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Final term EXAMINATION
Total Marks: 60
SEMESTER Fall 2005
Subject code – MTH101 (Fall 2005) Duration:2 Hour

Maximum Time Allowed: ( 2 Hour)

Please read the following instructions carefully before attempting any of the questions:

1. Attempt all questions. Marks are written adjacent to each question.

2. Do not ask any questions about the contents of this examination from anyone.
a. If you think that there is something wrong with any of the questions, attempt it
to the best of your understanding.
b. If you believe that some essential piece of information is missing, make an
appropriate assumption and use it to solve the problem.
c. Write all steps, missing steps may lead to deduction of marks.

3. You should copy the data directly from MATHTYPE into the exam software copying
MATHTYPE images from MICROSOFT WORD to the exam application may cause some
problem of visibility.
4. The duration of this examination is 120 minutes.
5. This examination is closed book, closed notes, closed neighbors.
6. Calculator is allowed

**WARNING: Please note that Virtual University takes serious note of unfair means.
Anyone found involved in cheating will get an `F` grade in this course.
For Teacher’s use only
Question Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Total

Marks

Question No: 1 Marks=2

∑2
m =3
m +1
= --------------

a- 100
b- 112
c- 120
d- 140

Question No: 2 Marks=2

1 5

If ∫ f (x)dx = −2 and ∫ f (x)dx = 1

0 0
5
Then ∫ f (x)dx = ?
1
a- -1
b- -3
c- 3
d- -2

Question No: 3 Marks=2

x
dt
Let F(x) = ∫t
1
, the point where the graph of F(x)

crosses the x-axis is

a- 0
b- 1
c- -1
d- None of the above.
 Question No: 4 Marks=2

The series
5 5 5
5+ + 2 + ... + k −1 + ...
4 4 4
a- Convergent Series
b- Divergent Series.
c- Cannot be determined.
d- None of the above.

Question No: 5 Marks=5

Find a non zero value for the costant k so that

⎧ tan kx
⎪ , x<0
f(x) = ⎨ x
⎪3x + 2k 2 , x ≥ 0
⎩
will be continuous at x=0.

Note: In order to get maximum marks do all necessary steps.

Question No: 6 Marks=10

dy
Find if
dx
dt 1
2y3t + t 3 y = 1 and =
dx Cost

Note: In order to get maximum marks do all necessary steps.

Question No: 7 Marks=5

Prove (by subsitution); If m and n are positive inegers, then

1 1

∫ x (1− x) dx = ∫ x (1− x) dx
m n n m

0 0

Note: In order to get maximum marks do all necessary steps.

Question No: 8 Marks=7

Find the volume of the solid that results when the region enclosed
by the curves x = Csc y, y = π /4, y = 3π /4, x = 0
is revolved about the y-axis.
Note: In order to get maximum marks do all necessary steps.

Question No: 9 Marks=8

Find the arc length in the second quadrant of the curve

1
x 2/3 + y2 / 3 = a 2 / 3 from the point x= -a to x = - a (a>0).
8

Note: In order to get maximum marks do all necessary steps.

Question No: 10 Marks=10

Find limit
⎛ x +1 ⎞
lim x ln ⎜ ⎟
x →+∞
⎝ x −1 ⎠

Note: In order to get maximum marks do all necessary steps.

Question No: 11 Marks=7

Determine whether the series converges or diverges

2 + ( −1)
k
+∞

∑
k =1 5k

Note: In order to get maximum marks do all necessary steps.

 www. vujannat.ning.com

MTH101 Calculus and Analytical Geometry

Final Term Examination – Spring 2005
Time Allowed: 150 Minutes

Maximum Time Allowed: ( 150 mins)

Please read the following instructions carefully before

attempting any of the questions:
1. Pasting the equations of math type from word file into
software may cause some visibility problem, so please
note that do not copy equations of math type into
software from word file. Paste the equations from math
type directly into software.

2. Do not ask any questions about the contents of this

examination from anyone.
a. If you think that there is something wrong with any of
the questions, attempt it to the best of your understanding.
b. If you believe that some essential piece of information is
missing, make an appropriate assumption and use it to
solve the problem.
c. Write all steps, missing steps may lead to deduction of
marks.

**WARNING: Please note that Virtual University takes serious

note of unfair means. Anyone found involved in cheating will
get an `F` grade in this course.

Total Marks: 65 Total Questions: 12

Question No. 1 Marks : 05

Using the concept of infinite series expresses 4.212121…………………. as a ratio of
integers.

Note: In order to get the maximum marks you have to show all the necessary steps

Question No. 2 Marks : 03

If
y = f (x)
then
f ′( x0 )
is the function whose value at x0 is the average rate of change of y with respect to x at the
point x0.

