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CBSE Questions for Class 11 Engineering Maths Limits And Derivatives Quiz 9 - MCQExams.com
CBSE
Class 11 Engineering Maths
Limits And Derivatives
Quiz 9
If
π
∫
0
x
f
(
s
i
n
x
)
d
x
=
A
π
/
2
∫
0
f
(
s
i
n
x
)
d
x
,
then A is _____________.
Report Question
0%
0
0%
π
0%
π
/
4
0%
2
π
lim
x
→
∞
[
n
n
2
+
1
2
+
n
n
2
+
2
2
+
n
n
2
+
3
2
+
.
.
.
.
+
1
n
5
]
Report Question
0%
π
/
4
0%
tan
−
1
(
2
)
0%
π
/
2
0%
tan
−
1
(
3
)
Find the derivative of
√
tan
x
with respect to
x
using the first principle.
Report Question
0%
s
e
c
2
x
2
√
t
a
n
x
0%
s
e
c
x
2
√
t
a
n
x
0%
s
e
c
2
x
√
t
a
n
x
0%
s
e
c
2
x
2
√
t
a
n
Explanation
y
=
√
tan
x
⇒
d
y
d
x
=
1
2
√
tan
x
×
sec
2
x
=
sec
2
x
2
√
tan
x
Hence, the answer is
sec
2
x
2
√
tan
x
.
The value of
n
l
i
m
→
∞
1.
n
+
2.
(
n
−
1
)
+
3.
(
n
−
2
)
+
.
.
.
+
n
.1
1
2
+
2
2
+
.
.
.
+
n
2
is
Report Question
0%
1
0%
−
1
0%
1
√
2
0%
1
2
Explanation
Now,
1.
n
+
2
(
n
−
1
)
+
3.
(
n
−
2
)
+
.
.
.
.
+
n
.1
The general term of the above series is
T
r
=
r
(
n
−
r
+
1
)
=
r
(
n
+
1
−
r
)
Sum of series
=
S
=
n
∑
r
=
1
T
r
=
n
∑
r
=
1
r
(
n
+
1
−
r
)
S
=
n
∑
r
=
1
n
r
+
n
∑
r
=
1
r
−
n
∑
r
=
1
r
2
=
n
n
∑
r
=
1
r
+
n
∑
r
=
1
r
−
n
∑
r
=
1
r
2
=
n
n
(
n
+
1
)
2
+
n
(
n
+
1
)
2
−
n
(
n
+
1
)
(
2
n
+
1
)
6
Similarly,
1
2
+
2
2
+
3
2
+
.
.
.
.
.
+
n
2
=
n
(
n
+
1
)
(
2
n
+
1
)
6
lim
n
→
∞
1.
n
+
2
(
n
−
1
)
+
3.
(
n
−
2
)
+
.
.
.
+
n
.1
1
2
+
2
2
+
3
2
+
.
.
.
.
.
.
.
.
n
2
lim
n
→
∞
n
2
(
n
+
1
)
2
+
n
(
n
+
1
)
2
−
n
(
n
+
1
)
(
2
n
+
1
)
6
n
(
n
+
1
)
(
2
n
+
1
)
6
lim
n
→
∞
n
2
+
1
2
−
2
n
+
1
6
2
n
+
1
6
lim
n
→
∞
(
3
n
+
3
−
2
n
−
1
2
n
+
1
)
=
lim
n
→
∞
(
3
+
3
n
−
2
−
1
n
2
+
1
n
)
=
3
−
2
2
=
1
2
The value of
lim
x
→
0
1
+
sin
x
−
cos
x
+
log
(
1
−
x
)
x
3
, is
Report Question
0%
−
1
0%
1
/
2
0%
−
1
/
2
0%
1
The value of
lim
θ
→
0
+
sin
√
θ
√
sin
θ
is equal to
Report Question
0%
0
0%
1
0%
−
1
0%
4
f
(
x
)
=
sin
x
and
f
′
(
π
)
Report Question
0%
−
1
0%
