MTH301 Calculus II Solved Final Term Papers For Final …api.ning.com/files/jPrLLXuRuu5jd2j3F2z5eKb50... · Virtualians Social Network Prepared by: Irfan Khan The level curves of

Virtualians Social Network www.virtualians.pk Prepared by: Irfan Khan

MTH301 Calculus II Solved Final Term

Papers For Final Term Exam Preparation

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

Laplace transform of „t „ is

► 1

s

► 1

s2

► e�s

► s



Symmetric equation for the line through (1,3,5) and (2,-2,3) is

► x �2 �

y 2 �

z �3

3 5

► x 2 �

y 3 �

z 5

5 2

► x �1�

y �3�

z �5

5 2

► x 1

y3

z�5 5 5



The level curves of f(x, y) = y Cscx are parabolas.

► True.

► False.


The equation z r is written in

► Rectangular coordinates

► Cylindrical coordinates

► Spherical coordinates

► None of the above



Intersection of two straight lines is --------------

► Surface

► Curve

► Plane

► Point


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

► One-dimensional

► Two-dimensional

► Three-dimensional

► Dimensionless



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


What are the parametric equations that correspond to the

following vector equation?

^ ^

r (t) sin2 t i(1cos 2t) j

► x sin2 t , y 1cos 2t , z 0


► y sin2 t , x 1cos 2t , z 0


► x sin2 t , y 1cos 2t , z 1

► x sin2 t , y cos 2t , z 1




^ ^ ^

r(t) 2t 1i3 t jsin 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


What is the derivative of following vector-valued function?

r (t) (cos 5t, tan t , 6 sin t) r(t)


sin 5t , sec t , 6 cos t

► 5


► r(t) (

sin 5t , sec t , 6 cos t)

5

► r(t) (5sin 5t, sec2 t , 6 cos t)

► r(t) (sin 5t, sec2 t , 6 cos t)



r (t)

t 4 ,

t 1 , 3

t

2

r(t) 4t

3 ,

1 ,

6

► t 1

t3

r(t) 4t3 ,

1 ,

6

► 2

t 1

t3

r(t) 4t

4 ,

1 ,

6

► 2 t 1 t

3

r(t) 4t

3 ,

1 ,

6

► 2

t 1 t3



The following differential is exact dz (x

2 y y) dx x dy


n

2

►

► True

► False


Which one of the following is correct Wallis Sine formula when n

is even and n 2 ?

2 n1 n3

n5 page #182 5 3 1

► sin 0

x dx 2

n n 2

n 4 6 4 2

► 2 n1 n3

n5

6 4 2 sin

n x dx

0 n n 2 n 4 7 5 3

► 2

n n2n4

6 4 2 sin

n x dx

0 2 n 1 n 3 n 5 5 3 1

n n n2 n4 sin x dx

6 4 2

0 n 1 n 3 n 5 5 3 1


Match the following equation in polar co-ordinates with its graph.


r cosa

where a is an arbitrary cons tant

►

►

►

►


If the equation of a curve, in polar co-ordinates, remains

unchanged after replacing (r, ) by (r, ) then the curve is


said to be symmetric about which of the following?


► Initial line

► Y-axis

► Pole



unchanged after replacing (r, ) by (r, ) then the curve is said

to be symmetric about which of the following?

► Initial line

► y-axis

► Pole


What is the amplitude of a periodic function defined by

f (x) sin x

? 3

► 0

► 1


1

► 3

► Does not exist


What is the period of a periodic function defined by

f (x) 4 cos 3x ?

► 4

► 3

2

► 3

►


Match the following periodic function with its graph. 3 0 x 4

f (x)

0 4 x 6


►


► Page #198

►

►


What is the period of periodic function whose graph is as

below?

