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Sample Question Paper for 9210-100
Graduate Diploma in Engineering
Engineering mathematics
Duration: three hours
You should have the
following 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
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.
© The City and Guilds of London Institute
Section A
1 a
Function u(x, y) is such that
w 2u w 2u
wx 2 wy 2
0.
(4 marks)
If x s 2 t 2
and y 2 st ,
find
b
w 2u w 2 u
.
ws 2 wt 2
The cost C ( x, y ) of making a single unit of product in a production facility is given by
C ( x, y ) 8 x 3 12 xy 3 y 2
where x and y are the material and labor costs respectively, required to make a
single unit.
c
i
Locate the stationary points of C
(4 marks)
ii
Determine their nature.
(4 marks)
A plate with the measurements shown in Figure Q1c is to be constructed
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.
x
y
y
z
z
z
z
Figure Q1c
2
(8 marks)
2
a
F and a Fscalar function M it is given that
For a vector
For afunction
vector function
’.(MF )
(’M ).F M’.F
3 where r xi yj zk
i
Show
i that
S ’. r
ii
Using
above
equations
show that
ii the
Using
thetwo
above
t
’.(r r )
2
³³∫∫
5r
2
(2 marks)(2 marks)
(6 marks)(6 marks)
2
2
((r 2 r))..nndsds, ,
iii Hence
Hence
evaluate
iii evaluate
Hence evaluate
ss
(6 marks)(6 marks)
where Swhere
is the surface
a cylinder
S is the on
surface
on a x2 + y2 = Ǡ2, 0 ч z ч Ǡ .
b
If EbandIfH are respectively the electrical and magnetic field vectors in a charge-free,
current-free
electro-magnetic
field in free
it isthen
known
theythat
satisfy
current-free
electro-magnetic
fieldspace,
in freethen
space,
it isthat
known
theythe
satisfy the
equations,
equations,
’.E
0
’.H
0
’xE
’xH
1 wE
c wt
1 wH
c wt
where c is the velocity of light in free space.
Show that
wH
wt
i
’ xi
ii
’x’
ii xE
1 w2E
and
c wt 2
1 w2E
c 2 wt 2
(3 marks)(3 marks)
(3 marks)(3 marks)
2
3
See next page
a
i
State the Cauchy-Riemann equations for an analytic function f(z)
(2 marks)
where
u ( x, y ) jv ( x, y ) , z
f ( z)
ii
Given that u ( x, y )
x jy
x sin( x ) cosh(Dy ) y cos( x) sinh( y )
(2 marks)
x cos( x) sinh( y ) y sin( x ) cosh( y )
v ( x, y )
Find D such that
f ( z)
u ( x, y ) jv ( x, y )
is an analytic function in the z-plane.
iii By assigning the values of D obtained in part ii above to u ( x, y ) , show
that f ( z ) can be expressed as f ( z )
[Note: cosh( y )
b
(4 marks)
z sin( z ) .
cos( jy ) and j sinh( y ) sin( jy ) ]
C is the closed curve obtained by joining points ( -1, 0) and (1, 0) along the x- axis
with the semi circle z
1 and with y > 0. Given z
x jy and with integration
carried out in the counter-clock wise sense,
z sin( z )
dz .
4
1)
f x sin( x )
dx .
ii Hence evaluate ∫³
f ( x 4 1)
i evaluate
³∫c ( z
(6 marks)
C
8
3
(6 marks)
4
Calculate the Laplace Transform of sin 2 t .
(2 marks)
b Calculate the inverse Laplace Transform of
(6 marks)
4 a
3
2
(S 1)(S 2 9)
c Determine the Laplace Transform L{u (t )} of the unit step function u (t ) with
(2
(2marks)
marks)
u (t ) 1 , t t 0 and u (t )
0 , t 0.
d In the flight of a helicopter, the pitch angle T is controlled by adjusting the rotor
angle G , where T and G satisfy the differential equation,
with T
5 a
b
d 2T
dT
0.4
6G
2
dt
dt
dT
0 and
0 , at t 0 . Also taking G
dt
i
obtain the equation satisfied by I , where I
ii
Hence determine T as a function of t .
L{T (t )} .
(5 marks)
(5 marks)
Solve using the Z Transform method, the difference equation
yn 2 4 yn 0 , y0 1 , y1 0
(6 marks)
Obtain the half range Fourier sine series expansion for
(6 marks)
4 x (S x)
f ( x)
c
u (t ) , as defined in part c,
S2
, 0d x dS .
