Paper no 1 to 7
Paper1
Q1. Suppose
f ( x, y)  x3e xy
. Which one of the following is correct?
f
 3x 2 e xy  x3 ye xy
x
f
 3 x 2 ye xy
x
f
 3 x 2 e xy  x 4 e xy
x
f
 3x 2 e xy
x
Q2. Let R be a closed region in two dimensional space. What does the double integral
over R calculates?
Area of R.
Radius of inscribed circle in R.
Distance between two endpoints of R.
None of these
Q3. What is the distance between points (3, 2, 4) and (6, 10, -1)?
7 2
2 6
34
7 3
Q4. -------------------- planes intersect at right angle to form three dimensional space.
Three
4
8
12
Q5. There is one-to-one correspondence between the set of points on co-ordinate line and
-----------Set of real numbers
Set of integers
Set of natural numbers
Set of rational numbers
Q6. Let the function
f
xx
, f yy and f xy 
f ( x, y) has continuous second-order partial derivatives
in some circle centered at a critical point
D  f xx ( x0 , y0 ) f yy ( x0 , y0 )  f xy 2 ( x0 , y0 )
If
D  0 then ---------------
f
has relative maximum at
( x0 , y0 )
has relative minimum at
( x0 , y0 )
f
f
(x , y )
has saddle point at 0 0
No conclusion can be drawn.
( x0 , y0 ) and let
Q7.
If R  {( x, y ) / 0  x  2 and 0  y  3}, then

(1  ye xy )dA 
R
2
3
 
0
0
2
3
 
0
0
3
0
 
2
0
2
3
 
0
(1  ye xy )dydx
(1  ye xy )dxdy
(1  ye xy )dxdy
(4 xe2 y )dydx
2
Q8. Suppose
f ( x, y)  2 xy where x  t 2  1 and y  3  t
true?
df
 6t  4t 2  2
dt
df
 6t  2
dt
df
 4t 3  6t  6
dt
df
 6t 2  12t  2
dt
. Which one of the following is
Q9. Let i , j and k be unit vectors in the direction of x-axis, y-axis and z-axis

respectively. Suppose that
a  2i  5 j  k

. What is the magnitude of vector
6
30
30
28
Q10. A straight line is --------------- geometric figure.
One-dimensional
Two-dimensional
Three-dimensional
Dimensionless
Q11.
If R  {( x, y ) / 0  x  2 and 1  y  4}, then

(6 x 2  4 xy 3 )dA 
R
4
2
 
1
0
2
4
 
0
1
4
2
 
1
0
4
1
 
2
0
(6 x 2  4 xy 3 )dydx
(6 x 2  4 xy 3 )dxdy
(6 x 2  4 xy 3 )dxdy
(6 x 2  4 xy 3 )dxdy
a
?
Q12. Which of the following formula can be used to find the Volume of a
a , b and c
parallelepiped with adjacent edges formed by the vectors
 
a b c 
a b  c 
a  b c 
?
a bc
f ( x, y)  y  x
Q13. The function
is continuous in the region --------- and
discontinuous elsewhere.
x y
x y
x y
Q14. What is the relation between the direction of gradient at any point on the surface to
the tangent plane at that point ?
parallel
perpendicular
opposite direction
No relation between them.
Q15. Suppose
f ( x, y)  x3e xy
. Which one of the statements is correct?
f
 3x3e xy
y
f
 x3e xy
y
f
 x 4e xy
y
f
 x 3 ye xy
y
Q16. Two surfaces are said to intersect orthogonally if their normals at every point
common to them are ---------perpendicular
parallel
in opposite direction
Q17. Let the function
f
xx
, f yy and f xy 
f ( x, y) has continuous second-order partial derivatives
in some circle centered at a critical point
D  f xx ( x0 , y0 ) f yy ( x0 , y0 )  f xy 2 ( x0 , y0 )
f ( x , y )  0 then f has --------------If D  0 and xx 0 0
Relative maximum at
Relative minimum at
( x0 , y0 )
( x0 , y0 )
(x , y )
Saddle point at 0 0
No conclusion can be drawn.
( x0 , y0 ) and let
Q18.
If R  {( x, y ) / 0  x  2 and  1  y  1}, then

