Loading [MathJax]/jax/output/CommonHTML/jax.js
MCQExams
0:0:1
CBSE
JEE
NTSE
NEET
Practice
Homework
×
CBSE Questions for Class 11 Commerce Applied Mathematics Differentiation Quiz 14 - MCQExams.com
CBSE
Class 11 Commerce Applied Mathematics
Differentiation
Quiz 14
If
y
=
x
(
x
x
)
, then
d
y
d
x
is
Report Question
0%
y
[
x
x
(
log
e
x
)
log
x
+
x
x
]
0%
y
[
x
x
(
log
e
x
)
log
x
+
x
]
0%
y
[
x
x
(
log
e
x
)
log
x
+
x
x
−
1
]
0%
y
[
x
x
(
log
e
x
)
log
x
+
x
x
−
1
]
Explanation
y
=
x
(
x
x
)
Take log on both sides
log
y
=
log
x
(
x
x
)
log
y
=
x
x
log
x
....
(
i
)
Let
x
x
=
z
∴
x
log
x
=
log
z
Differentiate both sides with respect to
x
1
z
d
z
d
x
=
log
x
+
x
x
d
z
d
x
=
z
[
1
+
log
x
]
d
z
d
x
=
x
x
[
log
e
+
log
x
]
d
z
d
x
=
x
x
log
e
x
Equation
(
i
)
⟹
log
y
=
z
log
x
Differentiate both sides with respect to
x
1
y
d
y
d
x
=
log
x
⋅
d
z
d
x
+
z
x
∴
d
y
d
x
=
y
[
x
x
(
log
e
x
)
log
x
+
x
x
−
1
]
If
y
=
x
(
log
x
)
log
(
log
x
)
, then
d
y
d
x
is
Report Question
0%
y
x
(
(
ln
x
x
−
1
)
+
2
ln
x
ln
(
ln
x
)
)
0%
y
x
(
log
x
)
log
(
log
x
)
(
2
log
(
log
x
)
+
1
)
0%
y
x
ln
x
[
(
l
n
x
)
2
+
2
ln
(
ln
x
)
]
0%
y
x
log
y
log
x
[
2
log
(
log
x
)
+
1
]
Explanation
y
=
x
(
log
x
)
log
(
log
x
)
Taking log of both sides, we get
∴
log
y
=
(
log
x
)
(
log
x
)
log
(
log
x
)
... (1)
Taking log of both sides, we get
log
(
log
y
)
=
log
(
log
x
)
+
log
(
log
x
)
log
(
log
x
)
Differentiating w.r.t.
x
, we get
1
log
y
.
1
y
d
y
d
x
=
1
x
log
x
+
2
log
(
log
x
)
log
x
1
x
=
2
log
(
log
x
)
+
1
x
log
x
or
d
y
d
x
=
y
x
.
log
y
log
x
[
2
log
(
log
x
)
+
1
]
Substituting the value of
y
from (1), we get
d
y
d
x
=
y
x
(
log
x
)
log
(
log
x
)
(
2
log
(
log
x
)
+
1
)
The derivative of
√
x
(
x
+
4
)
3
2
(
4
x
−
3
)
4
3
w.r.t
x
is
Report Question
0%
√
x
(
x
+
4
)
3
2
(
4
x
−
3
)
4
3
{
1
2
x
+
3
2
(
x
+
4
)
−
16
3
(
4
x
−
3
)
}
0%
√
x
(
x
+
4
)
4
3
(
4
x
−
3
)
3
2
{
1
2
x
+
3
2
(
x
+
4
)
−
16
3
(
4
x
−
3
)
}
0%
√
x
(
x
+
4
)
5
4
(
4
x
−
3
)
3
4
{
1
2
x
+
3
2
(
x
+
4
)
−
16
3
(
4
x
−
3
)
}
0%
None of these
Explanation
Let
y
=
√
x
(
x
+
4
)
3
2
(
4
x
−
3
)
4
3
Taking
log
of both sides, we get
log
y
=
1
2
log
x
+
3
2
log
(
x
+
4
)
−
4
3
log
(
4
x
−
3
)
Differentiating both sides w.r.t.
x
, we get
1
y
d
y
d
x
=
1
2
x
+
3
2
1
x
+
4
−
4
3
×
1
4
x
−
3
×
4
⇒
d
y
d
x
=
y
{
1
2
x
+
3
2
(
x
+
4
)
−
16
3
(
4
x
−
3
)
}
=
√
x
(
x
+
4
)
3
2
(
4
x
−
3
)
4
3
{
1
2
x
+
3
2
(
x
+
4
)
−
16
3
(
4
x
−
3
)
}
f
n
(
x
)
=
e
f
n
−
1
(
x
)
for all
n
ϵ
N
and
f
0
(
x
)
=
x
, then
d
d
x
{
f
n
(
x
)
}
is
Report Question
0%
f
n
(
x
)
d
d
x
{
f
n
−
1
(
x
)
}
0%
f
n
(
x
)
f
n
−
1
(
x
)
0%
f
n
(
x
)
f
n
−
1
(
x
)
⋯
f
2
(
x
)
.
f
1
(
x
)
0%
none of these
Explanation
d
d
x
{
f
n
(
x
)
}
=
d
d
x
{
e
f
n
−
1
(
x
)
}
⇒
e
f
n
−
1
(
x
)
d
d
x
{
f
n
−
1
(
x
)
}
=
f
n
(
x
)
d
d
x
{
f
n
−
1
(
x
)
}
⇒
f
n
(
x
)
.