0
0%
1
0%
None of these
Explanation
f
(
x
)
=
sin
x
f
′
(
x
)
=
d
d
x
sin
x
f
′
(
x
)
=
cos
x
f
′
(
π
)
=
cos
π
=
−
1
If
x
+
y
=
sin
(
x
−
y
)
then
d
y
d
x
is equal to
Report Question
0%
1
2
0%
0
0%
−
1
0%
1
3
Explanation
Given,
x
+
y
=
sin
(
x
−
y
)
x
+
y
=
sin
x
cos
y
−
cos
x
sin
y
1
+
d
y
d
x
=
sin
x
(
−
sin
y
)
+
cos
y
(
cos
x
)
−
cos
x
(
cos
y
)
−
(
−
sin
x
)
sin
y
1
+
d
y
d
x
=
−
sin
x
sin
y
+
cos
x
cos
y
+
sin
x
sin
y
−
cos
x
cos
y
1
+
d
y
d
x
=
0
∴
d
y
d
x
=
−
1
If
y
=
√
s
e
c
x
−
1
s
e
c
x
+
1
then
d
y
d
x
=
Report Question
0%
1
2
s
e
c
2
x
2
0%
s
e
c
2
x
2
0%
1
2
t
a
n
x
2
0%
t
a
n
x
2
Explanation
Given,
y
=
√
sec
x
−
1
sec
x
+
1
=
√
sec
x
−
1
sec
x
+
1
×
√
sec
x
−
1
sec
x
−
1
=
√
(
s
e
c
x
−
1
)
2
(
sec
x
−
1
)
(
sec
x
+
1
)
=
√
(
s
e
c
x
−
1
)
2
sec
2
x
−
1
=
√
(
s
e
c
x
−
1
)
2
tan
2
x
=
s
e
c
x
−
1
tan
x
=
s
e
c
x
tan
x
−
1
tan
x
=
csc
x
−
cot
x
∴
y
=
csc
x
−
cot
x
Now,
d
y
d
x
=
d
d
x
(
csc
x
−
cot
x
)
=
−
cot
(
x
)
csc
(
x
)
−
(
−
csc
2
(
x
)
)
=
−
cot
(
x
)
csc
(
x
)
+
csc
2
(
x
)
=
(
1
sin
(
x
)
)
2
−
1
sin
(
x
)
⋅
cos
(
x
)
sin
(
x
)
=
1
sin
2
(
x
)
−
cos
(
x
)
sin
2
(
x
)
=
1
−
cos
(
x
)
sin
2
(
x
)
=
1
−
cos
(
x
)
1
−
cos
2
(
x
)
=
−
1
−
cos
(
x
)
(
cos
(
x
)
+
1
)
(
cos
(
x
)
−
1
)
=
1
cos
(
x
)
+
1
=
1
2
cos
2
(
x
2
)
−
1
+
1
=
1
2
s
e
c
2
x
2
Let
f
:
(
0
,
∞
)
→
R
be a differentiable function such that
f
′
(
x
)
=
2
−
f
(
x
)
x
for all
x
∈
(
0
,
∞
)
and
f
(
1
)
≠
1
. Then
Report Question
0%
lim
x
→
0
+
f
′
(
1
x
)
=
1
0%
lim
x
→
0
+
x
f
(
1
x
)
=
2
0%
lim
x
→
0
+
x
2
f
′
(
x
)
=
0
0%
|
f
(
x
)
|
≤
2
for
all
X
∈
(
0
,
2
)
Explanation
Here,
f
′
(
x
)
=
2
−
f
(
x
)
x
or
d
y
d
x
+
y
x
=
2
[ i.e. linear differential equation in
y
]
Integrating Factor,
I
F
=
e
∫
1
x
d
x
=
e
log
x
=
x
∴
Required solution is
y
.
(
I
F
)
=
∫
Q
(
I
F
)
d
x
+
C
⇒
y
(
x
)
=
∫
2
(
x
)
d
x
+
C
⇒
y
x
=
x
2
+
C
∴
y
=
x
+
C
x
[
∵
C
≠
0
, as
f
(
1
)
≠
1
]
(a)
lim
x
→
0
+
f
′
(
1
x
)
=
lim
x
→
0
+
(
1
−
C
x
2
)
=
1
∴
Option (a) is correct.