( x  2 y 2 )dA 
R
1
2
 
1
0
2
1
 
0
1
1
2
 
1
0
2
0
 
1
( x  2 y 2 )dydx
( x  2 y 2 )dxdy
( x  2 y 2 )dxdy
( x  2 y 2 )dxdy
1
Q19.
f ( x, y , z ) 
x2 y
 xyz
z
If
then what is the value of f (1, 1, 1) ?
f (1, 1, 1)  1
f (1, 1, 1)  2
f (1, 1, 1)  3
f (1, 1, 1)  4
Q20.
If R  {( x, y ) / 0  x  4 and 0  y  9}, then

(3x  4 x xy )dA 
R
9
4
0
0
4
9
0
4
9
0
4
0
4
9
0
0
 
 
 
 
(3x  4 x xy )dydx
(3x  4 x xy )dxdy
(3x  4 x xy )dxdy
(3x  4 x xy )dydx
y2
Let f ( x, y )  2  x 
4
Find the gradient of f
2
Q-
2MARKS
Q - Let the function f ( x, y ) is continuous in the region R, where R is a rectangle as
shown below.
complete the following equation

R
f ( x, y) dA  

f ( x, y) _____
2MARKS
Q.Find all critical points of the function
f ( x, y)  4 xy  x3  2 y 2
4

1
2
0
(6 x2  4 xy3 )dx dy
Evaluate
Q-Evaluate the following double integral.
2
  3  2 x  3 y  dx dy
3MARKS
y
1
x2
Q- Let
. If x changes from 3 to 3.3, find the approximate change in the value of y
using differential dy.
3MARKS
Paper2
MIDTERM EXAMINATION
Fall 2009
MTH301- Calculus II
Question No: 1 ( Marks: 1 )
- Please choose one
Let x be any point on co-ordinate line. What does the inequality -3 < x < 1 means?
► The set of all integers between -3 and 1
► The set of all natural numbers between -3 and 1.
► The set of all rational numbers between -3 and 1
► The set of all real numbers between -3 and 1
Question No: 2 ( Marks: 1 )
- Please choose one
Which of the following number is associated to each point on a co-ordinate line?
Question No: 3 ( Marks: 1 )
- Please choose one
Which of the following set is the union of set of all rational and irrational numbers?
Question No: 4 ( Marks: 1 )
- Please choose one
 is an example of ----------ers
Question No: 5 ( Marks: 1 )
- Please choose one
Which of the following is associated to each point on a plane?
Question No: 6 ( Marks: 1 )
- Please choose one
-------------------- planes intersect at right angle to form three dimensional space.
► Three
► Four
► Eight
► Twelve
Question No: 7 ( Marks: 1 )
- Please choose one
At each point of domain, the function ----------------
► Is defined
► Is continuous
► Is infinite
► Has a limit
Question No: 8 ( Marks: 1 )
- Please choose one
What is the general equation of parabola whose axis of symmetry is parallel to y-axis?
2
► y  ax  b
(a  0)
2
► x  ay  b
(a  0)
2
► y  ax  bx  c
(a  0)
2
► x  ay  by  c
(a  0)
Question No: 9 ( Marks: 1 )
The spherical co-ordinates
- Please choose one
  ,  ,   , of a point are
  
 2, , 0 
 4 
ordinates of this point?
 0, 0,
2

►
►
►
 2, 0, 0
 0, 0, 2

2, 0, 2

►
Question No: 10 ( Marks: 1 )
- Please choose one
. What are the rectangular co-
Let
f ( x, y)  y 2 x 4 e x  2
.
5 f
y 3 x 2
Which method is best suited for evaluation of
?
► Normal method of finding the higher order mixed partial derivatives
► Chain Rule
► Laplacian Method
► Euler’s method for mixed partial derivative
Question No: 11 ( Marks: 1 )
Suppose
f ( x, y)  x3e xy
- Please choose one
. Which one of the following is correct?
f
 3 x 2 e xy  x 3 ye xy
► x
f
 3 x 2 e xy  x 4 e xy

x
►
f
 3x 2 e xy
► x
f
 3 x 2 ye xy
► x
Question No: 12 ( Marks: 1 )
Suppose
f ( x, y)  x3e xy
- Please choose one
. Which one of the statements is correct?
f
 3x3e xy
► y
f
 x3e xy