d
d
x
{
e
f
n
−
2
(
x
)
}
=
f
n
(
x
)
.
e
f
n
−
2
(
x
)
d
d
x
{
f
n
−
2
(
x
)
}
⇒
f
n
(
x
)
f
n
−
1
(
x
)
d
d
x
{
f
n
−
2
(
x
)
}
⋯
⇒
f
n
(
x
)
f
n
−
1
(
x
)
⋯
f
2
(
x
)
d
d
x
{
f
1
(
x
)
}
⇒
f
n
(
x
)
f
n
−
1
(
x
)
⋯
f
2
(
x
)
d
d
x
{
e
f
0
(
x
)
}
⇒
f
n
(
x
)
f
n
−
1
(
x
)
⋯
f
2
(
x
)
e
f
0
(
x
)
d
d
x
{
f
0
(
x
)
}
Use
e
f
0
(
x
)
=
f
1
(
x
)
and
f
0
(
x
)
=
x
Hence,
d
d
x
{
f
n
(
x
)
}
=
f
n
(
x
)
f
n
−
1
(
x
)
.
.
.
f
2
(
x
)
f
1
(
x
)
Hence, option C is correct.
If
f
(
x
)
=
Π
100
n
=
1
(
x
−
n
)
n
(
101
−
n
)
, then find
f
(
101
)
f
′
(
101
)
Report Question
0%
1
4950
0%
102
5050
0%
101
4950
0%
1
5050
Explanation
Given,
f
(
x
)
=
Π
100
n
=
1
(
x
−
n
)
n
(
101
−
n
)
⇒
l
o
g
f
(
x
)
=
{
n
(
101
−
n
)
{
Π
100
n
=
1
l
o
g
(
x
−
n
)
}
}
{
H
e
r
e
,
Π
changes to
∑
when taken log
}
⇒
l
o
g
f
(
x
)
=
100
∑
n
=
1
n
(
101
−
n
)
l
o
g
(
x
−
n
)
Differentiating both the sides, we get
f
′
(
x
)
f
(
x
)
=
100
∑
n
=
1
n
(
101
−
n
)
⋅
1
x
−
n
∴
f
′
(
101
)
f
(
101
)
=
100
∑
n
=
1
n
(
101
−
n
)
(
101
−
n
)
=
100
∑
n
=
1
n
=
5050
⇒
f
(
101
)
f
′
(
101
=
1
5050
If
y
=
√
x
+
√
y
+
√
x
+
√
y
+
…
∞
, then
d
y
d
x
=
Report Question
0%
y
2
−
x
2
y
3
−
2
x
y
−
1
0%
x
2
−
x
2
x
3
−
2
x
y
−
1
0%
x
2
−
x
2
x
3
−
2
x
y
2
−
1
0%
None of these
Explanation
y
=
√
x
+
√
y
+
√
x
+
√
y
+
…
∞
=
√
x
+
√
y
+
y
or
y
2
=
x
+
√
2
y
Differentiating w.r.t.
x
, we get
or
2
y
d
y
d
x
=
1
+
1
√
2
y
×
d
y
d
x
or
d
y
d
x
[
2
y
−
1
√
2
y
]
=
1
or
d
y
d
x
=
√
2
y
2
y
√
2
y
−
1
=
y
2
−
x
2
y
3
−
2
x
y
−
1
The value of
f
′
(
3
)
is
Report Question
0%
8
0%
10
0%
12
0%
18
Explanation
lim
x
→
0
f
(
x
)
x
=
f
(
x
+
y
)
−
f
(
y
)
−
x
2
y
−
x
y
2
x
Applying L'Hospital's Rule,
=
lim
x
→
0
f
(
x
)
x
=
f
′
(
x
+
y
)
−
2
x
y
−
y
2
1
=
1
(
g
i
v
e
n
)
=
f
′
(
y
)
−
y
2
1
=
1
(
g
i
v
e
n
)
Let
y
=
3
,
f
′
(
y
)
−
y
2
=
1
f
′
(
3
)
−
3
2
=
1
f
′
(
3
)
=
10
If for all
x
,
y
the function f is defined by;
f
(
x
)
+
f
(
y
)
+
f
(
x
)
⋅
f
(
y
)
=
1
and
f
(
x
)
>
0
.When
f
(
x
)
is differentiable
f
′
(
x
)
=
,
Report Question
0%
−
1
0%
1
0%
0
0%
cannot be determined
Explanation
Here,
f
(
x
)
+
f
(
y
)
+
f
(
x
)
⋅
f
(
y
)
=
1
.....(i)
Substitute
x
=
y
=
0
, we get
2
f
(
0
)
+
{
f
(
0
)
}
2
=
1
⇒
{
f
(
0
)
}
2
+
2
f
(
0
)
−
1
=
0
f
(
0
)
=
−
2
±
√
4
+
4
2
=
−
1
−
√
2
and
−
1
+
√
2
As
f
(
0
)
>
0
⇒
f
(
0
)
=
√
2
−
1
[neglecting
f
(
0
)
=
−
1
−
√
2
as f(0) is positive]
Again, putting
y
=
x
is Eq, (i),
2
f
(
x
)
+
{
f
(
x
)
}
2
=
1
On differentiating w.r.t. x,
2
f
′
(
x
)
+
2
f
(
x
)
f
′
(
x
)
=
0
2
f
′
(
x
)
{
1
+
f
(
x
)
}
=
0
⇒
f
′
(
x
)
=
0
because
f
(
x
)
>
0
Thus,
f
′
(
x
)
=
0
when
f
(
x
)
>
0
The function
f
(
x
)
=
e
x
+
x
being differentiable and one to one, has a differentiable inverse
f
−
1
(
x
)
, then find
d
d
x
(
f
−
1
(
x
)
)
at the point
f
(
l
o
g
e
2
)
.
