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VIRTUAL
UNIVERSITY STUDENT
MTH301 FINAL TERM
SOLVED PAPER
SOLUTON
MUHAMMAD ASIF
2012
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MASIF2024@GMAIL.COM
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FINALTERM EXAMINATION
MTH301
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MASIF2024@GMAIL.COM
Question No: 2
( Marks: 1 ) - Please choose one
All axes are negative in ----------- octant
► Second
► Fourth
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► Seventh
► Eighth
Question No: 3
( Marks: 1 ) - Please choose one
In three dimensional space, intersection of two surfaces is a --------------
► Plane
► Sphere
► Curve
► None of these
Question No: 4
( Marks: 1 ) - Please choose one
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2 f 2 f 2 f


0
2 x 2 y 2 z
For a function f ( x, y, z ) , the equation
► Gauss Equation
► Euler’s equation
► Laplace’s Equation
► Stoke's Equation
Question No: 5
Is the function
f ( x, y , z ) 
►
►
( Marks: 1 ) - Please choose one
f ( x, y , z )
continuous at origin? If not, why?
x3
 sin z
y
f ( x, y , z )
f (0, 0, 0)
is continuous at origin
is not defined
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is known as -------------
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lim
►
f (0, 0, 0)
( x , y , z ) (0, 0, 0)
is defined but
does not exist
lim
f (0, 0, 0)
►
not equal.
f ( x, y , z )
( x , y , z ) (0, 0, 0)
f ( x, y , z )
is defined and
Question No: 6
exists but these two numbers are
( Marks: 1 ) - Please choose one
Match the following vector-valued function with its graph.
^
^
^
r (t )  cos t i  sin t j  2 k
and
0  t  2
►
►
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►
►
Question No: 7
( Marks: 1 ) - Please choose one
Match the following vector-valued function with its graph.
^
^
^
r (t )  cos t i  sin t j  3 k
and
0  t  2
►
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►
►
►
Question No: 8
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( Marks: 1 ) - Please choose one
Match the vector-valued function with its graph.
^
^
^
r (t )   a cos t  i   a sin t  j   ct  k
where a and c are positive constants and
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0  t  2
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►
►
►
►
Question No: 10
( Marks: 1 ) - Please choose one
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What is the derivative of following vector-valued function?


r (t )  et , t 2 , sec 2t
2









r (t )  2tet , 2t , 2sec 2t tan 2t
2

►
r (t )  tet , 2t , sec 2t tan 2t
2

►
r (t )  2tet , 2t , tan 2t
2

►
r (t )  t 2et , 2t , sec 2t tan 2t
2

►
Question No: 11
( Marks: 1 ) - Please choose one
The following differential is exact
dz  ( x 2  y 2 ) dx  2 xy dy
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► True
► False
because
dz  ( x 2  y 2 ) dx  2 xy dy
p
 ( x2  y 2 )
y
 2y
Q
 2 xy
x
 2 y
hence
p Q

this is not exact diffrential
y x
That is why this is false
Question No: 13
( Marks: 1 ) - Please choose one
r 2  a 2 cos 2
a0
If
, then the equation, in polar co-ordinates, of the form
represent
which of the following family of curves?
► Leminscate
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► Cardiods
► Rose curves
► Spiral
Question No: 14
( Marks: 1 ) - Please choose one

Which of the following condition must be satisfied for a vector field F to be a
conservative vector field?

►
F 0

►
grad F  0

►
div F  0

►
curl F  0
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REASON
In conservative vector field the line of integral between the two points must be be

curl F  0
conservative the the
Question No: 15
( Marks: 1 ) - Please choose one
Match the following periodic function with its graph.
2  x
f ( x)  
1
0 x3
3 x 5
►
►
►
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►
Question No: 16
( Marks: 1 ) - Please choose one
Match the following periodic function with its graph.
3
4 x

f ( x )  7  x
3


0 x4
4  x  10
10  x  13
►
►
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►
►
Question No: 17
( Marks: 1 ) - Please choose one
What is the period of periodic function whose graph is as below?
►
►


BECAUSE it is Odd funtion
►
2
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►
2
Question No: 18
( Marks: 1 ) - Please choose one
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Let L denotes the Laplace Transform.
If
L{F (t )}  f ( s )
where
L{t F (t )}  
s
is a constant, then which of the following equation holds?
d
{ f ( s )}
ds
►
►
►
L{t F (t )}  f ( s  t )
L{t F (t )}  f ( s )

L{t F (t )}   f ( s ) ds
s
►
BEcause
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Suppose
, then
The limit (the improper integral) only exists if
Question No: 19
The function
( Marks: 1 ) - Please choose one
f ( x)  x 2 cos 2 x
is -------------
► Neither even nor odd
► Odd function
► Even function
BECAUSE
f ( x)  x 2 cos 2 x
f(x)=(E)(E)=EVEN FUNTION
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. So
is only defined for
.
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Question No: 21
Let the function
f
xx
( Marks: 1 ) - Please choose one
f ( x, y ) has continuous second-order partial derivatives
, f yy and f xy 
in some circle centered at a critical point ( x0 , y0 ) and let
D  f xx ( x0 , y0 ) f yy ( x0 , y0 )  f xy 2 ( x0 , y0 )
If
D  0 then ---------------
►
►
►
f
f
f
has relative maximum at ( x0 , y0 )
has relative minimum at ( x0 , y0 )
has saddle point at ( x0 , y0 )
► No conclusion can be drawn.
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Question No: 22
( Marks: 1 ) - Please choose one
The path of integration of a line integral must be -------------
► straight and single-valued
► continuous and single-valued
► straight and multiple-valued
► continuous and multiple-valued
Question No: 24
( Marks: 1 ) - Please choose one
What is the Laplace Inverse Transform of
 1 
L1 
  t 1
 s  1
►
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1
s 1
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 1 
t
t
L1 
 e  e
s