y
►
f
 x 4e xy

y
►
f
 x3 ye xy
► y
Question No: 13 ( Marks: 1 )
Suppose
- Please choose one
f ( x, y)  2 xy where x  t 2  1 and y  3  t
. Which one of the following is true?
df
 6t  4t 2  2
dt
►
df
 6t  2
dt
►
df
 4t 3  6t  6
dt
►
df
 6t 2  12t  2
dt
►
Question No: 14 ( Marks: 1 )
- Please choose one
Let i , j and k be unit vectors in the direction of x-axis, y-axis and z-axis respectively. Suppose

that
a  2i  5 j  k

. What is the magnitude of vector
a
?
►6
► 30
►
►
30
28
Question No: 15 ( Marks: 1 )
- Please choose one
f ( x, y )
Is the function
f ( x, y ) 
►
continuous at origin? If not, why?
xy
x  y2
2
f ( x, y )
is continuous at origin
lim
f ( x, y)
( x , y )(0, 0)
►
does not exist
lim
►
( x , y )(0, 0)
f (0, 0)
f ( x, y)
is defined and
Question No: 16 ( Marks: 1 )
exists but these two numbers are not equal.
- Please choose one
Let R be a closed region in two dimensional space. What does the double integral over R
calculates?
► Area of R.
► Radius of inscribed circle in R.
► Distance between two endpoints of R.
► None of these
Question No: 17 ( Marks: 1 )
- Please choose one
What is the relation between the direction of gradient at any point on the surface to the tangent
plane at that point ?
► parallel
► perpendicular
► opposite direction
► No relation between them.
Question No: 18 ( Marks: 1 )
- Please choose one
Two surfaces are said to intersect orthogonally if their normals at every point common to them
are ---------► perpendicular
► parallel
► in opposite direction
Question No: 19 ( Marks: 1 )
Let the function
f
xx
f ( x, y)
, f yy and f xy 
- Please choose one
has continuous second-order partial derivatives
in some circle centered at a critical point
D  f xx ( x0 , y0 ) f yy ( x0 , y0 )  f xy 2 ( x0 , y0 )
If
D0
and
f xx ( x0 , y0 )  0 then f has ---------------
► Relative maximum at
► Relative minimum at
► Saddle point at
( x0 , y0 )
( x0 , y0 )
( x0 , y0 )
► No conclusion can be drawn.
( x0 , y0 ) and let
Question No: 20 ( 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) ?
► 3, 2 and 9
► 1, -4 and -1
► 2, -3 and 20
► 0.5, -3 and 5/4
Question No: 21 ( Marks: 1 )
If

R  R1  R2
, where
R1
and
- Please choose one
R2
are no overlapping regions then
f ( x, y)dA   f ( x, y)dA 
R1
R2

f ( x, y)dA
R
►

f ( x, y)dA   f ( x, y)dA
R1
R2
►

R
►
f ( x, y)dV

f ( x, y)dA   f ( x, y)dA
R1
R2
►
Question No: 22 ( Marks: 1 )
- Please choose one
If R  {( x, y ) / 2  x  4 and 0  y  1}, then

(4 xe2 y )dA 
R
1
4
 
0
2
1
4
(4 xe2 y )dydx
►
 
0
2
4
2
(4 xe2 y )dxdy
►
 
1
0
4
2
(4 xe2 y )dxdy
►
 
1
(4 xe2 y )dydx
0
►
Question No: 23 ( Marks: 1 )
- Please choose one http://edu.goldendream.info
If R  {( x, y ) / 0  x  2 and 0  y  3}, then

R
(1  ye xy )dA 
2
3
 
0
0
2
3
(1  ye xy )dydx
►
 
0
0
3
0
(1  ye xy )dxdy
►
 
2
0
2
3
(1  ye xy )dxdy
►
 
0
(4 xe2 y )dydx
2
►
Question No: 24 ( Marks: 1 )
- Please choose one
Which of the following is geometrical representation of the equation
dimensional space?
► Parabola
► Straight line
► Half cylinder
► Cone
Question No: 25 ( Marks: 3 )
y  x2
, in three
 