Report Question
0%
1
3
0%
1
0%
3
0%
0
Explanation
Let
y
=
e
x
+
x
on differentiating w.r.t y,
1
=
(
e
x
+
1
)
d
x
d
y
⇒
d
x
d
y
=
1
1
+
e
x
∴
(
d
x
d
y
)
x
=
l
o
g
e
2
=
1
1
+
e
l
o
g
2
=
1
3
or
[
d
d
x
(
f
−
1
(
x
)
)
]
x
=
l
o
g
2
=
1
3
The value of
f
(
9
)
is
Report Question
0%
240
0%
356
0%
252
0%
730
Explanation
f
(
0
)
=
0
So,
lim
x
→
0
f
(
x
)
x
=
f
′
(
0
)
=
1
(Using L'Hospital's rule)
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
+
x
2
y
+
x
y
2
Differentiating w.r.t.
x
, keeping
y
constant, we get
f
′
(
x
+
y
)
=
f
′
(
x
)
+
2
x
y
+
x
2
Substitute
x
=
0
. Then
f
′
(
y
)
=
y
2
+
1
or
f
′
(
x
)
=
x
2
+
1
∴
f
(
x
)
=
x
3
3
+
x
+
c
f
(
0
)
=
0
=
c
∴
f
(
x
)
=
x
3
3
+
x
f
′
(
3
)
=
10
and
f
(
9
)
=
243
+
9
=
252
Given,
f
(
x
)
=
−
x
3
3
+
x
2
sin
1.5
a
−
x
sin
a
⋅
sin
2
a
−
5
a
r
c
sin
(
a
2
−
8
a
+
17
)
, then
Report Question
0%
f
(
x
)
is not defined at
x
=
s
i
n
8
0%
f
′
(
s
i
n
8
)
>
0
0%
f
′
(
x
)
is not defined at
x
=
s
i
n
8
0%
f
′
(
s
i
n
8
)
<
0
Explanation
f
(
x
)
=
−
x
3
3
+
x
2
sin
1.5
a
−
x
sin
a
⋅
sin
2
a
−
5
a
r
c
sin
(
a
2
−
8
a
+
17
)
f
(
x
)
=
−
x
3
3
+
x
2
sin
6
−
x
sin
4
sin
8
−
5
sin
−
1
(
(
a
−
4
)
2
+
1
)
f
′
(
x
)
=
−
x
2
+
2
x
sin
6
−
sin
4.
sin
8
.......(
a
=
4
considering the domain of inverse sine function)
f
′
(
sin
8
)
=
−
sin
2
8
+
2
sin
6
sin
8
−
sin
4
sin
8
=
sin
8
(
−
sin
8
+
2
sin
6
−
sin
4
)
=
−
sin
8
(
sin
8
+
sin
4
−
2
sin
6
)
=
−
sin
8
(
2
sin
6
cos
2
−
2
sin
6
)
=
2
sin
8
sin
6
(
1
−
cos
2
)
Now,
s
i
n
8
>
0
(Since,
2
π
<
8
<
3
π
)
s
i
n
6
<
0
(Since,
π
<
6
<
2
π
)
(
1
−
cos
2
)
>
0
Hence,
f
′
(
sin
8
)
<
0
Let f be a twice differentiable such that
f
″
(
x
)
=
−
f
(
x
)
and
f
′
(
x
)
=
g
(
x
)
. If
h
(
x
)
=
{
f
(
x
)
}
2
+
{
g
(
x
)
}
2
, where
h
(
5
)
=
11
. Find
h
(
10
)
Report Question
0%
1
0%
10
0%
11
0%
100
Explanation
h
′
(
x
)
=
2
f
(
x
)
×
f
′
(
x
)
+
2
g
(
x
)
g
′
(
x
)
g
(
x
)
=
f
′
(
x
)
g
′
(
x
)
=
f
′
′
(
x
)
=
−
f
(
x
)
h
′
(
x
)
=
2
f
(
x
)
f
′
(
x
)
−
2
g
(
x
)
f
(
x
)
h
′
(
x
)
=
2
f
(
x
)
g
(
x
)
−
2
g
(
x
)
f
(
x
)
=
0
∴
h
′
(
x
)
=
0
⇒
d
d
x
h
(
x
)
=
0
⇒
∫
d
h
(
x
)
=
∫
0
×
d
x
⇒
h
(
x
)
=
0
+
c
⇒
h
(
x
)
=
c
now,
h
(
5
)
=
11
∴
h
(
10
)
=
h
(
5
)
=
11
Derivative of
(
x
cos
x
)
x
with respect to
x
is
Report Question
0%
(
x
cos
x
)
x
[
(
log
x
+
1
)
−
{
log
cos
x
+
x
cos
x
.
