# Spring_2010_FinalTerm_OPKST_MTH301 by HinaNosheen

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```									                               FINALTERM EXAMINATION

Spring 2010

MTH301- Calculus II

Time: 90 min

Marks: 60

Student Info

Student ID:

Center:

Exam Date:

For Teacher's Use Only

Q No.         1     2    3       4         5        6    7    8    Total

Marks

Q No.         9     10   11      12        13       14   15   16

Marks

Q No.        17     18   19      20        21       22   23   24

Marks

Q No.        25     26   27      28        29       30   31   32

Marks

Q No.        33     34   35      36        37       38   39

Marks
Question No: 1       ( Marks: 1 ) - Please choose one

Q 1.   Intersection of two straight lines is --------------

►Surface

►Curve

a.         ►Plane

b.         ►Point

Question No: 2       ( Marks: 1 ) - Please choose one

Plane is a --------------- surface.

►One-dimensional

►Two-dimensional
►Three-dimensional

►Dimensionless

Question No: 3       ( Marks: 1 ) - Please choose one

Let w = f(x, y, z) and x = g(r, s), y = h(r, s), z = t(r, s) then by chain rule

w

r

w x w y w z
     
► x r y r z r

w x w y w z
     
r r r r r r
►

w x x w y y w z z
        
x r s y r s z r s
►
w r w r w r
     
r x r y r z
►

Question No: 4         ( Marks: 1 ) - Please choose one

What are the parametric equations that correspond to the following vector equation?

               ^                 ^
r (t )  sin 2 t i  (1  cos 2t ) j

x  sin 2 t ,       y  1  cos 2t , z  0
►

y  sin 2 t ,       x  1  cos 2t , z  0
►

x  sin 2 t ,       y  1  cos 2t , z  1
►

x  sin 2 t ,       y  cos 2t   , z 1
►

Question No: 5         ( Marks: 1 ) - Please choose one
What are the parametric equations that correspond to the following vector equation?

^        ^          ^
r (t )   2t  1 i  3 t j  sin 3t k

z  2t  1     ,       x  3 t    ,   y  sin 3t
►

y  2t  1     ,       x  3 t    ,   z  sin 3t
►

x  2t  1     ,       z  3 t    ,   y  sin 3t
►

x  2t  1     ,       y  3 t    ,   z  sin 3t
►

Question No: 6        ( Marks: 1 ) - Please choose one

What is the derivative of following vector-valued function?


r (t )  (cos5t , tan t , 6sin t )

 sin 5t                   
r (t )          , sec t , 6 cos t 
 5                        
►

              sin 5t
r (t )  (            , sec t , 6cos t )
5
►


r (t )  (5sin 5t , sec2 t , 6cos t )
►


r (t )  (sin 5t , sec2 t ,  6cos t )
►

Question No: 7             ( Marks: 1 ) - Please choose one

What is the derivative of following vector-valued function?


               3
r (t )   t 4 , t  1 , 2 
              t 


             1   6 
r (t )   4t 3 ,         , 3 
            t 1 t 
►

          1     6
r (t )   4t 3 ,       , 3
        2 t 1 t 
►


          1    6 
r (t )   4t 4 ,       , 3 
        2 t 1 t 
►


          1    6 
r (t )   4t 3 ,       , 3 
        2 t 1 t 
►

Question No: 8      ( Marks: 1 ) - Please choose one

The following differential is exact

dz  ( x 2 y  y ) dx  x dy

►True

►False
Question No: 9     ( Marks: 1 ) - Please choose one

Which one of the following is correct Wallis Sine formula when
n
is even and
n  2?


2
  n  1  n  3  n  5                 5 3 1

0
sin n x dx 
2     n      n  2  n  4

6 4 2
►


2
 n  1  n  3  n  5        6   4 2

0
sin n x dx 
n  n  2  n  4                  7   5 3
►


2
      n   n  2  n  4        6   4 2

0
sin n x dx 
2    n  1  n  3  n  5              5   3 1
►


2
 n   n  2  n  4        6    4 2

0
sin n x dx 
 n  1  n  3  n  5              5    3 1
►

Question No: 10      ( Marks: 1 ) - Please choose one
Match the following equation in polar co-ordinates with its graph.

r cos   a
where a is an arbitrary cons tan t

►

►

►

►
Question No: 11        ( Marks: 1 ) - Please choose one

(r ,  )
If the equation of a curve, in polar co-ordinates, remains unchanged after replacing              by
(r ,    )
then the curve is said to be symmetric about which of the following?