 3, , 
3 2

Consider the point
coordinatesof this point.
in spherical coordinate system. Find the rectangular
http://edu.goldendream.info
Question No: 26 ( Marks: 5 )
Consider a function
f ( x, y)  4 xy  x 4  y 4
.One of its critical point is (1, 1) . Find whether
(1, 1) is relative maxima, relative minima or saddle point of f ( x, y ) .
Question No: 27 ( Marks: 10 )
Find Directional derivative of the function,

u
f ( x, y)  x 2 y  4 y 3
(2, 1)
, at the point
3 1
i j
2
2
direction of vector,
Paper3
1. Every real number corresponds to ____________ on the co-ordinate line.




Infinite number of points
Two points (one positive and one negative)
A unique point
None of these
in the
2. There is one-to-one correspondence between the set of points on co-ordinate line and
__________.




Set of real numbers
Set of integers
Set of natural numbers
Set of rational numbers
3. Which of the following is associated to each point of three dimensional space?




A real number
An ordered pair
An ordered triple
A natural Number
4. All axes are positive in __________octant.




First
Second
Fourth
Eighth


5. The spherical co-ordinates of a point are  3,
 
,  .What are its cylindrical co-ordinates?
3 2
 3 3 
, , 0 
 2 2 
 






 
  3 cos
  3 sin


  3,
3
, 3 sin
,
3 2


,0
3 
, 3 cos


3


,0
3 
6. Suppose f  x, y   xy  2 y 2 where x  3t  1 and y  2t . Which one of the following is
true?



df
 4t  2
dt
df
 16t  t
dt
df
 18t  2
dt

df
 10t 2  8t  1
dt
7. 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
8. Magnitude of vector
is 2, magnitude of vector
to tail is 45 degrees. What is




is 3and angle between them when placed tail
?
4.5
6.2
5.1
4.2
9. Is the function
f  x, y  continuous at origin? If not, why?

f  x, y  is continuous at origin

f  0,0 is not defined

f  0,0 is defined but

f  0,0 is defined and
lim
f  x, y  does not exist
lim
f  x, y  exist but these two numbers are not equal.
 x , y  0,0 
 x , y  0,0 
10. Is the function f  x, y  continuous at origin? If not, Why?

f  x, y  is continuous at origin

f  0,0 is not defined

f  0,0 is defined but

f  0,0 is defined and
lim
f  x, y  does not exist
lim
f  x, y  exist but these two numbers are not equal.
 x , y  0,0 
 x , y  0,0 
11. Let R be a closed region in two dimensional space. What does the double integral over R
calculates?




Area of R
Radius of inscribed circle in R.
Distance between two endpoints of R.
None of these
12. Which of the following formula can be used to find the volume of a parallelepiped with
adjacent edges formed by the vectors
,
and
?




13. Two surfaces are said to be orthogonal at appoint of their intersection if their normals at that
point are_________.




Parallel
Perpendicular
In opposite direction
Same direction
14. By Extreme Value Theorem, if a function
R, then




f  x, y  has both __________ on R.
Absolute maximum and absolute minimum value
Relative maximum and relative minimum value
Absolute maximum and relative minimum value
Relative maximum and absolute minimum value
15. Let the function
f
xx
f  x, y  is continuous on a closed and bounded set
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 xy2  x0 , y0  if
D  0 and f xx  x0 , y0   0 then f has __________ .
 Relative maximum at  x0 , y0 
 Relative minimum at  x0 , y0 
 Saddle point at  x0 , y0 
 No conclusion can be drawn.
16. Let the function
f
xx
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 xy2  x0 , y0  if
if
D  0 then _________.

f has relative maximum at  x0 , y0 

f has relative minimum at  x0 , y0 

f has saddle point at  x0 , y0 
 No conclusion can be drawn.
17. If R  R1  R2 , where R1 and R2 are no over lapping regions then
 f  x, y dA   f  x, y dA 
R1
R2

 f  x, y dA
R

 f  x, y dA   f  x, y dA
R1

R2
 f  x, y dV
R
 f  x, y dV   f  x, y dA
R
18. If R 
R2
 x, y  / 0  x  2
and 1  y  4 , then
  6x
R
4 2