► 2

► 5

► 6


► 8



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

► 0

► 4

► 6

► 8


Let L denotes the Laplace Transform.

lim F(t)

If L{F (t)} f (s) where s is a constant.and exists then

which of the following equation holds? t 0 t

L F (t)

f (s a)

► t

L F (t)

f (s a)

► t



L F (t)

f (s) ds

► t

s

L F (t)

d

{ f (s)} ► t

ds


Which of the following is Laplace inverse transform of the

function

f (s) defined by

f (s) 3

2 ?

s 2 s

► 3te2t 2

► 3e2t 2t

► 3e2t 2

► None of these.


Let (x1, y

1, z

1 )

and (x2 , y

2 , z

2 )

be any two points in three dimensional space. What does the formula (x x )

2 ( y y )

2 (z z )

2

calculates?

2 1 2 1 2 1

► Distance between these two points


► Midpoint of the line joining these two points

► Ratio between these two points


Let the functions P(x, y) and Q(x, y) are finite and continuous

inside and at the boundary of a closed curve C in the xy-

plane. If P dx Q dy is an exact differential then

Ñ P dx Q dy C

► Zero

► One

► Infinite


What is Laplace transform of the function F (t) if F (t) t ?

Lt1

► s

Lt 1

► s2


► Lt es

► Lt s


What is the value of L{e5t

} if L denotes laplace transform?

L{e5t

} ►

1

s 5

L{e5t

}

►

s

s2 25

L{e5t

} ►

5

s2 25

L{e5t

} 5!

► s6


Evaluate the line integral (3x 2 y) dx (2x y) dy C

where C is

the line segment from (0, 0) to (0, 2).

► 1


► 0


► 2

► -2


Evaluate the line integral (2x y) dx (x2 y) dy

C

where C is the

line segment from (0, 0) to (2, 0).

► 0

► -4

► 4

► Do not exist


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


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

(6x2 4xy

3 )dA

R

4 2

► 1 0

(6x

2 4xy

3 )dydx

2 4

► 0 1

(6x

2 4xy

3 )dxdy

4 2

► 1 0

(6x

2 4xy

3 )dxdy

4 1

► 2 0

(6x

2 4xy

3 )dxdy



Which of the following is true for a periodic function whose

graph is as below?

► Even function

► Odd function

► Neither even nor odd function


Which of the following is true for a function whose graph is

given above.

► An odd function


► An even function

► Neither even nor odd


At each point of domain, the function ----------------

► Is defined

► Is continuous

► Is infinite

► Has a limit

Question No: 31 ( Marks: 2 )

Determine whether the following differential is exact or not.

dz 4x3 y

3 dx 3x

4 y

2 dy


Solution:


dz 4x3 y

3 dx 3x

4 y

2 dy

p 12x

3 y

2

y

Q 12x

3 y

2

X

p Q

y X

yes


Evaluate

sin nx dx

where n is an integer other than zero.

Solution:

sin nx dx

cosnx

� n

cosn

cosn

� n n

1

(cos ncos n) n

0


Find Laplce transform of the function F (t) if

F (t) e3t

Solution:


2

L(e3t

) e3t e

st

0

e( s3)t

.dt

0

e( s3)t

{ }lim 0 (s 3)

1

s 3 (

1 )

e( s3)t

1

(0 1) s 3

1

......Ans s 3


Determine the Fourier co-efficient

defined below: f (x) 2x 1 0 x 2

Solution:

a0 of the periodic function

a 1

o

f (x)dx

f (x) (2x 1)

(0, 2)

2

0 (2x 1)dx

2

x x0

6




dz (3x2e

2 y 2 y

2 e

3x ) dx (2x

3 e

2 y 2 ye

3x ) dy

Solution:

dz = Pdx + Qdy

Therefore,

For dz to be an exact differential it must satisfy P

y

= Q x

But this test fails becuase P

y

Not Exact

Q x


2

Use Wallis sine formula to evaluate 0

sin3

x sin5

xdx

Solution:


2 sin3

xdx 0

n1

. n

31

3

2 3

2 sin5

xdx 0

n1

. n3

n n 2

51

. 53

5 5 2

4

. 2

5 3

2

sin3

x sin5

xdx 0

2

4 . 2

3 5 3


Evaluate the following line integral which is independent of path.