►Initial line

►y-axis

►Pole

Question No: 12        ( Marks: 1 ) - Please choose one

(r ,  )
If the equation of a curve, in polar co-ordinates, remains unchanged after replacing              by
(r ,  )
then the curve is said to be symmetric about which of the following?

►Initial line

►y-axis
►Pole

Question No: 13      ( Marks: 1 ) - Please choose one

x
f ( x)  sin
3
What is the amplitude of a periodic function defined by                       ?

►0

►1

1
3
►

►Does not exist

Question No: 14      ( Marks: 1 ) - Please choose one

f ( x)  4cos3x
What is the period of a periodic function defined by                      ?

4
►


3
►

2
3
►


►

Question No: 15     ( Marks: 1 ) - Please choose one

Match the following periodic function with its graph.

3       0 x4
f ( x)  
0       4 x6

►
►

►

►

Question No: 16    ( Marks: 1 ) - Please choose one

What is the period of periodic function whose graph is as below?

►2
►5

►6

►8

Question No: 17    ( Marks: 1 ) - Please choose one

What is the period of periodic function whose graph is as below?

►0

►4

►6

►8
Question No: 18        ( Marks: 1 ) - Please choose one

Let L denotes the Laplace Transform.

 F (t ) 
lim          
t 0
 t 
L{F (t )}  f ( s)           s
If                     where       is a constant.and                     exists then which of the following equation
holds?

 F (t ) 
L          f (s  a)
 t 
►

 F (t ) 
L          f (s  a)
 t 
►


 F (t ) 
L           f (s) ds
 t  s
►

 F (t )     d
L           { f ( s )}
 t          ds
►

Question No: 19        ( Marks: 1 ) - Please choose one
f ( s)
Which of the following is Laplace inverse transform of the function                         defined by
3    2
f ( s)         
s2   s
?

3te2t  2
►

3 e 2t  2t
►

3 e 2t  2
►

►None of these.

Question No: 20                ( Marks: 1 ) - Please choose one

( x1 , y1 , z1 )         ( x2 , y2 , z2 )
Let                      and                      be any two points in three dimensional space. What does the formula
( x2  x1 )2  ( y2  y1 )2  ( z2  z1 )2
calculates?

►Distance between these two points
►Midpoint of the line joining these two points

►Ratio between these two points

Question No: 21      ( Marks: 1 ) - Please choose one

P( x, y)         Q( x, y)
Let the functions              and              are finite and continuous inside and at the boundary of a

 P dx  Q dy 
closed curve C in the xy-plane. If                       is an exact differential then

C
  P dx  Q dy  

►Zero

►One

►Infinite

Question No: 22      ( Marks: 1 ) - Please choose one
What is Laplace transform of the function F (t ) if F (t )  t ?

1
L t 
►           s

1
L t 
►           s2

L t  e s
►

L t  s
►

Question No: 23          ( Marks: 1 ) - Please choose one

L{e5t }
What is the value of                if L denotes laplace transform?

1
L{e5t } 
►                s 5

s
L{e5t } 
►               s  25
2
5
L{e5t } 
►               s  25
2

5!
L{e5t } 
►               s6

Question No: 24         ( Marks: 1 ) - Please choose one


C
(3x  2 y) dx  (2 x  y) dy

Evaluate the line integral                                       where C is the line segment from (0, 0) to (0,
2).

►1

►0

►2

►-2
Question No: 25     ( Marks: 1 ) - Please choose one


C
(2 x  y) dx  ( x 2  y) dy

Evaluate the line integral                                      where C is the line segment from (0, 0) to (2,
0).

►0

►-4

►4

► not exist
Do

Question No: 26     ( Marks: 1 ) - Please choose one

Which of the following are direction ratios for the line joining the points (1, 3, 5) and (2,  1, 4) ?

