9210-100 Sample Question Paper v1-0

8/12/2019 9210-100 Sample Question Paper v1-0

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Sample Question Paper for 9210-100Graduate Diploma in Engineering

Engineering mathematics

D uration: three hours

You should have thefollowing for this examination one answer book drawing instruments non-programmable calculator

The following data are attached percentage points of the

chi-squared distribution percentage points of the

t-distribution table of the standard Normal

probability distribution

The City and Guilds of London Institute

General instructions This paper consists of nine questions in three sections A, B and C. Answer five questions, at least one from each section. An electronic calculator may be used but candidates must show sufficient steps to justify their answers.

Drawings should be clear, in good proportion and in pencil. All questions carry equal marks. The maximum marks for each section within a

question are shown.


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Section A

1 a Function u(x, y) is such that 022

2

2

yu

xu

. (4 marks)

If 22 t s x and st y 2 ,

find 22

2

2

t u

su

.

b The cost ),( y xC of making a single unit of product in a production facility is given by23 3128),( y xy x y xC

where x and y are the material and labor costs respectively, required to make a

single unit.

i Locate the stationary points of C (4 marks)

ii Determine their nature. (4 marks)

c A plate with the measurements shown in Figure Q1c is to be constructed (8 marks)

so that the outer perimeter has a length of 30 cm. Use the Lagrange multiplier

method to find lengths for x, y and z in order that the area of the plate is the

maximum possible.

z z

zz

y y

x

Figure Q1c


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2 a For a vector function F and a scalar function it is given that F F F .).().(

i Show that 3. r where zk yj xir (2 marks)

ii Using the above two equations show that (6 marks)22 5).( r r r

iii Hence evaluate s

r r )( 2 . n ds , (6 marks)

where S is the surface on a cylinder x 2 + y2 = 2, 0 z .

b If E and H are respectively the electrical and magnetic field vectors in a charge-free,

current-free electro-magnetic field in free space, then it is known that they satisfy the

equations,

. 0 E

. 0 H

t H

c E

1

t E

c H

1

where c is the velocity of light in free space.

Show that

i x 221t E

ct H

and (3 marks)

ii 22

2

1t E

c xE x (3 marks)

Hence evaluate s (r 2 r ) . n ds ,

(2 marks)


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3 a i State the Cauchy -Riemann equations for an analytic function f(z) (2 marks)

where

),(),()( y x jv y xu z f , jy z

ii Given that x y xu ),( )sin( x y y )cosh(

)cos( x )sinh( y (2 marks)

x y xv ),( )cos( x y y )sinh( )sin( x )cosh( y

Find such that),(),()( y x jv y xu z f

is an analytic function in the z-plane.

iii By assigning the values of obtained in part ii above to ),( y xu , show (4 marks)

that )( z f can be expressed as z z f )( )sin( z .

[Note: )cos()cosh( jy y and j )sin()sinh( jy y ]

b C is the closed curve obtained by joining points ( -1, 0) and (1, 0) along the x- axis

with the semi circle 1 z and with y > 0. Given jy z and with integration

carried out in the counter-clock wise sense,

i evaluate C

dz z

z z

)1(

)sin(4 . (6 marks)

ii Hence evaluate dx x

x x)1()sin(

4 . (6 marks)

c

8


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4 a Calculate the Laplace Transform of t 2sin . (2 marks)

b Calculate the inverse Laplace Transform of (6 marks)

)9)(1(

322 S S

c Determine the Laplace Transform )}({ t u L of the unit step function )(t u with (2

marks)

1)( t u , 0t and 0)( t u , 0t .

d In the flight of a helicopter, the pitch angle is controlled by adjusting the rotorangle , where and satisfy the differential equation,

64.022

dt d

dt d

with 0 and 0dt d

, at 0t . Also taking )(t u , as defined in part c,

i obtain the equation satisfied by , where )}({ t L . (5 marks)

ii Hence determine as a function of t . (5 marks)

5 a Solve using the Z Transform method, the difference equation (6 marks)042 nn y y , 10 y , 01 y

b Obtain the half range Fourier sine series expansion for (6 marks)

2

)(4)(

x x

x f

, x0 .

c The temperature ),( t x of a rod of length at a point distan ce x from one end

at time t , satisfies the differential equation

2

22

dxd

cdt d

, x0 , 0t ,

with 0),0( t , and 0),( t , for all 0t .

i Show by the use of the variable s separable method that the solution for is (4 marks)

1

)sin(),(22

nn

t nc nx Bet x

ii By taking )()0,( x f x as defined in part b find the solution for ),( t x . (4 marks)

(2 marks)


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6

Circular currents together with the resistances and voltage in the circuit

shown in Figure Q6a satisfy the equations,

0)(

0)()(

)(

33232

322121

211

i Rii R

ii Rii R

V ii R

i By taking R1 = 1 , R2 = 4 , R3 = 2 and V = 1 volt, write down (2 marks)

the system of equations as

A bi , where ],,[ 321 iiiiT .

and with the matrix A , having positive diagonal terms.

ii Use a matrix factorization method to evaluate the currents i1, i2 and i3. (6 marks)

b Starting from point (1, 1) obtain the next iteration point in a search for (6 marks)the minimum point of the function 2)1()1(2 22 xy y x by use of

the steepest gradient method.