The temperature T ( x, t ) of a rod of length S at a point distance x from one end
at time t, satisfies the differential equation
with T (0, t )
i
dT
dt
d 2T
, 0 d x d S , t t 0,
dx 2
0 , and T (S , t ) 0 , for all t t 0 .
c2
Show by the use of the variables separable method that the solution for ƍ is
T ( x, t )
f
¦e
c 2 n 2t
(4 marks)
Bn sin( nx )
n 1
ii
By taking T ( x,0)
f ( x ) as defined in part b find the solution for T ( x, t ) .
5
(4 marks)
See next page
Section B
x1
6 a
i2
i1
K1
R2
M
M
V
R1R
x3
x2
M1
i3
K2
R3
Figure
Q6a
Figure
Q6a
Figure
Q6c
Figure
Q6c
Circular currents together with the resistances and voltage in the circuit
shown
s
in Figure Q6a satisfy the equations,
R1 (i1 i2 ) V
R1 (i2 i1 ) R2 (i2 i3 )
R2 (i3 i2 ) R3i3
0
0
By taking R1 = 1 ƺ, R2 = 4 ƺ, R3 = 2 ƺ and V = 1 volt, write down
i
(2 marks)
the system of equations as
Ai
b , where iT
[i1 , i2 , i3 ] .
and with the matrix A , having positive diagonal terms.
ii
b
Use a matrix factorization method to evaluate the currents i1, i2 and i3.
Starting ffrom point (1, 1) obtain the next iteration point in a search for
(6 marks)
(6 marks)
the minimum point of the function 2( x 1) 2 ( y 1) 2 xy 2 by use of
the steepest gradient method.
c
Two masses
masses each
each of
of mass
mass M coupled with two springs of spring constants
Two
K1 and K2 can move on a smooth trolley of mass M1 which also can move
on a smooth horizontal table. If the trolley and the two masses have
displacements x1, x2 and x3 as show n in Figure Q6c, then it is known
displacements
that the displacements satisfy the equations of motion.
M1x1 K1 ( x2 x1 )
M x2 K1 ( x2 x1 ) K 2 ( x3 x2 )
M1
Ta
Taking
2, M
M x3
K 2 ( x3 x2 )
1 , K1
K 2 1 and xi
Z 2 xi , i = 1, 2, 3, it can be
shown that the system of equations can be written as
Ax
Z2x ,
6
(6 marks)
marks)
(6
1
§1
¨
2
¨2
where A ¨ 1 2
¨ 0 1
¨
©
xT
[ x1 ,
·
0¸
¸
1¸
1 ¸¸
¹
x 2 , x3 ] and Z is the frequency of oscillations of the system.
Given [ x ( 0) ]T
[1 0 1] , perform two iterations to determine the maximum
frequency of vibrations Z .
7
See next page
Section C
7 a
Liquid is poured 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
dx
dt
1
1
[Q D x 2 ]
A
where D is a constant.
Taking Q = 0.3, A = 1, and D = 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
dx
dt
f (t , x) , at (t0 , x0 ) , the RK2 method is given by
k1
h f (t 0 , x 0 ),
k2
h f (t 0 h , x 0 k1 ),
t1
x1
b
t0 h,
x 0 0 . 5 * ( k1 k 2 )]
The temperature u ( x, t ) in a rod at distance x along the rod and time t satisfies
the differential equation
wu
wt
w 2u
, 1 d x d 1, t t 0
wx 2
with, U(X, 0) = 3( 1 - | x | ) 1 d x d 1 ,
and where the two ends of the rod are kept at u ( r1, t )
Find, u ( xi , t j ) for xi
i
1
1
i, t j
j,
3
27
0, r 1, r 2
j 1, 2, 3
by the use of a suitable scheme.
8
0, t t 0
(8 marks)
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.
Machine
Number
Mean
Standard
of
Diameter Deviation
Items
(cm)
(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 normally
distributed 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
9
See next page
Given that
x
2.0
y
2.2
¦ (x
i
x)( yi y )
¦ (x
i
x) 2
2.828
y)2
4. 0
¦(y
i
3.14
i
Find the correlation coefficient between x and y values.
ii
If the equation y
(4 marks)
a bx , 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 a
0.043 , 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
the average monthly cost incurred by the shut downs.
(4 marks)
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 day
status
Following day Status
Dry
Showery
Heavy Rain
Dry
14
4
2
Showery
6
12
2
Heavy Rain
2
6
2
Table Q9b
Determine the probability transition matrix for a Markov chain model of the weather
conditions.
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
ii 2 days.
(3 marks)
(3 marks)
10
Data Attachments
9210-100