  6x
2
 4 xy 3  dydx
2
 4 xy 3  dydx
2
 4 xy 3  dydx
1 0
2 4

  6x
0 1
4 2

  6x
1 0
2
 4 xy3 dA 
4 1

  6x
2
 4 xy 3  dydx
2 0
19. If R 
 x, y  / 2  x  4 and 0  y  1, then   4xe dA 
2y
R
1 4

   4 xe  dydx
2y
0 2
1 4

   4 xe  dxdy
2y
0 2
4 2

   4 xe  dxdy
2y
1 0
4 2

   4 xe  dydx
2y
1 0
20. If R 
 x, y  / 0  x  4
and 0  y  9, then
  3x  4 x

xy dA 
R
   3x  4 x
9 4


xy dydx
0 0
   3x  4 x
9 4


xy dxdy
0 0
   3x  4 x
9 0


xy dxdy
4 0
   3x  4 x
4 9


xy dydx
0 0
21. Suppose that the surface f  x, y, z  has continuous partial derivatives at the point
 a, b, c  Write down the equation of tangent plane at this point.
22. Evaluate the following double integral
23. Evaluate the following double integral
  12 xy
2
 8 x3  dydx .
   3  2 x  3 y  dxdy
24. Let f  x, y, z   xy 2e z Find the gradient of f .
2
25. Find, Equation of Tangent plane to the surface f  x, y, z   x2  y 2  z  9 at the
point 1, 2, 4  .
26. Use the double integral in rectangular co-ordinates to compute area of the region bounded by
the curves y  x 2 and y 
x.
Paper4
MIDTERM EXAMINATION
Spring 2010
MTH301- Calculus II (Session - 3)
Time: 60 min
Marks: 40
Question No: 1
( Marks: 1 )
- Please choose one
Which of the following number is associated to each point on a co-ordinate line?
► An integer
► A real number
► A rational number
► A natural number
Question No: 2
( Marks: 1 )
If a  0 , then the parabola
y  ax 2  bx  c
► Positive
x -
► Negative
x - direction
y
► Positive
► Negative
Question No: 3
y
- Please choose one
opens in which of the following direction?
- direction
- direction
( Marks: 1 )
- Please choose one
Rectangular co-ordinate of a point is
(1, 3,  2)
. What is its spherical co-ordinate?
 3 

