is initially quiescent and
2
Q9. (a)
dt dt
then switched on. Use Laplace transform to find the subsequent time variation of x.
(b) Describe qualitatively what the result from (a) represents.
(c) Use Laplace transform to solve the differential equation:
y'" + y = xcos2x
y(O) = 0, y'(O) = 0
. fractiOn
Hint: Use the parttal . s
- 4
2
= -1 ( - - -5 + 5 + 24 J•
(s 2 +1)(s 2 +4) 2 9 s 2 +1 s 2 +4 (s 2 +4) 2
QlO. (a) Obtain the Fourier series for the switching function P(t) shown in the figure:
P(r)·
l
-2 -~ -1 _.!. 0 .!. 1 l. 2 ·t
2 2 2 l
(b) Determine the steady-state forced vibration of the system below, if the applied force
P(t) is given in (a).
A spring-mass system acted upon by an alternating square-wave torcc.
Qll. A 2-DOF dynamic system has the following properties:
K 1 =3k
K 2 =2k
M 1 =M 2 =m
The modal natural frequencies of the system are given by
[TURN OVER]
The normalized mode shape matrix is
(a) Use the modal analysis transformation to determine a solution for the free-vibration
response of this simple 2-DOF system with initial conditions prescribed as:
[ ~! (0)]
~2 (0)
= [0]
~
[ ~1 (0)] [0].
~2(0)
=
0
(b) Plot and comment upon the system responses ~~ (t), ~ 2 (t), over the range 0 < t < 8/m,
where m= ~k/m.
Section C
Q12.
I 0 results for drained angle of shearing resistance, cf>, of a sand are measured as follows:
30.8 34 34.9 35.8 36.3 36.9 37.1 37.6 38.3 39.9
(i = 1) (i = 10)
(a) Find the sample mean value m, standard deviationS and standardized variate u, of cp.
(b) Taking a sample probability value P; = i I (n + 1) and assuming Normal distribution,
calculate the theoretical quantile E, for each sample.
(c) Prepare a Q-Q plot of u, versus Er
(d) Discuss whether the assumption ofNormal distribution is reasonable for the test
results.
[TURN OVER]
Ql3. (a) A discrete random variable x takes only three values: 1, 2 and 3. Assume all
values have equal probability. Calculate the population mean value J-1.,
variance cr 2 and standard deviation a of x.
(b) If we sample the population of x with replacement with a sample size of 2 (i.e. two
samples are taken each time), there are only 9 possible outcomes for each sampling as
follows:
(1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (3,1) (3,2) (3,3)
Using this special case, verify that ~ is an unbiased estimator of a2, but s is not an
unbiased estimator of the population standard deviation a.
Note: The formula for sample variance is as follows:
s2 = _l_L (xt -mi
n-l t
where n is the sample size and m is the sample mean value.
END OF PAPER