)(

)()(

)(

2323

2321212

12111

x x K x M

x x K x x K x M

x x K x M

T aking 21 M , 1 , 121 K K and ii x x2

, i = 1, 2, 3, it can be

shown that the system of equations can be written as A x x 2 ,

Section B

6 a

Figure Q6a Figure Q6c

R 1 R 2 R 3

i 1 i 2 i 3

V

x1

x2 x3 M1 M M

K 1 K 2

c Two masses each of mass M coupled with two springs of spring constants (6 marks)

K 1 and K 2 can move on a smooth trolley of mass M 1 which also can move

on a smooth horizontal table. If the trolley and the two masses have

displacements x 1 , x 2 and x 3 as show n in Figure Q6c, then it is known

that the displacements satisfy the equations of motion.


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where

110

121

021

21

A

],,[ 321 x x x xT

and is the frequency of oscillations of the system.

Given ]101[][ )0( T x , perform two iterations to determine the maximum

frequency of vibrations .


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Section C

7 a Liquid ispoured into a cylindrical tank of uniform cross section A at a rate Q.

The tank has an orifice at the base, causing the level of liquid in the tank x(t) to

satisfy the differential equation

][1

2

1

xQ Adt dx

where is a constant.

Taking Q = 0.3, A = 1, and = 0.01, with x(0) = 0.5. Determine x

i at t = 0.1, 0.2, by the use of the Euler Method, (6 marks)

ii at t = 0.1, by the use of the second order Runge-Kutta (RK2) method. (6 marks)

[For the equation ),( xt f dt dx

, at ),( 00 xt , the RK2 method is given by

)](*5.0

,

),,(

),,(

2101

01

1002

001

k k x x

ht t

k xht f hk

xt f hk

b The temperature ),( t xu in a rod at distance x along the rod and time t satisfies (8 marks)

the differential equation

,22

x

u

t

u ,11 x 0t

with, U(X, 0) = 3( 1 - | x | ) 11 x ,

and where the two ends of the rod are kept at ,0),1( t u 0t

Find, ),( ji t xu for ,31

i xi ,271

jt j

,0i ,1 2

,1 j ,2 3 by the use of a suitable scheme.


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8 a A factory has utilized two machines A and B to produce pistons for engines

with an intended diameter of 10.00 cm. A sample of pistons produced by each

machine gives the results in Table Q8a.

MachineNumber

ofItems

MeanDiameter

(cm)

StandardDeviation

(cm)

A 9 10.02 0.02

B 9 10.01 0.01

Table Q8a

i Examine statistically whether there is a significant difference between the (6 marks)

diameters of the pistons produced by the two machines ;

[Assume that diameters of items produced both machines are normallydistributed with equal variances.]

ii Find the mean and standard deviation of the 18 items obtained by combining (2 marks)

the two samples.

iii The factory also produces cylinders with internal diameter 10.03 cm and (6 marks)

standard deviation 0.02 cm.

Calculate the percentage of pistons that can fit to the cylinders.Assume that diameters of cylinders and also of pistons are normally

distributed and the mean and standard deviation of diameters of the

pistons are those found in part ii.

b The effort in person-months(E) required to complete a number of software

development projects with Lines of Code in units of 1000 (KLOC) is shown

below in Table Q8b.

KLOC(x)

1.0 1.4 2.1 2.5 3.1

E (y) 0.6 2.0 2.4 2.8 3.2

Table Q8b


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Given that

0.2 x

2.2 y

14.3))(( y y x x ii

828.2)( 2 x xi

0.4)( 2 y yi

i Find the correlation coefficient between x and y values. (4 marks)

ii If the equation bxa y , has been obtained by the use of (2 marks)

Least Squares Curve Fitting to the set of results for x and y, and

given that 043.0a , find the value of b.

9 a A machine in a production unit has to be shut down in the event of a mechanical

failure or an electrical failure. These occur at an average of 1 mechanical failure

and 2 electrical failures per month. Given that the failures are described by Poisson

distributions, find the probabilities that in a month there will be,

i no shut downs (3 marks)

ii at least one shut down. (3 marks)

iii If each shut down costs the company 10 000 units of currency, estimate (4 marks)the average monthly cost incurred by the shut downs.

b Daily conditions regarding rainfall at a hydroelectric power station has been (4 marks)

classified as dry, showery and heavy rain and the Table Q9b summarises the

changes in conditions over a period of 50 days.

Current daystatus

Following day Status

Dry Showery Heavy Rain

Dry 14 4 2

Showery 6 12 2

Heavy Rain 2 6 2Table Q9b

Determine the probability transition matrix for a Markov chain model of the weatherconditions.

c Using the results from part (b) and given that the current conditions are showery,

find the probabilities of having different conditions after

i 1 day (3 marks)ii 2 days. (3 marks)


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Data Attachments9210-100


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9210-100 Sample Question Paper v1-0