 2 2, ,

3 2 

 3 

 2 2, ,

2 4 

 3 

 2 2, ,

3 4 

►
 3 

 2, ,

3 4 

Question No: 4
( Marks: 1 )
- Please choose one
If a function is not defined at some point, then its limit ----------- exist at that point.
Always
Never
May
Question No: 5
Suppose
( Marks: 1 )
f ( x, y)  x3e xy
- Please choose one
. Which one of the statements is correct?
f
 3x3e xy
y
f
 x3e xy
y
f
 x 4e xy
y
f
 x3 ye xy
y
Question No: 6
If
( Marks: 1 )
f ( x, y)  x 2 y  y 3  ln x
2 f
x 2
then
=
2xy 
2y 
►
1
x2
1
x2
- Please choose one
2xy 
2y 
Question No: 7
Suppose
1
x2
1
x2
( Marks: 1 )
- Please choose one
f ( x, y)  xy  2 y 2 where x  3t  1 and y  2t
true?
df
 4t  2
dt
df
 16t  t
dt
df
 18t  2
dt
df
 10t 2  8t  1
dt
Question No: 8
Is the function
0
f ( x, y )  
1
( Marks: 1 )
f ( x, y )
- Please choose one
continuous at origin? If not, why?
If x  0 and y  0
Otherwise
. Which one of the following is
f ( x, y )
f (0, 0)
is continuous at origin
is not defined
lim
f (0, 0)
( x , y )(0, 0)
is defined but
does not exist
lim
f (0, 0)
f ( x, y)
( x , y )(0, 0)
f ( x, y)
is defined and
exists but these two numbers
are not equal.
Question No: 9
( Marks: 1 )
- Please choose one
What is the relation between the direction of gradient at any point on the surface to the
tangent plane at that point ?
Question No: 10
( Marks: 1 )
- Please choose one
Two surfaces are said to intersect orthogonally if their normals at every point common
to them are ---------► perpendicular
► parallel
► in opposite direction
Question No: 11
( Marks: 1 )
- Please choose one
By Extreme Value Theorem, if a function
bounded set R, then
f ( x, y)
f ( x, y)
is continuous on a closed and
has both ---------------- on R.
► Absolute maximum and absolute minimum value
► Relative maximum and relative minimum value
Question No: 12
Let the function
f
xx
( Marks: 1 )
f ( x, y)
, f yy and f xy 
- Please choose one
has continuous second-order partial derivatives
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
and
f xx ( x0 , y0 )  0 then f has ---------------
► Relative maximum at
► Relative minimum at
► Saddle point at
( x0 , y0 )
( x0 , y0 )
( x0 , y0 )
► No conclusion can be drawn.
Question No: 13
Let the function
f
xx
( Marks: 1 )
f ( x, y)
, f yy and f xy 
- Please choose one
has continuous second-order partial derivatives
in some circle centered at a critical point
D  f xx ( x0 , y0 ) f yy ( x0 , y0 )  f xy 2 ( x0 , y0 )
If
D0
►
►
then ---------------
f
has relative maximum at
( x0 , y0 )
has relative minimum at
( x0 , y0 )
f
( x0 , y0 ) and let
►
f
has saddle point at
( x0 , y0 )
► No conclusion can be drawn.
Question No: 14
( Marks: 1 )
- Please choose one
f ( x, y)  y  x
The function
is continuous in the region --------- and discontinuous
elsewhere.
►
►
►
x y
x y
x y
Question No: 15
( Marks: 1 )
- Please choose one
Plane is an example of --------------------► Curve
► Surface
► Sphere
► Cone
Question No: 16
( Marks: 1 )
R  R1  R2
R1
If

R1
, where
and
- Please choose one
R2
f ( x, y)dA   f ( x, y)dA 
R2
are no overlapping regions then

f ( x, y)dA
R
►

f ( x, y)dA   f ( x, y)dA
R1
R2
►

f ( x, y)dV
R
►

f ( x, y)dA   f ( x, y)dA
R1
R2
Question No: 17
( Marks: 1 )
►
- Please choose one
If R  {( x, y ) / 0  x  2 and 1  y  4}, then

(6 x 2  4 xy 3 )dA 
R
4
2
 
1
0
2
4
(6 x 2  4 xy 3 )dydx
►
 
0
►
1
(6 x 2  4 xy 3 )dxdy
4
2
 
1
0
4
1
(6 x 2  4 xy 3 )dxdy
►
 
2
(6 x 2  4 xy 3 )dxdy
0
►
Question No: 18
( Marks: 1 )
- Please choose one
If R  {( x, y ) / 0  x  2 and  1  y  1}, then

( x  2 y 2 )dA 
R
1
2
 
1
0
2
1
( x  2 y 2 )dydx
►
 
0
1
1
2
( x  2 y 2 )dxdy
►
 
1
0
2
0
( x  2 y 2 )dxdy
►
 
1
( x  2 y 2 )dxdy
1
►
Question No: 19
( Marks: 1 )
- Please choose one
If R  {( x, y ) / 0  x  2 and 0  y  3}, then

(1  ye xy )dA 
R
2
3
 
0
0
2
3
(1  ye xy )dydx
►
 
0
0
3
0
(1  ye xy )dxdy
►
 
2
0
2
3
(1  ye xy )dxdy
►
 
0
(4 xe2 y )dydx
2
►
Question No: 20
( Marks: 1 )
- Please choose one
If R  {( x, y ) / 0  x  4 and 0  y  9}, then

