advanced civil engineering mathematics 2012_13 main-round examination paper(8)

8/11/2019 Advanced Civil Engineering Mathematics 2012_13 Main-round Examination Paper(8)

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Advanced Civil Engineering Mathematics (CBE2026) Page 1 of 7

AY 2012/2013 Autumn Semester Main-round Exam [51301F/L4] FT

H o n g K o n g I n s t i t u t e

o f

V o c a t i o n a l E d u c a t i o n

HD in Civil Engineering

(51301F)

A d v a n c e d

C i v i l E n g i n e e r i n g

M a t h e m a t i c s

( C B E 2 0 2 6 )

Notes :

1.

Answer ALLquestions in Section Aand any

FOUR 4)questions inSection B. AY 2012/13

2.

Section A is worth a total of FORTY 40)marks Autumn Semester

and each question in Section Bis worth Main-round Examination

FIFTEEN 15)marks.3.

All works must be clearly shown.

4. This question paper hasSEVEN 7)pages. 15 January 2013

5. This question paper contains EIGHT 8) 1:30 pm 4:30 pm

questions in Section A and FIVE 5)questions in

Section B.


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You are required to answer ALLquestions in Section Aand any FOURquestions in

Section B.

Section A (40 marks)

You are required to answer ALLquestions in this section.

A.1 State the relationship between m and n. It is given that

= (0 4 61 3 27 5 2) and ||= ||. (4 marks)

A.2 Solve that following equation by using inverse matrix method.

7 11 = 177

13 15 = 269 (5 marks)A.3 Show that = satisfies the differential equation

= . (5 marks)

A.4 A series of three resistors are connected in parallel, its equivalent

resistance can be presented by

=

++ , find the percent

change inRifx,y, andzincrease by 1 , 0 and -2 , and

3,2,1,, zyx . (5 marks)

A.5 Find the general solution of the following differential equation

= + (5 marks)

A.6 Find the particular solution of the following differential equation.a 4 7 = 0 with conditions u(0) = 1 and 0= 1.Where a is a constant. (5 marks)

A.7 Find the double integral of a function , = over theplane region R bounded by thex-axis and the curve = with 0 . (6 marks)


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A.8 Apply Newton-Raphson method to find the root of the equation

= sin near to t= 2. Give your answer to 5 decimal places. (5 marks)


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Section B (60 marks)

You are required to answer any FOURquestions in this section

Each question is worth FIFTEENmarks.

B.9 (a) The height of an object thrown into the air is known to be

given by a quadratic function of time t. If the object is at height

of , 7 and 2 are 0.5, 1 and 2 respectively, determine the

coefficients of them. (7 marks)

(b) Find the range of value(s) of a for the following system of

linear equation which has an unique solution.

=

2 3 1 = 03 4 1 = (4 marks)(c) Find the the eigenvalues and eigenvectors of the matrix

. (4 marks)

B.10 A meteorologist uses the following formula to find the lowestpressure (in hPa) of cyclone into hurricane Katrina by introducing

Cartesian Plane.

, = 7570You should correct your answer to 6 decimal places if necessary

unless otherwise specified.

(a)

Find the critical centre of the pressure. (13 marks)

(b) Hence determine the pressure of cyclone. Correct your answer

to the nearest hPa. (2 marks)


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B.11 (a) (i) In a coal investigation, four points are positioned to find

the depth of the stratum. Based on the information from

surveyor, these points are (1, 4), (2, 15) and (4, 85) and

(0,1) in Cartesian Plane from origin respectively. The

outcrop pattern has been determined as approximate as a

cubic function. Find the equation of the function. (5 marks)

(ii) After approximating function, they would like to find the

volume of the coal. By using Simpsons Rule, estimate the

volume of it with 16 strips. It is known that the thickness

and length of the coal are 2 kmand 10 km respectively.

And x and y 0. (5 marks)(b) Find the general solution of the following differential equation.

1 1 = 0 (5 marks)


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B.12

The figure on above shows three tanks A, Band C. Water is

circulated among the three tanks. Let x, yand z (in litres) be

the volume of water contained in tanks A, Band Crespectivelyat time t(in hours). At t = 0, tank A contains 1000 litres of

water, while tanks Band C are empty. Water flows from tank

A to B at a rate of 5xlitres per hour and from tank B to C

at a rate of 3ylitres per hour. At the same time, water is pumped from

tank C back to B through another pipe at rate ofzlitres per

hour.

(a) Find x in terms of t, (2 marks)

(b) Set up two simultaneous differential equation for y and z.

Hence

(i) show that 4 = 15000.

(ii) find z in terms of t. (10 marks)

(c) Describe the long term behaviour of the amount of water in

each of three tanks A, B and C. (3 marks)


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B.13 A lamina, which is bounded by = , = 0 and = 5, islocated into Cartesian Plane. The density of the plate, = 1.Determine the following:

(a) the mass of the lamina. (3 marks)

(b) the location of the centroid of the lamina from thexaxis and

yaxis and ; (6 marks)

(c) the moment of inertia,IxxandIyyof the lamina. (6 marks)

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advanced civil engineering mathematics 2012_13 main-round examination paper(8)

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