Question No. 3 Marks : 10

Determine whether the series converges or diverges. If it converges, find the sum

∞ k −1
⎛ =3 ⎞
∑ ⎜− ⎟
k =1 ⎝ 4⎠
Note: In order to get the maximum marks you have to show all the necessary steps

Question No. 4 Marks : 07

Find the area of the region enclosed by :

y = x 3 + 5x2 + 3, y = x2 + 7x + 3,
x = 0, x = 3.
Note: In order to get the maximum marks you have to show all the necessary steps

Question No. 5 Marks : 05

Find the slope of the curve

y = (2x 2 + 7x − 2)2 at the point x= -2.
Note: In order to get the maximum marks you have to show all the necessary steps

Question No. 6 Marks : 03

 ∞
1
∑ n
The series n = 1 is
o Converges
o Absolutely converges
o Diverges
o None of the above

Question No. 7 Marks : 03

If the lim ( f(x) / g(x) ) is not an indeterminate form then L ' Hopital's rule cannot be
applied
o True
o False

Question No. 8 Marks : 03

Two non vertical lines with slopes m1 and m2 respectively are parallel if and only if

o m1 m 2 = 1
o m1 m 2 = − 1
m1
=1
o m2
m1
= −1
o m2

Question No. 9 Marks : 08

Solve the following integral

z
∫ 3
z +1
2
dz

by substitution method.
Note: In order to get the maximum marks you have to show all the necessary steps

Question No. 10 Marks : 05

Find the limit using L' Hopital's Rule

 sin x
lim+
x→π x −π
Note: In order to get the maximum marks you have to show all the necessary steps

Question No. 11 Marks : 10

⎧x 2 +1 , 0 < x ≤1
⎪
f (x) = ⎨x , 1< x ≤ 4
⎪2x +1 , 4≤ x
Is ⎩ continuous at x = 1?

Write all necessary steps and justify your answer.

Note: In order to get the maximum marks you have to show all the necessary steps.

Question No. 12 Marks : 03

Which of the following statement is true

Let f be a function that is defined at all points in the interval [a, b].

o If f is continuous on [a, b], then f is integrable on [a, b]

o If f is bounded on [a, b] and has only finitely many points of discontinuity
on [a, b], then f is integrable on [a, b]
o If f is not bounded on [a, b], then f is not integrable on [a, b].
o All of above are true
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Connecting VU Students

MTH101-Calculus and Analytical Geometry

Final Term Examination – Spring 2006
Time Allowed: 150 Minutes

Please read the following instructions carefully before attempting any

of the questions:
1. Pasting the equations of math type from word file into software
may cause some visibility problem, so please note that do not
copy equations of math type into software from word file. Paste
the equations from math type directly into software.

2. Do not ask any questions about the contents of this examination from
anyone.
a. If you think that there is something wrong with any of the
questions, attempt it to the best of your understanding.
b. If you believe that some essential piece of information is missing,
make an appropriate assumption and use it to solve the problem.
c. Write all steps, missing steps may lead to deduction of marks.

**WARNING: Please note that Virtual University takes serious note of

unfair means. Anyone found involved in cheating will get an `F` grade in
this course.

Question No. 1 Marks : 10

1
y = x3 , 0 ≤ x ≤
Find the area of the surface generated by revolving the curve 2 about the x-axis.

Note: In order to get the maximum marks you have to show all the necessary steps.

Question No. 2 Marks : 10

Find the limit using L’ Hopital’s Rule

⎛ x +1 ⎞
x

lim ⎜ ⎟
x →+∞ x + 2
⎝ ⎠ .

Note: In order to get the maximum marks you have to show all the necessary steps.
 Question No. 3 Marks : 10

Determine whether the series converges or diverges. If it converges, find the sum
∞ k −1
⎛ =3 ⎞
∑ ⎜− ⎟
k =1 ⎝ 4 ⎠

Note: In order to get the maximum marks you have to show all the necessary steps.

Question No. 4 Marks : 2

Two non vertical lines with slopes m1 and m2 respectively are parallel if and only if
m1m2 = 1
m1m2 = −1
m1 / m2 = 1
m1 / m2 = −1

Question No. 5 Marks : 5

What is the difference between differentiation and integration?

Note: In order to get the maximum marks you have to show all the necessary steps.

Question No. 6 Marks : 2

∞
1
The series ∑n
n −1

Converges
Absolutely converges
Diverges
Non of the other

Question No. 7 Marks : 2

Acceleration of a moving particle is given as a ( t ) = 9t 2 − 7t + 3 . We can find

for the moving particle.

Velocity function
Position function
Both (a) & (b)
None of the other
 Question No. 8 Marks : 2

100

∑k =?
k −3

5047
5050
5053
None of the other
Question No. 9 Marks : 4

If the distance traveled by the car is y = f(x) given in the function below, then find the velocity
dy
dx
of the car.
Where
lny = e y sinx
Note: In order to get the maximum marks you have to show all the necessary steps.

Question No. 10 Marks : 10

⎧x 2 +1 , 0 < x ≤1
⎪
Is f (x) = ⎨ x , 1< x ≤ 4 continuous at x = 1?
⎪ 2x +1 , 4≤ x
⎩
Write all necessary steps and justify your answer.

Note: In order to get the maximum marks you have to show all the necessary steps.