(3x  4 x xy )dA 
R
9
4
0
0

►
(3x  4 x xy )dydx
4
9
0
4
9
0
4
0
4
9
0
0
 
(3x  4 x xy )dxdy
►

(3x  4 x xy )dxdy
►

(3x  4 x xy )dydx
►
Question No: 21
( Marks: 2 )
Evaluate the following double integral.
  2xy  y  dx dy
3
Question No: 22
( Marks: 2 )
y2
4
Find the gradient of f
Let f ( x, y )  2  x 2 
Question No: 23
( Marks: 3 )
Evaluate the following double integral.
  3  2 x  3 y  dx dy
2
Question No: 24
( Marks: 3 )
Let f ( x, y, z )  yz 3  2 x 2
Find the gradient of f .
Question No: 25
( Marks: 5 )
Find Equation of a Tangent plane to the surface
(2,  1, 2)
Question No: 26
( Marks: 5 )
Evaluate the iterated integral
   xy  dy dx
4
2
x
x
2
Paper5
f ( x, y, z )  x 2  3 y  z 3  9
at the point
Paper6
MIDTERM EXAMINATION
Spring 2010
MTH301- Calculus II
Question No: 1 ( Marks: 1 ) - Please choose one
Which of the following is the interval notation of real line?
►(-∞ , +∞)
►(-∞ , 0)
►(0 , +∞)
Question No: 2 ( Marks: 1 ) - Please choose one
What is the general equation of parabola whose axis of symmetry is parallel to y-axis?
2
►y  ax  b
(a  0)
2
►x  ay  b
(a  0)
2
►y  ax  bx  c
(a  0)
2
►x  ay  by  c
(a  0)
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Question No: 3 ( Marks: 1 ) - Please choose one
Which of the following is geometrical representation of the equation
dimensional space?
►A point on y-axis
►Plane parallel to xy-plane
►Plane parallel to yz-axis
►Plane parallel to xz-plane
Question No: 4 ( Marks: 1 ) - Please choose one
Suppose
f ( x, y)  x3e xy
. Which one of the statements is correct?
y4
, in three
f
 3x3e xy
►y
f
 x3e xy
►y
f
 x 4e xy
►y
f
 x3 ye xy
►y
Question No: 5 ( Marks: 1 ) - Please choose one
If
f ( x, y)  x 2 y  y 3  ln x
2 f
x 2
then
=
2xy 
1
x2
►
2y 
1
x2
►
2xy 
1
x2
►
2y 
1
x2
►
Question No: 6 ( 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
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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: 7 ( Marks: 1 ) - Please choose one
Is the function
f ( x, y )
continuous at origin? If not, why?
 3x 2 y
if ( x, y )  0

f ( x, y )   x 2  y 2
0
if ( x, y)  0

►
►
f ( x, y )
f (0, 0)
is continuous at origin
is not defined
lim
►
f (0, 0)
( x , y )(0, 0)
is defined but
f ( x, y)
does not exist
lim
►
f (0, 0)
( x , y )(0, 0)
f ( x, y)
is defined and
exists but these two numbers are not equal.
Question No: 8 ( Marks: 1 ) - Please choose one
Let R be a closed region in two dimensional space. What does the double integral over R
calculates?
►Area of R.
►Radius of inscribed circle in R.
►Distance between two endpoints of R.
►None of these
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Question No: 9 ( Marks: 1 ) - Please choose one
Two surfaces are said to be orthogonal at a point of their intersection if their normals at that
point are --------►Parallel
►Perpendicular
►In opposite direction
Question No: 10 ( Marks: 1 ) - Please choose one
Two surfaces are said to intersect orthogonally if their normals at every point common to them
are ---------►perpendicular
►parallel
►in opposite direction
Question No: 11 ( Marks: 1 ) - Please choose one
Let the function
f
xx
f ( x, y)
, f yy and f xy 
has continuous second-order partial derivatives
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
and
f xx ( x0 , y0 )  0 then f has ---------------
►Relative maximum at
( x0 , y0 )
►Relative minimum at
( x0 , y0 )
►Saddle point at
( x0 , y0 )
►No conclusion can be drawn. http://edu.goldendream.info
Question No: 12 ( Marks: 1 ) - Please choose one
Let the function
f
xx
f ( x, y)
, f yy and f xy 
has continuous second-order partial derivatives
in some circle centered at a critical point
D  f xx ( x0 , y0 ) f yy ( x0 , y0 )  f xy 2 ( x0 , y0 )
If
D0
and
f xx ( x0 , y0 )  0 then f has ---------------
►Relative maximum at
►Relative minimum at
►Saddle point at
( x0 , y0 )
( x0 , y0 )
( x0 , y0 )
►No conclusion can be drawn.
( x0 , y0 ) and let
Question No: 13 ( Marks: 1 ) - Please choose one
f ( x, y)
Let the function
f
xx
, f yy and f xy 
has continuous second-order partial derivatives
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
has ---------------
►Relative maximum at
( x0 , y0 )
►Relative minimum at
( x0 , y0 )
►Saddle point at
( x0 , y0 )
►No conclusion can be drawn
Question No: 14 ( Marks: 1 ) - Please choose one
Let
( x1 , y1 , z1 )
and
( x2 , y2 , z2 )
be any two points in three dimensional space. What does the
( x2  x1 )  ( y2  y1 )  ( z2  z1 )2
2
2
formula
calculates?
►Distance between these two points
►Midpoint of the line joining these two points
►Ratio between these two points
Question No: 15 ( Marks: 1 ) - Please choose one
f ( x, y)  y  x
The function
elsewhere.
is continuous in the region --------- and discontinuous
x y
►
x y
►
x y
►
Question No: 16 ( Marks: 1 ) - Please choose one
Plane is an example of --------------------►Curve
►Surface
►Sphere
►Cone
Question No: 17 ( Marks: 1 ) - Please choose one
If