(3,2)

(0,0)

(2xe y ) dx (x

2e

y ) dy

Solution:


z 2xe y x ye

p z

2e y

x

Q z

x2e

y

y (3,2)

2 y

(0,0)

z 6e2 18e

2

z 24e2

2ey dx

x2 e

y dy


Determine the Fourier coefficients

of period 2 defined by 4 (1 + t) -1 < t < 0

f (t)

0 0 < t < 1

Solution:

bn for a periodic function f (t)

bn 1

f (x) sin nxdx

1

1

4(1t) sin nxdx

1 1

1 4(1t)cosnx

n 1

4(1t)

cos n(1) cos n(1)n

4(1t)

(cos n cos n) n


Determine whether the following vector field F

or not.

is conservative

^ ^ ^

F (x, y, z) (4x z) i(3y z) j( y x) k

…………………………………..


ASSALAM O ALAIKUM

All Dearz fellows

ALL IN ONE MTH301

Final term PAPERS & MCQz

Created BY Farhan & Ali

BS (cs) 2nd sem

Hackers Group

From Mandi Bahauddin

Remember us in your prayers

[email protected]

[email protected]

FINALTERM EXAMINATION

Spring 2010

MTH301- Calculus II (Session - 2)

Time: 90 min

Marks: 60

Student Info

StudentID: $$

Center: OPKST

mailto:[email protected]



ExamDate: 19 Aug 2010


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



-------------------- planes intersect at right angle to form three

dimensional space.

► Three

► Four

► Eight

► Twelve


If the positive direction of x, y axes are known then ------------

the positive direction of z-axis.

► Horizontal rightward direction is

► Vertical upward direction is

► Left hand rule tells

► Right hand rule tells


What is the distance between points (3, 2, 4) and (6, 10, -1)? ► 7 2

► 2 6

► 34

► 7 3



The equation ax by cz d 0 , where a,b,c,d are real numbers, is

the general equation of which of the following?

► Plane

► Line

► Curve

► Circle


The spherical co-ordinates of a point are 3, ,

3 2 . What are its

cylinderical co-ordinates?

3 ,

3 , 0

► 2 2

► 3 cos , 3 sin , 0 3 3

► 3 sin , 3

, 3 cos 2 3

3,

, 0

► 3


Domain of the function f (x, y) y x2 is

► y x2

► y x2

► y x2

► Entire space




Match the following vector-valued function with its graph.

^ ^ ^

r(t) cos t isin t j2 k and 0 t 2

►

►

►

►




^ ^


► x sin2 t , ► y sin

2 t ,


y 1cos 2t x

1cos 2t

, z 0

, z 0


► x sin2 t , y 1cos 2t , z 1





^ ^ ^

r(t) 2t 1i3 t jsin 3t k

► z 2t 1 ,

► y 2t 1 ,

► x 2t 1 ,

x 3 t ,

x 3 t ,

z 3 t ,

y sin 3t

z sin 3t

y sin 3t

► x 2t 1 , y 3 t , z sin 3t


Is the following vector-valued function r (t) continuous at t 1? If

not, why?

t r (t)

1, t

2 , 2t

t 1

► r (t) is continuous at

► r (1) is not defined

t 1

► r (1) is defined but

lim r (t) t 1

does not exist

► r (1) is defined and lim r (t) exists but these two t 1

numbers are not equal.




is even and n 2 ?


2

►

2

►

► 2

n1 n3

n5

5 3 1 sin

n x dx

0 2 n

n 2 n 4 6 4 2

► 2 n1 n3 n5 6 4 2 sin

n x dx

0 n n 2

n 4 7 5 3

► 2 n n 2 n 4 6 4 2 sin

n x dx

0 2 n 1

n 3 n 5 5 3 1

n n n2 n4 sin x dx 6 4 2

0 n 1 n 3 n 5 5 3 1


Which one of the following is correct Wallis Cosine formula when n is odd and

n 3 ?