(
sin
x
)
}
]
0%
(
x
cos
x
)
x
[
(
log
x
+
1
)
+
{
log
cos
x
+
x
cos
x
.
(
−
sin
x
)
}
]
0%
(
x
cos
x
)
x
[
(
log
x
+
1
)
+
{
log
sin
x
+
x
cos
x
.
(
cos
x
)
}
]
0%
None of these
Explanation
Let
y
=
(
x
cos
x
)
x
.
Taking logarithm on both sides, we get
log
y
=
log
(
x
cos
x
)
x
=
x
log
(
x
cos
x
)
=
x
log
x
+
x
log
cos
x
Differentiating both sides with respect to
x
, we get
1
y
d
y
d
x
=
d
d
x
(
x
log
x
)
+
d
d
x
(
x
log
cos
x
)
d
y
d
x
=
y
[
(
1
+
log
x
)
+
(
log
cos
x
+
x
cos
x
d
d
x
(
cos
x
)
)
]
∴
d
y
d
x
=
(
x
cos
x
)
x
[
(
log
x
+
1
)
+
{
log
cos
x
+
x
cos
x
.
(
−
sin
x
)
}
]
Find the derivative of
|
x
|
+
a
0
x
n
+
a
1
x
n
−
1
+
a
2
x
n
−
2
+
.
.
.
.
+
a
n
−
1
x
+
a
n
Report Question
0%
x
|
x
|
+
n
a
0
x
n
−
1
+
(
n
−
1
)
a
1
x
n
−
2
+
(
n
−
2
)
a
2
x
n
−
3
+
.
.
.
.
+
a
n
−
1
0%
1
+
n
a
0
x
n
−
1
+
(
n
−
1
)
a
1
x
n
−
2
+
(
n
−
2
)
a
2
x
n
−
3
+
.
.
.
.
+
a
n
−
1
0%
x
|
x
|
+
n
a
0
x
n
−
1
+
(
n
−
1
)
a
1
x
n
−
2
+
(
n
−
2
)
a
2
x
n
−
3
+
.
.
.
.
+
a
n
0%
None of these
Explanation
We can write
|
x
|
as
√
x
2
so,
d
(
√
x
2
)
d
x
=
2
x
2
√
x
2
=
x
|
x
|
and,
d
(
a
0
x
n
)
d
x
=
n
a
0
x
n
−
1
Similarly,
d
(
a
n
−
1
x
)
d
x
=
a
n
−
1
and
d
(
a
n
)
d
x
=
0
Hence, option A is the correct option.
Equation
x
n
−
1
=
0
,
n
>
1
,
n
ϵ
N
, has roots
1
,
a
2
,
.
.
.
.
a
n
.
The value of
n
∑
r
=
2
1
2
−
a
r
, is
Report Question
0%
2
n
−
1
(
n
−
2
)
+
1
2
n
−
1
0%
2
n
(
n
−
2
)
+
1
2
n
−
1
0%
2
n
−
1
(
n
−
1
)
−
1
2
n
−
1
0%
none of these
Explanation
Equation
x
n
−
1
=
0
,
n
>
1
,
n
ϵ
N
, has roots
1
,
a
2
,
…
a
n
.
We have to find the value of
n
∑
r
=
2
1
2
−
a
r
x
n
−
1
=
(
x
−
1
)
(
x
−
a
2
)
…
(
x
−
a
n
)
→
{i}
Taking log on both the sides
log
(
x
n
−
1
)
=
log
(
x
−
1
)
+
log
(
x
−
a
2
)
+
⋯
+
log
(
x
−
a
n
)
Differentiating both the sides, we get
n
x
n
−
1
x
n
−
1
=
1
x
−
1
+
1
x
−
a
2
+
⋯
+
1
x
−
a
n
→
{ii}
Putting
x
=
2
in eqn {ii}
n
2
n
−
1
2
n
−
1
=
1
+
1
2
−
a
2
+
⋯
+
1
2
−
a
n
1
2
−
a
2
+
⋯
+
1
2
−
a
n
=
n
2
n
−
1
2
n
−
1
−
1
=
n
2
n
−
1
−
2
n
+
1
2
n
−
1
1
2
−
a
2
+
⋯
+
1
2
−
a
n
=
2
n
−
1
(
n
−
2
)
+
1
2
n
−
1
Let
f
(
x
)
=
√
x
−
1
+
√
x
+
24
−
10
√
x
−
1
,
1
≤
x
≤
26
be a real valued function, then
f
′
(
x
)
for
1
<
x
<
26
is
Report Question
0%
0
0%
1
√
x
−
1
0%
2
√
x
−
1
0%
1
Explanation
We have,
f
(
x
)
=
√
x
−
1
+
√
x
+
24
−
10
√
x
−
1
,
1
<
x
<
26
Rearrange the terms so as to get
(
a
−
b
)
2
=
a
2
−
2
a
b
+
b
2
f
(
x
)
=
√
x
−
1
+
√
(
x
−
1
)
+
25
−
10
√
x
−
1
∴
f
(
x
)
=
√
x
−
1
+
√
(
√
x
−
1
−
5
)
2
f
(
x
)
=
√
x
−
1
+
|
√
x
−
1
−
5
|
∴
f
(
x
)
=
√
x
−
1
−
(
√
x
−
1
−
5
)
[
∵
√
x
−
1
−
5
<
0
for
1
<
x
<
26
]
Now, differentiating w.r. to
x
, we get
∴
f
′
(
x
)
=
0
for all
x
∈
(
1
,
26
)
f
:
R
→
R
and
f
(
x
)
=
2
a
x
+
s
i
n
2
x
, then the set of values of a for which
f
(
x
)
is one-one and onto is
Report Question
0%
a
∈
(
−
1
2
,
1
2
)
0%
a
∈
(
−
1
,
1
)
0%
a
∈
R
−
(
−
1
2
,
1
2
)
0%
a
∈
R
−
(
−
1
,
1
)
Explanation
f
(
x
)
=
2
a
x
+
sin
2
x
Now
f
′
(
x
)
=
2
a
+
2
cos
2
x
=
2
(
a
+
cos
2
x
)
Now
1
≤
cos
2
x
≤
−
1
Hence if
a
∈
R
−
(
−
1
,
1
)
, then we won't get any critical point. Else we get critical points.