R  R1  R2
, where
R1
and
R2
are no overlapping regions then
f ( x, y)dA   f ( x, y)dA 
R1
R2

f ( x, y)dA
R
►

f ( x, y)dA   f ( x, y)dA
R1
R2
►

f ( x, y)dV
R
►

f ( x, y)dA   f ( x, y)dA
R1
R2
►
Question No: 18 ( Marks: 1 ) - Please choose one
If R  {( x, y ) / 0  x  2 and 1  y  4}, then

(6 x 2  4 xy 3 )dA 
R
4
2
 
1
(6 x 2  4 xy 3 )dydx
0
►
2
4
 
0
(6 x 2  4 xy 3 )dxdy
1
►
4
2
 
1
(6 x 2  4 xy 3 )dxdy
0
►
4
1
 
2
(6 x 2  4 xy 3 )dxdy
0
►
Question No: 19 ( Marks: 1 ) - Please choose one
If R  {( x, y ) / 0  x  2 and 0  y  3}, then

(1  ye xy )dA 
R
2
3
 
0
(1  ye xy )dydx
0
►
2
3
 
0
(1  ye xy )dxdy
0
►
3
0
 
2
(1  ye xy )dxdy
0
►
2
3
 
0
(4 xe2 y )dydx
2
►
Question No: 20 ( Marks: 1 ) - Please choose one
If R  {( x, y ) / 0  x  4 and 0  y  9}, then

(3x  4 x xy )dA 
R
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9
4
 
0
►
0
(3x  4 x xy )dydx
4
9
0
4
 
(3x  4 x xy )dxdy
►
9
0
 
4
(3x  4 x xy )dxdy
0
►
4
9
0
0

(3x  4 x xy )dydx
►
Question No: 21 ( Marks: 2 )
Following is the graph of a function of two variables
In its whole domain, state whether the function has relative maximum value or absolute
maximum value at point B. Also, justify your answer
Question No: 22 ( Marks: 2 )
Let the function f ( x, y ) is continuous in the region R, where R is bounded by graph of
functions g1 and g2 (as shown below). http://edu.goldendream.info
In the following equation, replace question mark (?) with the correct value.

?
f ( x, y) dA 
R
?
 
?
f ( x, y) _____?____
?
Question No: 23 ( Marks: 3 )
Evaluate the following double integral.
  3  2 x  3 y  dx dy
2
Question No: 24 ( Marks: 3 )
Let f ( x, y, z )  yz 3  2 x 2
Find the gradient of f .
Question No: 25 ( Marks: 5 )
Find, Equation of Normal line (in parametric form) to the surface
f ( x, y, z )  xy  2 yz  xz 2  10
at the point
(5, 5, 1)
Question No: 26 ( Marks: 5 )
Use double integral in rectangular co-ordinates to compute area of the region bounded by the
curves
y  xand  y  x 2
, in the first quadrant.
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paper7
Use double integral in rectangular co-ordinates to compute area of the region bounded by the
curves
y  x2
and
y x
.