FINALTERM EXAMINATION 2009
Calculus & Analytical Geometry-I
Gulshan Ali
(Hafizabad Campus) gulshanvu@yahoo/gmail.com www.vuzs.net
Time: 120 min Marks: 80
Question No: 1 ( Marks: 1 ) - Please choose one
If f is a twice differentiable function at a stationary point
x
and
f ''( x ) >
0 then f
has relative …………. At
0
0
x
0
► Minima
► Maxima
► None of these
Question No: 2 ( Marks: 1 ) - Please choose one
In the notation ∫
f ( xdx ) = F ( x ) + C C represents
► A polynomial
► A Constant
► A Variable
► None of these
Question No: 3 ( Marks: 1 ) - Please choose one
According to Power-Rule of differentiation, if
f ( x ) =
x n where
n
is a real number, then d dx
[ x
n
] =
►
nx − 1 ► nx n − 1 ► nx n +
1
Wis
1 ( n − 1) x n + Question No: 4 ( Marks: 1 ) - Please choose one
If
►
dy dx
=
► 2
► -2
► 0
► -3
Question No: 5 ( Marks: 1 ) - Please choose one
30
2 x − y = −
3 then
0
= ________
►
π
►
3
π
►
4
π
►
6
π 2
Question No: 6 ( Marks: 1 ) - Please choose one
If a function g is differentiable at a point x and a function f is differentiable at a point g(x), then the ________ is differentiable at point x .
► Composition (f o g)
► Quotient ( f / g )
► Product (f . g)
► Sum (f + g)
Question No: 7 ( Marks: 1 ) - Please choose one
Let a function
f
be defined on an interval, and let x
1
and x
2
denote points in that
( interval. If
f x 1 ) <
f ( x 2
) whenever
x 1 <
x 2
then which of the following statement is correct?
►
f
is an increasing function.
►
f
is a decreasing function.
►
f
is a constant function.
Question No: 8 ( Marks: 1 ) - Please choose one
If
f ′′ ( x ) <
0 on an open interval (a,b) then which of the following statement is correct?
►
f
is concave up on (a, b).
►
f
is concave down on (a, b)
►
f
is linear on (a, b).
Question No: 9 ( Marks: 1 ) - Please choose one
The sum
∑ n
f ( x *
k )
∆ x
k k
=
1
is known as:
► Riemann Sum
► General Sum
► Integral Sum
► Geometric Sum
Question No: 10 ( Marks: 1 ) - Please choose one
What does 'n' represent in Riemann Sum
∑ n
f ( x *
k )
∆ x
k k
=
1
?
► No. of Circles
► No. of Rectangles
► No. of Loops
► No. of Squares
Question No: 11 ( Marks: 1 ) - Please choose one
What is the area of the region in the following figure?
2 A = ∫
⌈ ⌊ ( x + 6 ) −
( x 2
)
⌉ ⌋ dx 0
►
►
∫ 2
( 6
) ( 2 )
x
►
A = ⌈ ⌊ x + −
x ⌉ ⌋ dx 2 ∫
( ) ( 2
)
0
►
A = ⌈ ⌊ x + 6 +
x ⌉ ⌋ dx A = ∫
x
⌈ ⌊ ( x + ) −
( x 2
)
⌉ ⌋ dx 0
Question No: 12 ( Marks: 1 ) - Please choose one
If
6
4
1
4 ∫
f ( x ) dx = 2 and
∫ 1
g ( x ) dx = 10 then which of the following is value of 4 ∫ [3 f ( x ) − g ( x )] dx 1
?
► 16
► 12
► -4
► -8
Question No: 13 ( Marks: 1 ) - Please choose one
∫ 1
2 x ( x 2 + 4) dx = ____________ 0
Wis
9 2
►
►
5 2
►
2 5
►
−
9 2
Question No: 14 ( Marks: 1 ) - Please choose one
Let f is a smooth function on [0, 3]. What will be the arc length L of the curve y = f(x) from
x = 0 to x = 3?
►
3 L = ∫
1 + [ f ( x )] 2 dy 0
►
b L = ∫
+ f x a
►
1 [ '( )] 2 3 L = ∫
1 + [ f '( x )] 2 dy 0
►
3 L = ∫
1 + [ f '( x )] 2 dx 0
Question No: 15 ( Marks: 1 ) - Please choose one
Let f be a smooth, nonnegative function on [1, 3]. What is the surface area S generated by revolving the portion of the curve y = f(x) between x = 1 and x = 3 about the x-axis?
2 S = ∫
2 1 + [ f ( x )] dx 0
►
►
3 S = ∫
2 π f ( x ) 1 + [ f '( x )] dx 0
►
2 S = ∫
2 1 + [ f '( x )] dx 0
►
S = ∫ 3
2 π f ( x ) 1 + [ f '( x )] 2 dx 1
Question No: 16 ( Marks: 1 ) - Please choose one
Let an object is displaced 2m by a force of 2N. What is the work done W?
