Mcqs Mth301 Solved

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Fall 2009 MTH301- Calculus II Time: 120 min Marks: 80

Question No: 1

π

( Marks: 1 ) - Please choose one

is an example of ----------►

Irrational numbers

►

Rational numbers

►

Integers

►

Natural numbers

Question No: 2


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

Surface

►

Curve

►

Plane

►

Parabola

Question No: 3


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An ordered triple corresponds to ----------- in three dimensional space. ►

A unique point

►

A point in each octant

►

Three points

►

Infinite number of points

Question No: 4


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

►

Direction cosines

►

Direction ratios

►

Direction angles

Question No: 5

Is the function


f( x, y)

continuous at origin? If not, why?

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

►

►

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: 6


Match the following vector-valued function with its graph. ^

^

r (t ) = 3 cos t i + 3 sin t j

and

0 ≤ t ≤ 2π

►

►

►



►

Question No: 7


Match the following vector-valued function with its graph. ^

^

^

2 3 r (t ) = t i + t j + t k

and

t ≥ 0

►

►

►



►

Question No: 8


What are the parametric equations that correspond to the following ve ctor equation? →

^

^

r (t ) = sin t i + (1 − cos 2t ) j 2

►

►

►

►

x= sin 2 t ,

y = 1 − cos 2 t , z = 0

y= sin 2 t ,

x= 1 − co cos 2 t ,

z= 0

x= sin 2 t ,

y= 1 − cos 2 t ,

z= 1

x= sin 2 t ,

y = cos 2 t

Question No: 9

, z= 1


→

Is the following vector-valued function

r (t )

continuous at

t = 0

? If not, why?

→

r (t ) = (4 cos t ,

t , 4s 4 sin t )

→

►

r (0)

is not defined


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lim r (t )

→

►

r (0)

t → 0

is defined but

does not exist →

lim r (t )

→

►

r (0)

t →0

is defined and

exists but these two numbers are not equal.

→

►

r (t )

is continuous at

Question No: 10

t = 0


What is the derivative of following vector-valued function? →

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

→

 sin5t  , sec t , 6 cos t   5 

r ′(t ) =  ► →

r ′(t ) = (

− sin5t 5

, sec t , 6 cos t )

► →

►

r ′(t ) = ( −5 sin 5t, se s ec 2 t , 6 cos t ) →

►

r ′(t ) = (sin 5t , sec t , − 6 co cos t )

Question No: 11

2


The following differential is exact


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►

True

►

False

Question No: 12


The following differential is exact dz = (3 x 2 + 4 xy) dx + (2 x2 + 2 y) dy

►

True

►

False

Question No: 13


Which one of the following is correct Wallis Sine formula when

n

is odd and

π

2

∫

n sin x dx =

0

π

2

( n − 1) ( n − 3) ( n − 5 ) 5 −−−−−−− n 6 ( n − 2) ( n − 4 )

3 1 4 2

► π

2

∫ 0

n sin x dx =

π

2

( n ) ( n − 2) ( n − 4 ) 6 −−−−−−− 5 ( n − 1) ( n − 3) ( n − 5 )

4 2 3 1

►


n≥3

?

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2

∫

sin n x dx =

( n − 1) ( n − 3) ( n − 5 ) 6 −−−−−−− n 7 ( n − 2) ( n − 4)

sin n x dx =

( n ) ( n − 2) ( n − 4) 6 −−−−−−− 5 ( n − 1) ( n − 3) ( n − 5 )

0

4 2 5 3

► π

2

∫ 0

4 2 3 1

►

Question No: 14


Which of the following is correct? π

2

3

∫ sin x dx = 16 4

0

► π

2

3π

∫ sin x dx = 16 4

0

► π

2

3

∫ sin x dx = 8 4

0

► π

2

∫ sin x dx = 4

0

2π 3

►


http://vujannat.ning.com Best Website TO Help VU sTudents Question No: 15


Match the following equation in polar co-ordinates with its graph. r = 4sin θ 2 1 1

2

3

4

-1

►

-2

4 3 2 1

►

-2 -1

1

2

2 1 -4 -3 -2 -1 -1 -2

► -2 -1 -1

1

2

-2 -3

►

-4

Question No: 16


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


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Initial line

►

y-axis

►

Pole

Question No: 17


f ( x) = sin

What is the period of a periodic function defined by

x

2 ?

π

2 ►

►

π

3π 2 ►

►

4π

Question No: 18


Match the following periodic function with its graph. 3 0< x<4  4 x  f( x) = 7 − x 4 < x< 10

−3  

10 < x < 13



►

►

►

►

Question No: 19


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

►

2

►

3

►

4

►

5



Question No: 20


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

►

0

►

2

►

►

π

2π

Question No: 21

( Marks: 1 ) - Please choose choose one

−3π    −2,  2   Polar co-ordinates of a point are . Which of the following is another possible polar co-ordinates representation of this point?

