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MA2006 2019-2020 Semester 1
Engineering Mathematics (Nanyang Technological University)
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MA2006
NANYANG TECHNOLOGICAL UNIVERSITY
SEMESTER 1 EXAMINATION 2019-2020
MA2006 – ENGINEERING MATHEMATICS
November/December 2019
Time Allowed: 2½ hours
INSTRUCTIONS
1.
This paper contains FOUR (4) questions and comprises FIVE (5) pages.
2.
Answer ALL questions.
3.
All questions carry equal marks.
4.
This is a CLOSED-BOOK examination.
5.
Mathematical tables and formulae are provided on pages 4 and 5.
1(a) For the two nn matrices A and B, given:
A + B + AB = 0 (all the elements of the matrix 0 are 0)
Prove: AB =BA.
(7 marks)
(b) Consider the following system of linear equations,
x1 + 2x2 + ax3 = 3
ax2  4x3 = 6
2x1 + 5x2 + ax3 = b
where a and b are unknown real constants. Find the values of a and b such that the
equations have
(i) unique solution,
(ii) no solution and
(iii) infinitely many solutions.
(9 marks)
(c) For the matrix
0  6
7

A   0  7  6 ,
 6  6 a 
where a is a real number. It is known that matrix A has an eigenvalue 11. Find the value of
a, all the eigenvalues and eigenvectors of matrix A.
(9 marks)
1
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In Question 2, Cartesian coordinates are denoted by the usual symbols x , y and z .
2(a) It is given that  and  are functions of x , y and z . If div(grad( ))  0 and
div(grad( ))  0 at all points ( x, y, z ) in space, while S is any given enclosed surface, use
Gauss theorem to prove that
 S (n  [ grad( )]  n  [ grad( )]) d  0 .
Gauss theorem: if T is the region enclosed by the closed surface S whose outward unit
normal vector is n and if u is a vector function of x , y and z such that div(u) exists in
T , then
 u  n d   div(u) dV .
S
T
(7 marks)
(b)
Evaluate each of the following line integrals:
(i)
 xds , where C is the portion of the curve x = 2t, y = t2 and z  t between (2,1,-1)
C
and (4,4,-2).
(4 marks)
(ii)
 F d r , where F = xy i + z j +y k, dr = dx i+ dy j + dz k and C is the directed
2
C
straight line segment from (0, 0, 0) to (1, 1, 1).
(5 marks)
(c) Let S be the portion of the surface z = 1+ x2 + y2 for z ≤ 10. Find the area of the surface S.
(9 marks)
2
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3(a) A periodic function is defined in one period as


0 x ;
1,
2
f ( x)  
 4 (  x),   x  .
2
 
(i)
Sketch f(x) over the interval   x   .
(3 marks)
(ii)
Find the Fourier series for f(x).
(10 marks)
(iii) By using the Fourier series obtained in 3(a) (ii), find the value of following infinite
series:
1 1
1
1  (1) n
1    
, n=1, 2,3,   are integers.
9 25 49
2n 2
(4 marks)
(b) If the function f(x) in 3(a) is not periodic but is only defined in the interval 0  x  , a
Fourier sine series can be obtained for f(x) by a proper expansion.
(i)
Sketch the expansion for f(x) over the interval   x  3 .
(4 marks)
(ii)
Find the Fourier sine series for f(x).
(4 marks)
4(a) Find the Laplace transform to the function f(t) defined as
0,
0t  ;
2


f (t )  t sin(t ),   t  ;
2

 .
0,
t

(9 marks)
(b) Find the inverse Laplace transform


s 2  s  20
L1 
.
2
 ( s  6)( s  2s  26) 
(6 marks)
(c) Solve the following ordinary differential equation for y(t) by using the Laplace transform.
(10 marks)
0, 0  t  1;

y  3 y  2 y  1, 1  t  2;
y (0)  y(0)  0.
0, t  2.