(b)
lim
x
→
0
+
x
f
(
1
x
)
=
lim
x
→
0
+
(
1
+
C
x
2
)
=
1
∴
Option (b) is correct.
(c)
lim
x
→
0
+
x
2
f
′
(
x
)
=
lim
x
→
0
+
(
x
2
−
C
)
=
−
C
≠
0
∴
Option (c) is correct.
(d)
f
(
x
)
=
x
+
C
x
,
C
≠
0
For
C
>
0
,
lim
x
→
0
+
f
(
x
)
=
∞
∴
Function is not bounded in
(
0
,
2
)
.
∴
Option (d) is incorrect.
If
L
=
lim
x
2
→
a
b
−
cos
(
x
2
−
a
)
(
x
2
−
a
)
sin
(
c
x
2
−
a
)
is non-
zero finite
(
a
>
0
)
,
then-
Report Question
0%
L = 2 , b = 1 , c = 1
0%
L
=
1
2
,
b
=
1
,
c
=
1
0%
L = 4 , b = - 1 , c = - 1
0%
L
=
1
4
,
b
=
−
1
,
c
=
−
1
The solution the differential equation
cos
x
sin
y
d
x
+
sin
x
cos
y
d
y
=
0
Report Question
0%
sin
x
sin
y
=
c
0%
cos
x
+
cos
y
=
c
0%
sin
x
+
sin
y
=
c
0%
sin
x
.
sin
y
=
c
Explanation
Given differential equation is
cos
x
sin
y
d
x
+
sin
x
cos
y
d
y
=
0
⇒
cos
x
sin
y
d
x
=
−
sin
x
cos
y
d
y
⇒
cos
x
sin
x
d
x
=
−
cos
y
sin
y
d
y
On integrating both sides, we get
∫
cot
x
d
x
=
∫
−
cot
y
d
y
log
sin
x
=
−
log
sin
y
+
log
C
⇒
log
sin
x
sin
y
=
log
C
⇒
sin
x
.
sin
y
=
C
For
x
>
y
,
lim
x
→
0
[
(
sin
x
)
1
/
x
+
(
1
x
)
sin
x
]
is :
Report Question
0%
0
0%
-1
0%
1
0%
2
If the function
f
(
x
)
satisfies the relation
f
(
x
+
y
)
=
y
|
x
−
1
|
(
x
−
1
)
f
(
x
)
+
f
(
y
)
with
f
(
1
)
=
2
, then
lim
x
→
1
f
′
(
x
)
is?
Report Question
0%
2
0%
−
2
0%
0
0%
Limit do not exixst
Let
f
:
R
→
R
be a differentiable function satisfying
f
′
(
3
)
+
f
′
(
2
)
=
0
.
Then
lim
x
→
0
(
1
+
f
(
3
+
x
)
−
f
(
3
)
1
+
f
(
2
−
x
)
−
f
(
2
)
)
1
x
is equal to
Report Question
0%
e
2
0%
e
0%
e
−
1
0%
1
Explanation
lim
x
→
0
(
1
+
f
(
3
+
x
)
−
f
(
3
)
1
+
f
(
2
−
x
)
−
f
(
2
)
)
1
x
(
1
∞
form
)
⇒
e
lim
x
→
0
f
(
3
+
x
)
−
f
(
2
−
x
)
−
f
(
3
)
+
f
(
2
)
x
(
1
+
f
(
2
−
x
)
−
f
(
2
)
)
using L'Hopital
⇒
e
lim
x
→
0
f
′
(
3
+
x
)
+
f
′
(
2
−
x
)
−
x
f
′
(
2
−
x
)
+
(
1
+
f
(
2
−
x
)
−
f
(
2
)
)
⇒
e
f
′
(
3
)
+
f
′
(
2
)
1
=
1
Evaluate the following limits.
lim
x
→
a
√
x
+
√
a
x
+
a
.
Report Question
0%
−
1
√
a
0%
1
a
0%
1
2
√
a
0%
1
√
a
Explanation
Evaluate
lim
x
→
a
√
x
+
√
a
x
+
a
not an indeterminant Form so, limit can be found by simply putting unit of
x
lim
x
→
a
√
x
+
√
a
x
+
a
=
2
√
a
2
a
=
1
√
a
Evaluate the following limits.
lim
x
→
0
3
x
+
1
x
+
3
.