Hence
f
′
(
x
)
does not change signs if
a
∈
R
−
(
−
1
,
1
)
.
Therefore
f
(
x
)
is a one one function in
R
−
(
−
1
,
1
)
.
If
y
and
z
are the functions of
x
and if
y
2
+
z
2
=
λ
2
, then
y
d
d
x
(
y
λ
)
+
d
d
x
(
z
2
λ
)
is equal to
Report Question
0%
z
λ
d
z
d
x
0%
z
λ
d
x
d
z
0%
λ
z
d
z
d
x
0%
None of these
Explanation
We have,
λ
=
√
y
2
+
z
2
Now,
y
d
d
x
(
y
λ
)
+
d
d
x
(
z
2
λ
)
=
y
d
d
x
(
y
√
y
2
+
z
2
)
+
d
d
x
(
z
2
√
y
2
+
z
2
)
=
y
[
√
y
2
+
z
2
d
d
x
−
y
(
y
d
y
d
x
+
z
d
z
d
x
)
√
y
2
+
z
2
y
2
+
z
2
]
+
[
(
√
y
2
+
z
2
)
2
z
d
z
d
x
−
z
2
(
y
d
y
d
x
+
z
d
z
d
x
)
√
y
2
+
z
2
y
2
+
z
2
]
=
1
(
y
2
+
z
2
)
3
2
×
[
y
d
y
d
x
(
y
2
+
z
2
)
−
y
2
(
y
d
y
d
x
+
z
d
z
d
x
)
+
2
z
d
z
d
x
(
y
2
+
z
2
)
−
z
2
(
y
d
y
d
x
+
z
d
z
d
x
)
]
=
1
(
y
2
+
z
2
)
3
2
[
z
(
y
2
+
z
2
)
d
z
d
x
]
=
z
√
y
2
+
z
2
d
z
d
x
=
z
λ
d
z
d
x
.
Let
h
(
x
)
be differentiable for all
x
and let
f
(
x
)
=
(
k
x
+
e
x
)
h
(
x
)
where
k
is some constant. If
h
(
0
)
=
5
,
h
′
(
0
)
=
−
2
and
f
′
(
0
)
=
18
, then the value of
k
is equal to
Report Question
0%
3
0%
4
0%
5
0%
1
Explanation
f
(
x
)
=
(
k
x
+
e
x
)
.
h
(
x
)
h
(
0
)
=
5
,
h
′
(
0
)
=
−
2
,
f
′
(
0
)
=
18
f
′
(
x
)
=
d
d
x
[
(
k
x
+
e
x
)
h
(
x
)
]
⇒
f
′
(
x
)
=
(
k
x
+
e
x
)
d
d
x
h
(
x
)
+
h
(
x
)
d
d
x
(
k
x
+
e
x
)
⇒
f
′
(
x
)
=
(
k
x
+
e
x
)
h
′
(
x
)
+
h
(
x
)
(
k
+
e
x
)
put
x
=
0
, we get
f
′
(
0
)
=
(
0
+
1
)
h
′
(
0
)
+
h
(
0
)
(
k
+
e
0
)
f
′
(
0
)
=
h
′
(
0
)
+
h
(
0
)
(
k
+
1
)
18
=
−
2
+
5
(
k
+
1
)
20
=
5
(
k
+
1
)
4
=
k
+
1
k
=
3
Let
f
be a differentiable function satisfying
f
(
x
)
+
f
(
y
)
+
f
(
z
)
+
f
(
x
)
f
(
y
)
f
(
z
)
=
14
for all
x
,
y
,
z
∈
R
Then,
Report Question
0%
f
′
(
x
)
<
0
for all
x
∈
R
0%
f
′
(
x
)
=
0
for all
x
∈
R
0%
f
′
(
x
)
>
0
for all
x
∈
R
0%
none of these
Explanation
We have,
f
(
x
)
+
f
(
y
)
+
f
(
z
)
+
f
(
x
)
f
(
y
)
f
(
z
)
=
14
for all
x
,
y
,
z
∈
R
.
.
.
(
i
)
Putting
x
=
y
=
z
=
0
, we get,
3
f
(
0
)
+
{
f
(
0
)
}
3
=
14
{
f
(
0
)
}
3
+
3
f
(
0
)
−
14
=
0
f
(
0
)
=
2.