► - 4
► 4
► 2
► 0
Question No: 17 ( Marks: 1 ) - Please choose one
Consider the improper integral
+∞
l ∫ a
= l
→∞
∫ α
if the limit exists then which of the following can be occured?
► Diverges
► Converges
► Test fail
Question No: 18 ( Marks: 1 ) - Please choose one
If f is continuous on (a, b] but does not have a limit from the right then the integral
defined by
f ( x ) dx lim f ( x )
dx
∫ b a f ( x ) dx = lim l → a
∫ b
l
f ( x )
dx
is called :
► Improper
► Proper
► Line
Question No: 19 ( Marks: 1 ) - Please choose one
− < For a sequence
then the sequence is known as :
► Increasing
► Decreasing
► Nondecreasing
► Nonincreasing
Question No: 20 ( Marks: 1 ) - Please choose one
For a sequence
{ a
n
}
if the difference between successive terms
a n +
1
a
n
0
a
a
+ >
then the sequence is known as:
► Increasing
► Decreasing
► Nondecreasing
► Nonincreasing
Question No: 21 ( Marks: 1 ) - Please choose one
Which of the following is true for the sequence
n 1 1 { a
n
}
if the ratio of successive terms
n
n ∞
=
?
► Nonincreasing
► Nondecreasing
► Increasing
► Decreasing
Question No: 22 ( Marks: 1 ) - Please choose one
If
{ }
n
0
f ( n )
=
a n
is the nth term of the sequence and f is differentiable and
f '( n ) ≤
0 then the sequence will be :
► Increasing
► Decreasing
► Nondecreasing
► Nonincreasing
Question No: 23 ( Marks: 1 ) - Please choose one
If Newton's Method is used to approximate the real solutions of the equation x 3 + x − 3 =
0 and the first guess x 1
=
1 , What is x
2
?
►
5 4
►
1 4
►
−
1 2
►
3 4
►
3 2
Question No: 24 ( Marks: 1 ) - Please choose one
Suppose that we apply Newton’s Method to approximate the real solutions of the
equation
x =
2 , then which of the following is value of
2
x 3 − 2 x 2 − 1 =
0 . If we start at
1
x
?
► 6
► 2.25
► 0
► 2
Question No: 25 ( Marks: 1 ) - Please choose one
If the sequence of partial sum of a series converges then what will the series show itself ?
► Diverges
► Converges
► Gives no information
Question No: 26 ( Marks: 1 ) - Please choose one
The series
u ∑
u k
ρ =
lim →∞
k
+ 1 k
u
k ρ be a series with positive terms and suppose that
>
1 , then which of the following is true?
if
► Converges
► Diverges
► May converges or diverges
► Gives no information
Question No: 27 ( Marks: 1 ) - Please choose one
The series
=
ρ =
1 , then which of the following is true?
if
► Converges
► Diverges
► May converges or diverges
► Gives no information
Question No: 28 ( Marks: 1 ) - Please choose one
The series
∑
u k
be a series with positive terms and suppose that
ρ lim u k
→∞
k
+ 1 u k
∑
u k
be a series with positive terms and suppose that
1 ρ
= lim k →∞ k
u k =
lim( k →∞
u k
)k if
ρ =
1 , then which of the following is true?
► Converges
► Diverges
► May converges or diverges
► Gives no information
Question No: 29 ( Marks: 1 ) - Please choose one
For an alternating series to be convergent which of the following condition must be satisfied?
►
=
►
lim k →∞
a
k
1
a > a > a > a >
►
1 2 3
...... k
....
a 1 ≤ a 2 ≤ a 3
...... ≤ a k
≤
....
► Gives no information
W]
Question No: 30 ( Marks: 1 ) - Please choose one
For an alternating series to be convergent which of the following condition must be satisfied?
►
a ≥ a ≥ a ...... ≥ a k
≥
....
►
1 2 3
lim k →∞
a
k
=
0
►
a ≤ a ≤ a ...... ≤ a k
≤
....
►
1 2 3
lim k →∞
a
k
=
1
Question No: 31 ( Marks: 1 ) - Please choose one
What is the base of natural logarithm?
► 2.71
► 10
► 5
► Any real number
Question No: 32 ( Marks: 1 ) - Please choose one
A function
F
is called an antiderivative of a function
f
on a given interval if _______ =
f ( x ) , for all
x
in that interval. F '( x ) ►
F ( x ) ► f '( x ) ► f ′′ ( x ) ►
Question No: 33 ( Marks: 1 ) - Please choose one
log b
ac =
_______
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