Report Question
0%
1
3
0%
2
3
0%
5
3
0%
None of these
Explanation
Evaluate
lim
x
→
0
3
x
+
1
x
+
3
as
→
0
,
E
x
p
r
e
s
s
i
o
n
→
1
3
, not indeterminant form
so, Limit can be foord, by simple substituting value
lim
x
→
0
3
x
+
1
x
+
3
=
1
3
Evaluate the following limits.
lim
x
→
0
x
2
/
3
−
9
x
−
27
.
Report Question
0%
1
3
0%
1
2
0%
1
5
0%
None of these
Evaluate the following limits.
lim
x
→
0
a
x
+
b
c
x
+
d
,
d
≠
0
.
Report Question
0%
a
c
0%
a
d
0%
b
d
0%
None of these
Explanation
lim
x
→
0
a
x
+
b
c
x
+
d
,
d
≠
0
as
x
→
0
, Expression
→
b
d
, which is a determined Form, so Limit= value
lim
x
→
0
a
x
+
b
c
x
+
d
=
b
a
Evaluate the following limits.
lim
x
→
1
√
x
2
−
1
+
√
x
−
1
√
x
2
−
1
,
x
>
1
.
Report Question
0%
√
2
+
1
√
2
0%
√
2
−
1
√
2
0%
√
2
+
1
2
0%
None of these
Evaluate the following limits.
lim
x
→
0
√
a
2
+
x
2
−
a
x
2
.
Report Question
0%
1
√
a
0%
1
√
2
a
0%
1
a
0%
1
2
a
Explanation
lim
x
→
0
√
a
2
+
x
2
−
a
x
2
As
x
→
0
, it is
0
0
Form,
So, using rationalisation
lim
x
→
0
(
√
a
2
+
x
2
−
a
)
x
2
×
(
√
a
2
+
x
2
+
a
)
√
a
2
+
x
2
+
a
)
lim
x
→
0
(
a
2
+
x
2
)
−
a
2
x
2
(
√
a
2
+
x
2
+
a
)
(
(
a
+
b
)
(
a
+
b
)
=
a
2
−
b
2
)
lim
x
→
0
x
2
x
2
(
√
a
2
+
x
2
+
a
)
lim
x
→
0
1
(
√
a
2
+
x
2
+
a
)
=
1
√
a
2
+
a
=
1
2
a
Evaluate the following limits.
lim
x
→
0
2
x
√
a
+
x
−
√
a
−
x
.
Report Question
0%
−
2
√
a
0%
√
a
0%
2
√
a
0%
None of these
Evaluate the following limits.
lim
x
→
a
x
−
a
√
x
−
√
a
.
Report Question
0%
2
√
a
0%
2
a
0%
2
a
1
3
0%
None of these
Explanation
lim
x
→
a
x
−
a
√
x
−
√
a
if
x
→
a
, expression
→
0
0
, an indetermined form
using factorisation
lim
x
→
a
(
√
x
−
√
a
)
(
√
x
+
√
a
)
(
√
x
−
√
a
)
=
2
√
a
Evaluate the following limits.
lim
x
→
0
√
a
+
x
−
√
a
x
√
a
2
+
a
x
.
Report Question
0%
1
2
√
a
0%
1
2
a
√
a
0%
1
2
a
0%
None of these
Explanation
lim
x
→
0
√
a
+
x
−
√
a
x
√
a
2
+
a
x
as
x
→
0
, it is
0
0
form
So, using rationalization
lim
x
→
0
√
a
+
x
−
√
a
x
√
a
2
+
a
x
(
√
a
+
x
+
√
a
)
(
√
a
+
x
+
√
a
)
lim
x
→
0
x
x
√
a
2
+
a
x
(
√
a
+
x
+
√
a
)
lim
x
→
0
1
√
a
2
+
a
x
(
√
a
+
x
+
√
a
)
=
1
2
a
√
a
Evaluate the following limits.
lim
x
→
4
2
−
√
x
4
−
x
.