Now, putting
y
=
z
=
x
in
(
i
)
, we get
3
f
′
(
x
)
+
3
{
f
(
x
)
}
2
f
′
(
x
)
=
0
for all
x
∈
R
{
{
f
(
x
)
}
2
+
1
}
f
′
(
x
)
=
0
for all
x
∈
R
f
′
(
x
)
=
0
for all
x
∈
R
Let
f
(
x
)
=
x
2
+
x
g
′
(
1
)
+
g
"
(
2
)
and
g
(
x
)
=
f
(
1
)
.
x
2
+
x
f
′
(
x
)
+
f
"
(
x
)
then
Report Question
0%
f
′
(
1
)
+
f
′
(
2
)
=
0
0%
g
′
(
2
)
=
g
′
(
1
)
0%
g
″
(
2
)
+
f
″
(
3
)
=
6
0%
none of these
Explanation
Let
g
′
(
1
)
=
a
and
g
′
(
2
)
=
b
--------(1)
Then
f
(
x
)
=
x
2
+
a
x
+
b
,
f
(
1
)
=
1
+
a
+
b
f
′
(
x
)
=
2
x
+
a
,
f
″
(
x
)
=
2
∴
g
(
x
)
=
(
1
+
a
+
b
)
x
2
+
(
2
x
+
a
)
x
+
2
=
x
2
(
3
+
a
+
b
)
+
a
x
+
2
⇒
g
′
(
x
)
=
2
x
(
3
+
a
+
b
)
+
a
Hence
g
′
(
1
)
=
2
(
3
+
a
+
b
)
+
a
--------(2)
g
′
(
2
)
=
4
(
3
+
a
+
b
)
+
a
--------(3)
From (1),(2) and (3)
a
=
2
(
3
+
a
+
b
)
+
a
and
b
=
2
(
3
+
a
+
b
)
⇒
3
+
a
+
b
=
0
and
b
+
2
a
+
6
=
0
∴
f
(
x
)
=
x
2
−
3
x
and
g
(
x
)
=
−
3
x
+
2
∴
f
′
(
1
)
+
f
′
(
2
)
=
0
g
″
(
2
)
+
f
″
(
3
)
=
6
If
√
y
+
x
+
√
y
−
x
=
c
(where
c
≠
0
), then
d
y
d
x
has the value equal to
Report Question
0%
2
x
c
2
0%
x
y
+
√
y
2
−
x
2
0%
y
−
√
y
2
−
x
2
x
0%
c
2
2
y
Explanation
√
y
+
x
+
√
y
−
x
=
c
Squaring both sides, we get
(
√
y
+
x
+
√
y
−
x
)
2
=
c
2
2
y
+
2
√
y
2
−
x
2
=
c
2
y
+
√
y
2
−
x
2
=
c
2
2
...........(1)
Differentiating bothe the sides, we get
d
y
d
x
+
1
2
√
y
2
−
x
2
(
2
y
d
y
d
x
−
2
x
)
=
0
d
y
d
x
+
y
√
y
2
−
x
2
d
y
d
x
−
x
√
y
2
−
x
2
=
0
d
y
d
x
(
1
+
y
√
y
2
−
x
2
)
=
x
√
y
2
−
x
2
d
y
d
x
=
x
y
+
√
y
2
−
x
2
..................Option (b)
On rationalizing we get,
d
y
d
x
=
x
(
y
−
√
y
2
−
x
2
)
x
2
=
y
−
√
y
2
−
x
2
x
........Option (c)
Putting
y
+
√
y
2
−
x
2
=
c
2
. we get
d
y
d
x
=
x
c
2
2
=
2
x
c
2
......... Option (a)
Equation
x
n
−
1
=
0
,
n
>
1
,
n
ϵ
N
, has roots
1
,
a
2
,
.
.
.
.
a
n
.
The value of
n
∑
r
=
2
1
1
−
a
r
, is
Report Question
0%
n
4
0%
n
(
n
−
1
)
2
0%
n
−
1
2
0%
none of these
Explanation
Equation
x
n
−
1
=
0
,
n
>
1
,
n
ϵ
N
, has roots
1
,
a
1
,
a
2
,
…
a
n
We have to find value of
n
∑
r
=
2
1
1
−
a
r
x
n
−
1
=
(
x
−
1
)
(
x
−
a
2
)
…
(
x
−
a
n
)
→
{i}
Taking log on both the sides
log
(
x
n
−
1
)
=
log
(
x
−
1
)
+
log
(
x
−
a
2
)
+
⋯
+
log
(
x
−
a
n
)
Differentiating both the sides, we get
n
x
n
−
1
x
n
−
1
=
1
x
−
1
+
1
x
−
a
2
+
⋯
+
1
x
−
a
n
→
{ii}
n
x
n
−
1
x
n
−
1
−
1
x
−
1
=
1
x
−
a
2
+
…
1
x
−
a
n
∵
(
x
n
−
1
)
=
(
x
−
1
)
(
1
+
x
+
x
2
+
x
3
+
x
4
+
⋯
+
x
n
−
1
)
∴
n
x
n
−
1
−
1
(
1
+
x
+
x
2
+
⋯
+
x
n
−
1
)
x
n
−
1
=
1
x
−
a
2
+
…
1
x
−
a
n
lim
x
→
1
(
n
x
n
−
1
−
1
(
1
+
x
+
x
2
+
⋯
+
x
n
−
1
)
x
n
−
1
)
=
lim
x
→
1
(
1
x
−
a
2
+
…
1
x
−
a
n
)
Applying L'Hospital's rule on L.H.S
lim
x
→
1
n
(
n
−
1
)
x
n
−
2
−
{
1
+
2
x
+
⋯
+
(
n
−
1
)
x
n
−
2
}
n
x
n
−
1
=
1
1
−
a
2
+
…
1
1
−
a
n
⇒
n
(
n
−
1
)
−
[
1
+
2
+
⋯
+
(
n
−
1
)
]
n
=
1
1
−
a
2
+
…
1
1
−
a
n
∴
1
1
−
a
2
+
…
1
1
−
a
n
=
n
−
1
2
Suppose,
A
=
d
y
d
x
of
x
2
+
y
2
=
4
at
(
√
2
,
√
2
)
,
B
=
d
y
d
x
of
s
i
n
y
+
s
i
n
x
=
s
i
n
x
⋅
s
i
n
y
at
(
π
,
π
)
and
C
=
d
y
d
x
of
2
e
x
y
+
e
x
e
y
−
e
x
=
e
x
y
+
1
at
(
1
,
1
)
, then
(
A
−
B
−
C
)
has the value equal to .....