► 2 and 9
3,

► -4 and -1
1,
► -3 and 20
2,

►0.5, -3 and 5/4

Question No: 27         ( Marks: 1 ) - Please choose one

If R  {( x, y ) / 0  x  2 and 1  y  4}, then


R
(6 x 2  4 xy 3 )dA 

4   2

 
1   0
(6 x 2  4 xy 3 )dydx

►

2   4

 
0   1
(6 x 2  4 xy 3 )dxdy

►

4   2

 
1   0
(6 x 2  4 xy 3 )dxdy

►
4   1

 
2   0
(6 x 2  4 xy 3 )dxdy

►

Question No: 28        ( Marks: 1 ) - Please choose one

Which of the following is true for a periodic function whose graph is as below?

►Even function

►Odd function

►Neither even nor odd function

Question No: 29        ( Marks: 1 ) - Please choose one
Which of the following is true for a function whose graph is given above

► odd function
An

► even function
An

►Neither even nor odd

Question No: 30      ( Marks: 1 ) - Please choose one

At each point of domain, the function ----------------
► defined
Is

► continuous
Is

► infinite
Is

► a limit
Has

Question No: 31      ( Marks: 2 )

Determine whether the following differential is exact or not.

dz  4 x3 y 3 dx  3x 4 y 2 dy

Solution:

dz  4 x3 y 3 dx  3x 4 y 2 dy
p
 12 x3 y 2
y
Q
 12 x3 y 2
X
p Q

y X
yes
Question No: 32        ( Marks: 2 )

Evaluate





sin nx dx

where n is an integer other than zero.

Solution:





sin nx dx


  cos nx 

 n     
  cos n cos n 
           
     n       n   
1
 ( cos n  cos n )
n
0

Question No: 33        ( Marks: 2 )

F (t ) if F (t )  e
3t

Find Laplce transform of the function

Solution:

L(e )   e3t  e  st
3t

0

  e  ( s 3)t .dt
0

e  ( s 3)t
{                }lim 0  
( s  3)
1            1
           ( ( s 3)t )
s 3 e
1
           (0  1)
s 3
1
          ......Ans
s 3

Question No: 34                   ( Marks: 3 )

a0
Determine the Fourier co-efficient                    of the periodic function defined below:

f ( x)  2 x  1                         0 x2

Solution:

1       

 
a            f ( x)dx


f ( x)  (2 x  1)
(0, 2)
2
  (2 x  1)dx
0
2
  x2  x
                      0

6
Question No: 35       ( Marks: 3 )

Determine whether the following differential is exact or not.

dz  (3x 2e2 y  2 y 2e3 x ) dx  (2 x3e2 y  2 ye3 x ) dy

Solution:

dz = Pdx + Qdy

Therefore,
P   Q
For dz to be an exact differential it must satisfy                =
y   x
P   Q
But this test fails becuase        
y   x
Not Exact

Question No: 36       ( Marks: 3 )



  sin       x  sin 5 x  dx
2
3

0

Use Wallis sine formula to evaluate
Solution:



   0
2
sin 3 xdx
n 1
      .
n
3 1

3
2

3


   0
2
sin 5 xdx
n 1 n  3
     .
n n2
5 1 5  3
     .
5 52
4 2
 .
5 3


  sin           x  sin 5 x  dx
2
3

0

2 4 2
         .
3 5 3

Question No: 37               ( Marks: 5 )

Evaluate the following line integral which is independent of path.

(3,2)


(0,0)
(2 xe y ) dx  ( x 2e y ) dy
Solution:

z
p     2e y                             2e dx
y

x
z
Q     x 2e y                           x e dy
2 y

y
(3,2)
z             2 xe y  x 2 ye y
(0,0)

z  6e 2  18e 2
z  24e 2

Question No: 38              ( Marks: 5 )

bn                             f (t )
Determine the Fourier coefficients                     for a periodic function            of period 2 defined by

4 (1 + t)                           -1 < t < 0
f (t )  
0                                       0<t<1

Solution:

1      
bn 
      f ( x) sin nxdx


1    1


            4(1  t ) sin nxdx
1

1  4(1  t ) cos nx 
1

                      
           n         1
4(1  t )
            cos n(1)  cos n(1) 
n
4(1  t )
            (cos n  cos n)
n
Question No: 39        ( 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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