► 2 n1 n3 n5 5 3 1 cos

n x dx

0 2 n

n 2 n 4 6 4 2

► 2 n n 2 n 4 6 4 2 cos

n x dx

0 2 n 1

n 3 n 5 5 3 1

n n n2n4 cos x dx 6 4 2

0 n 1

n 3 n 5 5 3 1

► 2 n1 n3 n5 6 4 2 cos

n x dx

0 n n 2 n 4 7 5 3





► Initial line

► y-axis

► Pole




If a 0 , then the equation, in polar co-ordinates, of the form r

2 a

2 cos 2 represent which of the following family of curves?

► Leminscate

► Cardiods

► Rose curves

► Spiral

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

x

What is the period of a periodic function defined by ( ) sin 2

?

► 2

► 3

► 2

► 4



► 2

► 5

► 6

► 8




► 0

► 2.25

► 3

► 6


Let L denotes the Laplace Transform.

If L{F (t)} f (s) where s is a constant, then which of the following

equation holds? d

► L{t F (t)} { f (s)} ds

► L{t F (t)}

► L{t F (t)}

f (s t)

f (s)

► L{t F (t)} s

f (s) ds


The graph of an odd function is symmetrical about ----------------

► x-axis

► y-axis

► origin


Consider the function f 0,

1 ,

1

f (x, y, z)

1 x 2 y

2 z

2 . What is the value of

2 2

f 0,

1 ,

1

1

►

2 2 2 f 0,

1 ,

1 2


► 2 2


f 0,

1 ,

1

1 ►

2 2

2

f 0,

1 ,

1 0

► 2 2


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


Sign of line integral is reversed when -----------

► path of integration is divided into parts.

► path of integration is parallel to y-axis.

► direction of path of integration is reversed.

► path of integration is parallel to x-axis.


Let the functions P(x, y) and Q(x, y) are finite and continuous

inside and at the boundary of a closed curve C in the xy-plane.

If P dx Q dy is an exact differential then

Ñ P dx Q dy C

► Zero

► One

► Infinite




2



L{e5t

} ►

L{e5t

} ►

L{e5t

} ►

1

s 5

s

s2 25

5

s2 25

L{e5t

} 5!

► s6


What is laplace transform of the function F (t) if F (t) sin 3t ?

L{sin 3t} ►

L{sin 3t} ►

L{sin 3t}

►

L{sin 3t} ►

3

s2 9

s

s2 9

1

s 3

3!

s4


If L denotes laplace transform then L{te

5t }

L{te5t

}

►

1

s2 5

L{te5t

} ►

L{te5t

}

1

s2 5

1 2

► s 5

L{te5t }

1

► s 5



Evaluate the line integral C

(3x 2 y) dx (2x y) dy

where C is the


► 1

► 0

► 2

► -2



(2x y) dx (x2 y) dy

where C is the


► -4

► -2

► 0

► 2


Divergence of a vector function is always a ------------

► Scalar

► Vector



t

k i

Which of the following is true for a function whose graph is given

above

► An odd function




Does the following limit exist? If yes find its value, if no give

reason lim

(e

^

5) (t

^ 1 ^

2t 3) 2t 2

t0

j


Define the periodic function whose graph is shown below.