Report Question
0%
1
2
0%
1
3
0%
1
0%
2
Explanation
A
:
d
d
x
(
x
2
+
y
2
=
4
)
at
(
√
2
,
√
2
)
)
2
x
+
2
y
d
y
d
x
=
0
d
y
d
x
=
−
x
y
=
−
√
2
√
2
=
−
1
B
:
d
d
x
(
sin
y
+
sin
x
=
sin
x
⋅
sin
y
)
at
(
π
,
π
)
cos
y
d
y
d
x
+
cos
x
=
sin
x
cos
y
d
y
d
x
+
sin
y
cos
x
d
y
d
x
(
cos
y
−
sin
x
cos
y
)
=
sin
y
cos
x
−
cos
x
d
y
d
x
=
cos
x
(
sin
y
−
1
)
cos
y
(
1
−
sin
x
)
B
=
−
1
(
0
−
1
)
−
1
(
1
−
0
)
=
−
1
C
:
d
d
x
(
2
e
x
y
+
e
x
e
y
−
e
x
=
e
x
y
+
1
)
at
(
1
,
1
)
[
2
e
x
y
(
x
d
y
d
x
+
y
)
]
+
e
x
e
y
d
y
d
x
+
e
y
e
x
−
e
x
=
e
x
y
+
1
(
x
d
y
d
x
+
y
)
d
y
d
x
=
y
e
x
y
+
1
−
2
y
e
x
y
−
e
x
+
y
+
e
x
2
x
e
x
y
+
e
x
+
y
−
x
e
x
y
+
1
.......... since
e
x
e
y
=
e
x
+
y
C
=
e
2
−
2
e
1
−
e
2
+
e
1
2
e
1
+
e
2
−
e
2
=
−
e
1
2
e
1
=
−
1
2
Therefore
A
−
B
−
C
=
1
2
If
f
(
x
)
=
√
x
+
2
√
2
x
−
4
+
√
x
−
2
√
2
x
−
4
, then the value of
10
f
′
(
102
+
)
is
Report Question
0%
−
1
0%
0
0%
1
0%
Does not exist
Explanation
f
(
x
)
=
√
x
+
2
√
2
x
−
4
+
√
x
−
2
√
2
x
−
4
f
(
x
)
=
√
(
√
x
−
2
+
√
2
)
2
+
√
(
√
x
−
2
−
√
2
)
2
f
(
x
)
=
|
√
x
−
2
+
√
2
|
+
|
√
x
−
2
−
√
2
|
For
√
x
−
2
to exist,
x
≥
2
Also
√
x
−
1
+
√
2
>
0
.............(always true)
But
√
x
−
1
−
√
2
≥
0
only if
x
≥
4
<
0
only if
x
<
4
Now
f
(
x
)
becomes
f
(
x
)
=
√
x
−
2
+
√
2
−
√
x
−
2
+
√
2
for
2
≤
x
<
4
f
(
x
)
=
√
x
−
2
+
√
2
+
√
x
−
2
−
√
2
for
x
≥
4
f
(
x
)
=
2
√
2
,
for
2
≤
x
<
4
f
(
x
)
=
2
√
x
−
2
for
4
≤
x
<
∞
f is continous
[
2
,
4
]
∪
[
4
,
∞
]
f
′
(
x
)
=
0
for
2
≤
x
<
4
f
′
(
x
)
=
1
√
x
−
2
for
≤
x
<
∞
f
′
(
102
+
)
=
1
√
102
−
2
=
1
10
10
f
′
(
102
+
)
=
1
Let
f
(
x
)
be a polynomial function of second degree.If
f
(
1
)
=
f
(
−
1
)
and
a
,
b
,
c
are in A.P
f
′
(
a
)
,
f
′
(
b
)
,
f
′
(
c
)
are in
Report Question
0%
G.P.
0%
H.P.
0%
A.G.P
0%
A.P.