3
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FORMULAE FOR VECTOR CALCULUS PART
 






1. a  a1i  a2 j  a3k , b  b1i  b2 j  b3k ,
 
a  b  a1b1  a2b2  a3b3 ,
  
i
j k



 
a  b  a1 a2 a3  a2b3  b2 a3 i  a3b1  b3 a1  j  a1b2  b1a2 k .
b1
b2
b3
f  f  f 
2. grad f  f  i 
j  k , for scalar function f ( x, y, z ).
z
y
x




3. V  px, y, z i  qx, y, z  j  r x, y, z k ,

 p q r
 ,

  V  div V 
x y z



i
j
k

 

  r q   p r    q p  
  V  curl V 
   i     j    k ,
x y z  y z   z x   x y 
p q
r
2
2
2
 dx   dy   dz 
4. ds  dx   dy   dz           dt ,
 dt   dt   dt 
2
2
2
2
2
2
 dx   dy   dz 
C f x, y, z ds  t1 f xt , yt , zt   dt    dt    dt  dt.





 





5. r  xi  yj  zk , dr  dx i  dy j  dz k , For F  px, y, z i  qx, y, z  j  r x, y, z k ,
 
t2 
dz 
dy
dx
C F  dr   pdx  qdy  r dz  t1  p dt  q dt  r dt dt.
6. Green's Theorem
 g f 
C f  x, y  dx  g  x, y  dy  R  x  y  dA
7. Surface integral:
t2
2
 f   f 








g
x
y
z
d
g
x
,
y
z
,
x
y
,
,
,
1

     dA, for the surface given by z  f x, y .
S
R
 x   y 
2
FORMULAE FOR SPECIAL FUNCTIONS
sin( x  y )  sin x cos y  cos x sin y
sin x sin y  [  cos( x  y )  cos( x  y )] 2
cos( x  y )  cos x cos y sin x sin y
cos x cos y  [cos( x  y )  cos( x  y )] 2
sin x cos y  [sin( x  y )  sin( x  y )] 2
sin 2 x  (1  cos 2 x ) 2,
cos 2 x  (1  cos 2 x ) 2
x y
x y
x y
x y
sin x  sin y  2sin
cos
,
sin x  sin y  2 cos
sin
2
2
2
2
x y
x y
x y
x y
cos x  cos y  2 cos
cos
, cos x  cos y  2sin
sin
2
2
2
2
x
x
x
x
sinh x  ( e  e ) 2, cosh x  ( e  e ) 2 , sinh x  i sin ix , cosh x  cos ix
4
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FORMULAE FOR FOURIER SERIES
Euler Formulae for Fourier series for a periodic function f ( x ) with a period P  2 L

f ( x )  a0   ( an cos
n 1
nx
nx
bn sin
)
L
L
1
1
1
nx
nx
f ( x ) dx , an   f ( x ) cos
dx , bn   f ( x ) sin
dx
where a0 

L L
L L
L
L
2L L
L
L
L
Complex form of Fourier series for f ( x )

f ( x )   cn einx / L where cn 
n 
L
1
f ( x ) e  inx / L dx , n  0, 1, 2,

2L L
LAPLACE TRANSFORM TABLE
F ( s )  L{ f ( t )}
f ( t )  L1 { F ( s )}
1
t n 1
( n  1)!
t n 1e at
( n  1)!
sin t

cos t
1
s
1
, ( n  1, 2,
sn
)
1
, ( n  1, 2,
( s  a )n
)
f ( t )  L1 { F ( s )}
F ( s )  L{ f ( t )}
sinh at
a
cosh at
1
s  a2
s
s2  a2
2
u ( t  a ) , Unit step function
 ( t  a ) , Unit impulse
function
1
s 2  2
s
2
s  2
LAPLACE TRANSFORM FORMULAE
F ( s )  L{ f ( t )}
f ( t )  L1 { F ( s )}
e as
s
e as
Remarks
f ( n ) (t )
s n F ( s )  s n 1 f (0) 
t
0 f ( )d 
F (s) s
Integration of function
eat f  t 
u (t  a ) f (t  a )
F (s  a)
Shift on s-axis
e as F ( s )
 F '( s )   dF / ds
Shift on t-axis
tf ( t )
 f ( n 1) (0)
Differentiation of function
Differentiation of transform
Differentiation of transform
n
t f (t )
( 1) F
f (t ) t
s F ( s ) ds
Integration of transform
L( f * g )  L( f ) L( g )
Convolution
t
0 f (t  ) g ( )d 
n
(n)
(s)

t
  f (  ) g ( t   )d 
0
f (t )  f (t  p )
p  st
0 e
f (t ) dt (1  e sp )
f ( t ) is periodic with period p
End of Paper
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