1


►
 1  t
L1 
 e
s

1


►
 1 
t
L1 
 e
s

1


►
BECAUE
As the Laplace transform is linear, the inverse Laplace transform is also linear. That is,
Of course, we also have
used.
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. Let us demonstrate how linearity can be
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Question No: 25
( Marks: 1 ) - Please choose one

(3x  2 y ) dx  (2 x  y ) dy
C
Evaluate the line integral
where C is the line segment from
(0, 0) to (2, 0).
►6
► -6
►0
► Do not exist
Question No: 26
( Marks: 1 ) - Please choose one

(2 x  y ) dx  ( x 2  y ) dy
C
Evaluate the line integral
0) to (0, 2).
► -4
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where C is the line segment from (0,
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► -2
►0
►2
Question No: 27
( Marks: 1 ) - Please choose one
Curl of vector function is always a --------------
► Scalar
► Vector
BECAUSE
In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a 3dimensional vector field. At every point in the field, the curl of that field is represented by
avector. The attributes of this vector (length and direction) characterize the rotation at that point.
Question No: 28
( Marks: 1 ) - Please choose one
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If p is the period of a function then that function is said to be periodic if
_______________
f ( x  p )  f ( x)
,
► For all values of x in the domain of f
► For positive values of x in the domain of f
► For negative values of x in the domain of f
BECAUSE
A function f is said to be periodic with period P if (for some nonzero constant P) we
have
for all values of x. If there exists a least positive[1] constant P with this property, it is
called the prime period. A function with period P will repeat on intervals of length P,
and these intervals are sometimes also referred to as periods.
Question No: 29
( Marks: 1 ) - Please choose one
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Which of the following is geometrical representation of the equation
dimensional space?
► Parabola
► Straight line
► Half cylinder
► Cone
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y  x2
, in three
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Question No: 30
( Marks: 1 ) - Please choose one
If a = 1 ,then the equation, in polar co-ordinates of the form r = a  represents which of
the following curve?
► cardioid
L{t}
► circle
► rose curve
► archimedian spiral
Question No: 33
( Marks: 2 )
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4
Find Laplace Transform of the function F (t ) if F (t )  t

L{t }   t 4e st dt
4
0

 t 4e st   4  st
4
L{t }  
  t e dt

 s 0 0
4 t 3e st  3 t 2e  st


0
0
s s
s s
4.3.2.  1
L{t 4 }  0 
(0  1)
s5
4.3.2.1
L{t 4 }  0 
(0  1)
5
s
4.3.2.1!
L{t 4 } 
s5
24
L{t 4 }  5
s
t4
L{t }  st
se
4
Question No: 34


0
2
 te  st dt
s
( Marks: 3 )

What is the arc-length of the curve
^
^
^
r (t )  (4  3t ) i  (2  2t ) j  (5  t ) k
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when
3t  4
?
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
^
^
^
r (t )  (4  3t ) i  (2  2t ) j  (5  t ) k
when
3t  4
solution :
here
x  4  3t
y  2  2t
z  5t
then
dx
dy
dz
3
,  2
, 1
dt
dt
dt
4
Arc length  
3
4
Arc length  
2
2
2
 dx   dy   dz 
        dt
 dt   dt   dt 
 3   2   1
2
2
2
dt
3
4
Arc length   15dt
3
Arc length  0
Question No: 35
because here we can ' nt put lim it
( Marks: 3 )
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Determine the Fourier co-efficient
f ( x)  2 x  1
a0
of the periodic function defined below:
0 x2
SOLUTION
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f ( x)  2 x  1
f ( x) 
0 x2

1
a0   {an cos nwx  bn sin nwx}
2
x 1
2
 (2 x  1) sin nwxdx 
0
(2 x  1) sin nwx
cos nwx
 (
) dx
 nw

nw
0
0
2
2
5sin nw2 cos nw2
1  sin nw2 

 2 2
 0
nw
 nw
n w  nw

2

T
w
w
2
5sin nw2 cos nw2
1  sin nw2 


 2 2
nw
 nw
n w  nw 
1
1
bn 
(1  5cos n )  2 2 sin n 
nw
nw
4
bn  
nw

Question No: 36
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Define the periodic function whose graph is shown below.
SOLUTION:
0<x<4
6<x<10
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Question No: 37
( Marks: 5 )
Evaluate the following line integral which is independent of path.
 
1, 
 2

(e x sin y ) dx  (e x cos y ) dy
(0,0)
 
1, 
 2

(e x sin y ) dx  (e x cos y ) dy
(0,0)
p  xy
p
 e x cos y
y
p
 e x cos y
x
p p
I    (  )dxdy
y x
R

1 2
I     (e x cos y  e x cos y )dxdy
0 0
I 0
Question No: 38
( Marks: 5 )
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Determine the fourier co-efficient
f ( x)  x 2 
Question No: 39
an
of the following function.
0  x  2 
( Marks: 5 )

Determine whether the following vector field F is conservative or not.

^
^
^
F ( x, y, z )  (4 x  z ) i  (3 y  z ) j  ( y  x) k
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solution:

^
^
^
F ( x, y, z )  (4 x  z ) i  (3 y  z ) j  ( y  x) k
the
vector field will be conservative if
f g h


x y z
then
f
4
x
g
3
y
h
0
z
there is no function hold the consevative field
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