Question No: 33 ( Marks: 2 ) F (t) t

4

Find Laplace Transform of the function F (t) if

Solution:


The Laplace transform of the given function will be:


f (t) t 4

L{t 4}

4!

s5



dz (4x3 y 2xy

3 ) dx (x

4 3x

2 y

2 ) dy


2

Use Wallis sine formula to evaluate 0

sin3

x sin5

xdx

Solution:

2

sin8

xdx

7. 5

. 3

. 1

0 8 6 4 2 2


Find Laplace transform of the function F (t) if

F (t) e2t

sin 3t

Solution:

Laplace transform will be


L(t) e2t ............... 1

1

s 2

L(t) sin 3t .......... 2

L(t)

L(t)

a

s2 3

2

a

s2 9

Combining,

L(t) ( 1

)( s 2

a )

s2 9


Using definite integral, find area of the region that is enclosed

between the curves y x2 and y x


Determine the fourier co-efficient bn of the following function.

f (x) x2 0 x 2




Determine whether the following vector field F

or not.

is conservative

^ ^ ^

F (x, y, z) (4x z) i(3y z) j( y x) k

ASSALAM O ALAIKUM

All Dearz fellows

ALL IN ONE MTH301

Final term PAPERS & MCQz

Created BY Farhan & Ali

BS (cs) 2nd sem

Hackers Group

From Mandi Bahauddin

Remember us in your prayers

[email protected]

[email protected]

FINALTERM EXAMINATION




Fall 2009


MTH301- Calculus II

Time: 120 min

Marks: 80


is an example of -----------

► Irrational numbers

► Rational numbers

► Integers

► Natural numbers


Straight line is a special kind of ----------------

► Surface

► Curve


► Plane


► Parabola


An ordered triple corresponds to ----------- in three dimensional

space.

► A unique point

► A point in each octant

► Three points

► Infinite number of points


The angles which a line makes with positive x ,y and z-axis are

known as ------------

► Direction cosines

► Direction ratios


► Direction angles


y)(0

y)(0


Is the function f (x, y) continuous at origin? If not, why?

f (x, y) 4xy sin 3x 2 y

► f (x, y) is continuous at origin

► f (0, 0) is not defined

► f (0, 0) is defined but ( x, lim

, 0) f (x, y) does not exist

► f (0, 0) is defined and ( x, lim

, 0) f (x, y) exists but these two




^ ^

r(t) 3cos t i3sin t j and 0 t 2


►


►

►

►



^ ^ ^

r(t) t it 2 jt

3 k and t 0


►

►

►

►




^ ^




► x sin2 t , y 1cos 2t , z 0

► y sin2 t , x 1cos 2t , z 0

► x sin2 t , y 1cos 2t , z 1



Is the following vector-valued function r (t) continuous at t 0 ? If

not, why?

r (t) (4 cos t,

t , 4 sin t)

► r (0) is not defined

► r (0) is defined but

lim r (t) t 0

does not exist

► r (0) is defined and

lim r (t) exists but these two numbers t0

are not equal.

► r (t) is continuous at t 0





r (t) (cos 5t, tan t , 6 sin t)

r(t)

sin 5t , sec t , 6 cos t

► 5

► r(t) (

sin 5t , sec t , 6 cos t)

5

► r(t) (5sin 5t, sec2 t , 6 cos t)

► r(t) (sin 5t, sec2 t , 6 cos t)


The following differential is exact dz (3x

2 y 2) dx (x

3 y) dy

► True

► False



The following differential is exact dz (3x

2 4xy) dx (2x

2 2 y) dy


n1 n3 n5 5 3 1

2 n n 2 n 4 6 4 2

2

►

► True

► False



is odd and n 3 ?

2

► sinn

x dx 0

► 2

n n2n4

6 4 2 sin

n x dx

0 2 n 1 n 3 n 5 5 3 1

► 2 n1 n3

n5

6 4 2 sin

n x dx

0 n n 2 n 4 7 5 3

n n n2 n4 sin x dx

6 4 2

0 n 1 n 3 n 5 5 3 1


Which of the following is correct?


2

2

2

2

► sin4

x dx 3

0 16

► sin4

x dx 3

0 16

► sin4

x dx 3

0 8

► sin4

x dx 2

0 3


Match the following equation in polar co-ordinates with its graph. r 4sin

2

1

-1

► -2

1 2 3 4

4

3

2

1

► -2 -1 1 2


2

1

-4 -3 -2 -1

-1

► -2

-2 -1 1 2

-1

-2

-3

► -4





► Initial line

► y-axis

► Pole


f x x

What is the period of a periodic function defined by ( ) sin 2

?