Explanation
Let
f
(
x
)
=
p
x
2
+
q
x
+
r
Differentiatin g w.r. to
x
, we get
f
′
(
x
)
=
2
p
x
+
q
Now,
f
(
−
1
)
=
f
(
1
)
∴
p
−
q
+
r
=
p
+
q
+
r
∴
q
=
0
∴
f
′
(
x
)
=
2
p
x
f
′
(
a
)
=
2
p
a
.
f
′
(
b
)
=
2
p
b
,
f
′
(
c
)
=
2
p
c
Since
a
,
b
,
c
are in
A
.
P
.
, therefore
2
b
=
a
+
c
4
p
b
=
2
p
a
+
2
p
c
2
f
′
(
b
)
=
f
′
(
a
)
+
f
′
(
c
)
∴
f
′
(
a
)
,
f
′
(
b
)
,
f
′
(
c
)
are in A.P
For the curve represented implicitly as
3
x
−
2
y
=
1
, the value of
lim
x
→
∞
(
d
y
d
x
)
is
Report Question
0%
equal to
1
0%
equal to
0
0%
equal to
log
2
3
0%
non existent
Explanation
y
=
l
o
g
(
3
x
−
1
)
l
o
g
2
y
′
=
3
x
l
o
g
3
(
3
x
−
1
)
l
o
g
2
At
x
→
∞
,
3
x
3
x
−
1
→
1
So,
y
′
=
l
o
g
3
l
o
g
2
Hence, option C is the correct option.
If
x
2
+
y
2
=
R
2
(
R
>
0
)
then
k
=
y
″
√
(
1
+
y
′
2
)
3
where
k
in terms of
R
alone is equal to
Report Question
0%
−
1
R
2
0%
−
1
R
0%
2
R
0%
−
2
R
2
Explanation
x
2
+
y
2
=
R
2
x
=
−
y
y
′
=>
y
′
=
−
x
y
y
″
=
−
y
+
x
y
′
y
2
=
−
y
2
−
x
2
y
3
=
−
R
2
y
3
k
=
y
″
√
(
1
+
y
′
2
)
3
=
−
R
2
y
3
√
(
1
+
x
2
y
2
)
3
=
−
R
2
y
3
√
(
R
2
y
2
)
3
=
−
R
2
R
3
=
−
1
R
If
cos
4
θ
x
+
sin
4
θ
y
=
1
x
+
y
then
d
y
d
x
=
Report Question
0%
x
y
0%
tan
2
θ
0%
0
0%
(
x
2
+
y
2
)
sec
2
θ
Explanation
Given
cos
4
θ
x
+
sin
4
θ
y
=
1
x
+
y
⇒
(
x
+
y
)
(
cos
4
θ
x
+
sin
4
θ
y
)
=
(
cos
2
θ
+
sin
2
θ
)
2
(
∵
cos
2
θ
+
sin
2
θ
=
1
)
∴
y
x
cos
4
θ
+
x
y
sin
4
θ
−
2
sin
2
θ
cos
2
θ
=
0
⇒
(
√
y
x
cos
2
θ
−
√
x
y
sin
2
θ
)
2
=
0
∴
tan
2
θ
=
y
x
⇒
y
=
x
tan
2
θ
⇒
d
y
d
x
=
tan
2
θ
Two functions
f
and
g
have first and second derivatives at
x
=
0
and satisfy the relations,
f
(
0
)
=
2
g
(
0
)
,
f
′
(
0
)
=
2
g
′
(
0
)
=
4
g
(
0
)
,
g
″
(
0
)
=
5
f
″
(
0
)
=
6
f
(
0
)
=
3
then
Report Question
0%
If
h
(
x
)
=
f
(
x
)
g
(
x
)
then
h
′
(
0
)
=
15
4
0%
If
k
(
x
)
=
f
(
x
)
.
g
(
x
)
sin
x
then
k
′
(
0
)
=
2
0%
lim
x
→
0
g
′
(
x
)
f
′
(
x
)
=
1
2
0%
None of these
Explanation
6
f
(
0
)
=
3
;
g
(
0
)
=
2
f
(
0
)
,
f
′
(
0
)
=
4
g
(
0
)
,
2
g
′
(
0
)
=
4
g
(
0
)
⇒
f
(
0
)
=
1
2
,
g
(
0
)
=
4
,
f
′
(
0
)
=
16
,
g
′
(
0
)
=
8
h
(
x
)
=
f
(
x
)
g
(
x
)
h
′
(
x
)
=
g
(
x
)
f
′
(
x
)
−
f
(
x
)
g
′
(
x
)
(
g
(
x
)
)
2
So,
h
′
(
0
)
=
4.16
−
1
2
.8
4.4
=
64
−
4
16
=
15
4
k
(
x
)
=
f
(
x
)
.
g
(
x
)
sin
x
k
′
(
x
)
=
f
(
x
)
.
g
(
x
)
c
o
s
x
+
s
i
n
x
(
f
(
x
)
g
′
(
x
)
+
g
(
x
)
f
′
(
x
)
)
k
′
(
0
)
=
1
2
.4
.1
+
0
=
2
lim
x
→
0
=
g
′
(
x
)
f
′
(
x
)
=
8
16
=
1
2
0:0:1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
0
Answered
1
Not Answered
29
Not Visited
Correct : 0
Incorrect : 0
Report Question
×
What's an issue?
Question is wrong
Answer is wrong
Other Reason
Want to elaborate a bit more? (optional)
Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers
<
>
Support mcqexams.com by disabling your adblocker.
×
Please disable the adBlock and continue.
Thank you.
Reload page