 −π   2,  4   ►

 −π   2, 2    ►

 −π   2,  3   ►



 3π   2,   4  ►

Question No: 22


f ( x) = x3 e

x

The function

is -------------

►

Even function

►

Odd function

►

Neither even nor odd

Question No: 23

( Marks: 1 ) - Please choose choose one

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

x-axis

►

y-axis

►

origin

Question No: 24


2 At which point the vertex of parabola, represented by the equation y = x − 4 x+ 3 , occurs?

►

(0, 3)



(2, (2, − 1)

►

( −2, 15) 15)

►

(1, 0)

Question No: 25


2 The equation y = x − 4 x+ 2 represents a parabola. Find a point at w hich the vertex of given parabola occurs?

►

( 2, − 2)

►

( −4, 34) 34)

►

(0, 0)

►

( −2, 14) 14)

Question No: 26


f( x, y)

Is the function xy f( x, y) = 2 x + y 2

►

f( x, y)

lim

( x , y ) → ( 0 , 0 )

►

continuous at origin? If not, why?

is continuous at origin f( x, y)

does not exist


http://vujannat.ning.com Best Website TO Help VU sTudents lim

►

( x , y ) → ( 0, 0 )

f (0, 0)

f( x, y)

is defined and

exists but these two numbers are not

equal.

Question No: 27


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.

Question No: 28


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

∫ e ►

− st

F (t ) dt dt

0

∞

∫ e ►

st

F (t ) dt

0

∞

∫ e

− st

F (t ) dt dt

► −∞ ∞

∫ e ►

− st

F (t ) dt dt

0



Question No: 29


t

What is the value of 5t L{e } =

►

5t L{e } =

►

5 L{e } =

5t L{e } =

►

if L denotes laplace transform?

1

s −5 s 2 s + 25

5

t

►

L{e5 }

s + 25 2

5!

s

Question No: 30

6


1

What is the Laplace Inverse Transform of s + 1 −1

 1   = t + 1  s + 1

L  ►

−1

 1  − t t = e+ e + 1 s  

L

►

−1

 1  t  =e  s + 1

L  ►


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 1  − t  =e  s + 1

L  ►

Question No: 31


5

What is Laplace Inverse Transform of s + 25 2

−1

 5   = sin 5t 2  s + 25 

L  ►

−1

 5   = cos 5t 2 25 s +  

L  ►

−1

 5   = sin 25t 2  s + 25 

L  ►

−1

 5   = cos 25t 2  s + 25 

L  ►

Question No: 32


What is L{−6}if L denotes Laplace Transform? L{−6} = ►

L{−6} = ►

L{−6} = ►

1

s+6

−6 s s s + 36 2



L{−6} = ►

−6 2 s + 36

Question No: 33


∫ (3 x+ 2 y)

dx+ (2 x− y) dy

C

Evaluate the line integral

where C is the the line segment

from (0, 0) to (2, 0). ►

6

►

-6

►

0

►

Do not exist

Question No: 34


∫ (2 x+ y)

dx + ( x2 − y) dy

C

Evaluate the line integral

where C is the line segment from

(0, 0) to (0, 2). ►

-4

►

-2

►

0

►

2



Question No: 35


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

Curve

►

Surface

►

Sphere

►

Cone

Question No: 36


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

∫∫ ( x+ 2 y ) dA= 2

R

1

2

∫ ∫ ( x+ 2 y ) ddyydx 2

−1 0

► 2

−1

dxdy ∫ ∫ ( x+ 2 y ) dx 2

0

1

►


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2

∫ ∫ ( x+ 2 y ) ddxxdy 2

−1 0

► 2

0

dxdy ∫ ∫ ( x+ 2 y ) dx 2

1

−1

►

Question No: 37


To evaluate the line integral, the integrand is expressed in terms of x, y, z with

►

►

►

►

dr = dx iˆ + dy ˆj

dr = dx iˆ + dy ˆj + dy kˆ dr = dx + dy + dz dr = dx + dy

Question No: 38


Match the following equation in polar co-ordinates with its graph. r=a wher wheree a is an an arbit arbitra rary ry const constan ant. t.



►

►

►

►

Question No: 39


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: 40


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

►

An odd function

►

An even function

►

Neither even nor odd

Question No: 1

( Marks: 2 )

- Please choose one

Laplace transform of ‘t ‘ is

►

1 s



1 s

►

e

2

−s

s

►

Question No: 2

( Marks: 2 )

- Please choose one

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

►

►

►

►

y + 2

x − 2 = −

x + 2 = −

3

y + 3

x − 1 = −

x + 1 =

Question No: 3

5

=−

y − 3 5

y + 3 5

=

=−

=−

z −3 5

z +5 2

z −5 2

z −5

( Marks: 1 )

5

- Please choose one

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

True.

►

False.

Question No: 4 The equation

( Marks: 1 )

z = r

- Please choose one

is written in

►

Rectangular coordinates

►

Cylindrical coordinates



Spherical coordinates

►

None of the above


Mcqs Mth301 Solved

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