► 2


►

3

► 2

► 4


Match the following periodic function with its graph.

3 x

4

f (x) 7 x

0 x 4

4 x 10

3 10 x 13

►

►

►


►




► 2

► 3

► 4

► 5



► 0

► 2


►


► 2


2,

3Polar co-ordinates of a point are

2 . Which of the following

is another possible polar co-ordinates representation of this

point?

► 2,

4

► 2,

2

► 2,

3

2,

3►

4


The function f ( x) x3 ex is -------------

► Even function

► Odd function




The graph of an even function is symmetrical about ---------------

► x-axis

► y-axis

► origin


At which point the vertex of parabola, represented by the y x

2 4x 3

equation , occurs?

► (0, 3)

► (2, 1)

► (2, 15)

► (1, 0)




y )(0

y)(0

The equation y x2 4x 2

represents a parabola. Find a point at

which the vertex of given parabola occurs?

► (2, 2)

► (4, 34)

► (0, 0)

► (2, 14)


Is the function f (x, y) continuous at origin? If not, why?

f (x,

y) xy

x2 y

2

► f (x, y) is continuous at origin

► ( x, lim

, 0) f (x, y) does not exist

► f (0, 0) is defined and ( x, lim

, 0) f (x, y) exists but these two



Sign of line integral is reversed when -----------


► path of integration is divided into parts.


► path of integration is parallel to y-axis.

► direction of path of integration is reversed.

► path of integration is parallel to x-axis.


What is Laplace transform of a function F(t)?

(s is a constant)

s

est

► 0

F (t ) dt

est

► 0

F (t) dt

►

est

F (t) dt

est

► 0

F (t ) dt






6

L{e5t

}

►

1

s 5

L{e5t }

►

s

s2 25

L{e5t }

►

5

s2 25

L{e5t

} 5!

► s


What is the Laplace Inverse Transform of

1

s 1

L1

1 t 1

► s 1

L1 1

e

t e

t

► s 1

L1 1

e

t

► s 1

L1 1

et

► s 1


L

L

L

L


5 2

What is Laplace Inverse Transform of s 25

1 5 2

sin 5t

► s 25

1 5 2

cos 5t

► s 25

1 5 2

sin 25t

► s 25

1 5 2

cos 25t

► s 25


What is L{6} if L denotes Laplace Transform?

L{6}

►

1

s 6

L{6} 6

► s

L{6}

►

s

s2 36

L{6}


► 6

s2 36




(3x 2 y) dx (2x y) dy

where C is the


► 6

► -6

► 0

► Do not exist



(2x y) dx (x2 y) dy

where C is the


► -4

► -2

► 0


► 2



Plane is an example of ---------------------

► Curve

► Surface

► Sphere

► Cone


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

(x 2 y2 )dA

R

1 2

► 1 0

(x 2 y2

)dydx

2 1 2

► 0 1

(x 2 y )dxdy


1 2

► 1 0

(x 2 y2

)dxdy

2 0

►

(x 2 y2 )dxdy

1 1


To evaluate the line integral, the integrand is expressed in terms

of x, y, z with

► dr dx i dy j

► dr dx i dy j dy k

► dr dx dy dz

► dr dx dy


Match the following equation in polar co-ordinates with its graph. r a

where a is an arbitrary constant.


►

►

►

►


Which of the following is true for a periodic function whose

graph is as below?


► Even function


► Odd function

► Neither even nor odd function


The graph of “saw tooth wave” given above is --------------

► An odd function




Documents

MTH301 Calculus II Solved Final Term Papers For Final …api.ning.com/files/jPrLLXuRuu5jd2j3F2z5eKb50... · Virtualians Social Network Prepared by: Irfan Khan The level curves of