Question No. 11 Marks : 8

Evaluate
z
dz
3
z 2 +1

Note: In order to get the maximum marks you have to show all the necessary steps.
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MIDTERM EXAMINATION
FALL 2007 Marks: 40
MTH101 - CALCULUS AND ANALYTICAL GEOMETRY Time: 120min
(Session - 7 )

StudentID/LoginID:

Student Name:

Center Name/Code:

Exam Date: Thursday, December 06, 2007

Please read the following instructions carefully before attempting any

of the questions:

1. Attempt all questions. Marks are written adjacent to each question.

2. Do not ask any questions about the contents of this examination

from anyone.

a. If you think that there is something wrong with any of the

questions, attempt it to the best of your understanding.

b. If you believe that some essential piece of information is

missing, make an appropriate assumption and use it to solve the
problem.

c. Write all steps, missing steps may lead to deduction of marks.

3. Calculator is allowed.

**WARNING: Please note that Virtual University takes serious note of unfair means.
Anyone found involved in cheating will get an `F` grade in this course.

For Teacher's use only

Question 1 2 3 4 5 6 7 8 9 10 Total
Marks
Question No: 1 ( Marks: 1 ) - Please choose one

The equation (x+2)2 + ( y-3)2=0 represents a graph of

► Straight line

► Circle

► Single point

► None of these

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

If graph of a function f(x) is then represents the graph of

► f(x+2)
2
► f(x+2) -1
2
► f(x-2) +1
2
► f(x-2) -1
Question No: 3 ( Marks: 1 ) - Please choose one

Which of the following statement is incorrect

► lim 2 = 2
x→+∞

► lim 2 = ∞
x→+∞

► lim x = + ∞
x→+∞
 ► 1
lim =0
x→+∞ x

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

1 1
lim xsin( ) = 0 lim sin( ) = 0
x→0 x x→0 x
As so

► True

► False

Question No: 5 ( Marks: 1 ) - Please choose one

A function not defined at a point can never be continuous at that point; it however may be
differentiable at that point.

► True

► False

Question No: 6 ( Marks: 6 )

Prove that (0, -2), (-4, 8), and (3, 1) lie on a circle with center (-2, 3).

Question No: 7 ( Marks: 6 )

Find a formula for the function f graphed in figure given below

Question No: 8 ( Marks: 7 )

Evaluate
 3 x +9
lim
x → − 3 x2 − 9

Question No: 9 ( Marks: 9 )

⎧ x−a
⎪ , when x ≠ a
f (x) = ⎨ (x − a)
⎪1 , when x = a
⎩

If , Check whether f(x) is continuous at x = a

Question No: 10 ( Marks: 7 )

y = 2x3 + 4x − 2
Find the slope of the curve at x = 3
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MTH101 Calculus And Analytical Geometry

Mid Term Examination – Spring 2006
Time Allowed: 90 Minutes

Maximum Time Allowed: ( 1.5 Hours)

Please read the following instructions carefully before attempting any

of the questions:

1. Attempt all questions. Marks are written adjacent to each question.

2. Do not ask any questions about the contents of this examination from
anyone.
a. If you think that there is something wrong with any of the
questions, attempt it to the best of your understanding.
b. If you believe that some essential piece of information is
missing, make an appropriate assumption and use it to solve
the problem.
c. Write all steps, missing steps may lead to deduction of
marks.

3. The duration of this examination is 90 minutes.

4. This examination is closed book, closed notes, closed neighbors.
5. Calculator is allowed

**WARNING: Please note that Virtual University takes serious note of

unfair means. Anyone found involved in cheating will get an `F` grade in
this course.
Question No. 1 Marks : 2

Determine whether the points A(9,2),B(7,2)

Lie on horizontal line.

Lie on vertical line.
Lie on origin.
None of the other.

Question No. 2 Marks : 2

If y = 6x5 − 4x 2 , then y′′(1) = ?

10
20
112
110

Question No. 3 Marks : 2

f(x) = ( 5/x ) + ( 2x / x + 4 ) has point of discontinuity at

x = 0 and x = 1
x = 0 and x = 4
x= 0 and x = -4
None of the other.

Question No. 4 Marks : 6

Evaluate lim ( x 2 + 5x − x)
x→+∞

Note: In order to get maximum marks do all necessary

steps.

Question No. 5 Marks : 10

Find the largest interval on which f is
(a) Increasing, (b) Decreasing: find the largest open interval on which f is
(c) Concave up, (d) Concave down: and
(e) Find the x-coordinates of all inflection points.
f (x) = 3x3 − 4x + 3
Note: In order to get maximum marks do all necessary steps.

Question No. 6 Marks : 2

Let f(x) = x 2 +1, then f(1/x) = ?

1 /(x 2 +1)
(1/ x 2 ) +1
x+1
None of the other.

Question No. 7 Marks : 10

Solve | x − 3 |2 −4 | x − 3 |= 12 for x.

Note: In order to get maximum marks do all necessary steps.

Question No. 8 Marks : 6

Find the x-coordinate of the point on the graph of y=x2 where the tangent line is
parallel to the secant line that cuts the curve at x=-1 and x=2

Note: In order to get maximum marks do all necessary steps.