Report Question
0%
1
4
0%
1
2
0%
1
3
0%
None of these
Explanation
lim
x
→
4
2
−
√
x
4
−
√
x
if
x
→
4
, expression
→
0
0
, an indetermined form
using factorisation
lim
x
→
4
2
−
√
x
2
2
−
(
√
x
)
2
=
lim
x
→
4
2
−
√
x
(
2
−
√
x
)
(
2
+
√
x
)
=
1
4
Evaluate the following limits.
lim
x
→
2
√
1
+
4
x
−
√
5
+
2
x
x
−
2
.
Report Question
0%
1
2
0%
1
3
0%
1
4
0%
1
5
Explanation
Evaluate
lim
x
→
2
√
1
+
4
x
−
√
5
+
2
x
(
x
−
2
)
as
x
→
2
, it is
0
0
From so,
using rationalisation
lim
x
→
2
(
√
1
+
4
x
−
√
5
+
2
x
)
(
1
+
4
x
)
+
√
5
+
2
x
(
x
−
2
)
(
√
1
+
4
x
+
√
5
+
2
x
)
lim
x
→
2
(
(
1
+
4
x
)
−
(
5
+
2
x
)
)
(
x
−
2
)
(
√
1
+
4
x
+
√
5
+
2
x
)
lim
x
→
2
2
(
x
−
2
)
(
x
−
2
)
(
6
)
=
1
3
Evaluate the following limits.
lim
x
→
2
x
−
2
√
x
−
√
2
.
Report Question
0%
2
√
3
0%
2
√
2
0%
2
√
5
0%
None of these
Explanation
lim
x
→
2
(
x
−
2
)
√
x
−
√
2
as
x
→
2
, it is
0
0
Form,
So, using Factorisation
lim
x
→
2
(
(
√
x
)
2
−
(
√
2
)
2
)
√
x
−
√
2
=
lim
x
→
2
(
√
x
−
√
2
)
(
√
x
+
√
2
)
(
√
x
−
√
2
)
lim
x
→
2
√
x
+
√
2
=
2
√
2
Evaluate the following limit.
lim
x
→
0
8
x
−
2
x
x
.
Report Question
0%
l
o
g
4
0%
l
o
g
6
0%
l
o
g
5
0%
None of these
Explanation
we know
lim
x
→
0
a
x
−
1
x
=
log
a
lim
x
→
0
8
x
−
2
x
x
=
lim
x
→
0
8
x
−
1
+
1
−
2
x
x
=
lim
x
→
0
8
x
−
1
x
−
lim
x
→
0
2
x
−
1
x
=
log
8
−
log
2
=
log
8
/
2
=
log
4
.
Evaluate the following limits.
lim
x
→
0
√
2
−
x
−
√
2
+
x
x
.
Report Question
0%
1
√
2
0%
−
1
√
3
0%
−
1
2
0%
−
1
√
2
Explanation
lim
x
→
0
√
2
−
x
−
√
2
+
x
x
As
x
→
0
, it is
0
0
from, so,
using rationalisation
lim
x
→
0
√
2
−
x
−
√
2
+
x
x
(
√
2
−
x
+
√
2
+
x
)
(
√
2
−
x
+
√
2
+
x
)
lim
x
→
0
(
2
−
x
)
−
(
2
+
x
)
x
(
√
2
−
x
+
√
2
+
x
)
lim
x
→
0
−
2
x
x
(
√
2
−
x
+
√
2
+
x
)
=
−
2
2
√
2
=
−
1
√
2
Evaluate the following limits.
If
lim
x
→
a
x
9
−
a
9
x
−
a
=
9
, find all possible values of a.
Report Question
0%
2
,
−
2
.
0%
1
,
−
1
.
0%
1
,
0
.
0%
None of these
Explanation
Using formula
lim
x
→
a
x
n
−
a
n
x
−
a
=
n
a
n
−
1
lim
x
→
a
x
9
−
a
9
x
−
a
=
9.9
8
=
9
a
8
=
1
⇒
a
=
±
1
0:0:1
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Practice Class 11 Engineering Maths Quiz Questions and Answers
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