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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
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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
Applying L'Hospital's Rule,
=\lim_{x\rightarrow 0}\frac{f(x)}{x}=\frac{f'(x+y)-2xy-y^{2}}{1}=1(given)
=\frac{f'(y)-y^{2}}{1}=1(given)
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)\cdot 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)\cdot f(y)=1
.....(i)
Substitute
x=y=0
, we get
2f(0)+\left \{f(0)\right \}^2=1\Rightarrow \left \{f(0)\right \}^2+2f(0)-1=0
f(0)=\frac {-2\pm \sqrt {4+4}}{2}=-1-\sqrt 2
and
-1+\sqrt 2
As
f(0) > 0\Rightarrow f(0)=\sqrt 2-1
[neglecting
f(0)=-1-\sqrt 2
as f(0) is positive]
Again, putting
y=x
is Eq, (i),
2f(x)+\left \{f(x)\right \}^2=1
On differentiating w.r.t. x,
2f'(x)+2f(x)f'(x)=0
2f'(x)\left \{1+f(x)\right \}=0\Rightarrow 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
\dfrac {d}{dx} (f^{-1}(x))
at the point
f(log_e 2)
.
Report Question
0%
\dfrac {1}{3}
0%
1
0%
3
0%
0
Explanation
Let
y=e^x+x
on differentiating w.r.t y,
1=(e^x+1)\dfrac {dx}{dy}\Rightarrow \dfrac {dx}{dy}=\dfrac {1}{1+e^x}
\therefore \left (\dfrac {dx}{dy}\right )_{x=log_e2}=\dfrac {1}{1+e^{log 2}}=\dfrac {1}{3}
or
\left [\dfrac {d}{dx}(f^{-1}(x))\right ]_{x=log 2}=\dfrac {1}{3}
The value of
f(9)
is
Report Question
0%
240
0%
356
0%
252
0%
730
Explanation
f(0)=0
So,
\displaystyle\lim _{x\rightarrow 0}{\frac{f(x)}{x}}=f^\prime(0)=1
(Using L'Hospital's rule)
f(x+y)=f(x)+f(y)+x^2y+xy^2
Differentiating w.r.t.
x
, keeping
y
constant, we get
f^\prime(x+y)=f^\prime(x)+2xy+x^2
Substitute
x=0
. Then
f^\prime(y)=y^2+1
or
f^\prime(x)=x^2+1
\displaystyle\therefore f(x)=\frac{x^3}{3}+x+c
f(0)=0=c
\displaystyle\therefore f(x)=\frac{x^3}{3}+x
f^\prime(3)=10
and
f(9)=243+9=252
Given,
f(x)=-\displaystyle \frac {x^3}{3}+x^2 \sin 1.5 a-x \sin a\cdot \sin 2a-5 arc \sin (a^2-8a+17)
, then
Report Question
0%
f(x)
is not defined at
x=sin 8
0%
f' (sin 8) > 0
0%
f' (x)
is not defined at
x=sin 8
0%
f'(sin 8) < 0
Explanation
f(x)=-\displaystyle \frac {x^3}{3}+x^2 \sin 1.5 a-x \sin a\cdot \sin 2a-5 arc \sin (a^2-8a+17)
f(x)=\displaystyle \frac { -{ x }^{ 3 } }{ 3 } +{ x }^{ 2 }\sin { 6 } -x\sin { 4\sin { 8 } -5\sin ^{ -1 }{ \left( { \left( a-4 \right) }^{ 2 }+1 \right) } }
f'(x)=-{ x }^{ 2 }+2x\sin { 6-\sin { 4.\sin { 8 } } }
.......(
a=4
considering the domain of inverse sine function)
f'\left( \sin { \quad 8 } \right) =-\sin ^{ 2 }{ 8 } +2\sin { 6 } \sin { 8 } -\sin { 4 } \sin { 8 }
=\sin { 8\left( -\sin { 8+2\sin { 6-\sin { 4 } } } \right) }
=-\sin { 8 } \left( \sin { 8 } +\sin { 4-2\sin { 6 } } \right)
=-\sin { 8\left( 2\sin { 6\cos { 2-2\sin { 6 } } } \right) }
=2\sin { 8 } \sin { 6\left( 1-\cos { 2 } \right) }
Now,
sin 8 > 0
(Since,
2\pi < 8 < 3\pi
)
sin 6 < 0
(Since,
\pi < 6 < 2\pi
)
(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)=\left \{f(x)\right \}^2+\left \{g(x)\right \}^2
, where
h(5)=11
. Find
h(10)
Report Question
0%
1
0%
10
0%
11
0%
100
Explanation
h'(x) = 2f(x) \times f'(x) + 2 g(x) g'(x)
g(x) = f'(x)
g'(x) = f^{\prime \prime} (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
\therefore h'(x)= 0
\Rightarrow \dfrac{d}{dx} h(x) = 0
\Rightarrow \displaystyle \int d \ h(x) = \int 0 \times dx
\Rightarrow h(x) = 0 + c
\Rightarrow h(x) = c
now,
h(5) = 11
\therefore h(10) = h(5) = 11
Derivative of
{(x\cos{x})}^x
with respect to
x
is
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0%
\displaystyle {(x\cos{x})}^x\left[(\log{x}+1)-\left\{\log{\cos{x}}+\frac{x}{\cos{x}}.(\sin{x})\right\}\right]
0%
\displaystyle {(x\cos{x})}^x\left[(\log{x}+1)+\left\{\log{\cos{x}}+\frac{x}{\cos{x}}.(-\sin{x})\right\}\right]
0%
\displaystyle {(x\cos{x})}^x\left[(\log{x}+1)+\left\{\log{\sin{x}}+\frac{x}{\cos{x}}.(\cos{x})\right\}\right]
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
\displaystyle\frac{1}{y}\frac{dy}{dx}=\frac{d}{dx}(x\log{x})+\frac{d}{dx}(x\log{\cos{x}})
\displaystyle \frac { dy }{ dx } =y\left[ \left( 1+\log { x } \right) +\left( \log { \cos { x\quad +\displaystyle \frac { x }{ \cos { x } } \displaystyle \frac { d }{ dx } \left( \cos { x } \right) } } \right) \right]
\therefore \displaystyle\frac{dy}{dx}={(x\cos{x})}^x\left[(\log{x}+1)+\left\{\log{\cos{x}}+\frac{x}{\cos{x}}.(-\sin{x})\right\}\right]
Find the derivative of
|x|+a_0x^n+a_1x^{n-1}+a_2x^{n-2}+....+a_{n-1}x+a_n
Report Question
0%
\frac {x}{|x|}+na_0x^{n-1}+(n-1)a_1x^{n-2}+(n-2)a_2x^{n-3}+....+a_{n-1}
0%
1+na_0x^{n-1}+(n-1)a_1x^{n-2}+(n-2)a_2x^{n-3}+....+a_{n-1}
0%
\frac {x}{|x|}+na_0x^{n-1}+(n-1)a_1x^{n-2}+(n-2)a_2x^{n-3}+....+a_{n}
0%
None of these
Explanation
We can write
|x|
as
\sqrt{x^{2}}
so,
\frac{\mathrm{d} (\sqrt{x^{2}})}{\mathrm{d} x}=\frac{2x}{2\sqrt{x^{2}}}=\frac{x}{|x|}
and,
\frac{\mathrm{d} (a_{0}x^{n})}{\mathrm{d} x}=na_{0}x^{n-1}
Similarly,
\frac{\mathrm{d} (a_{n-1}x)}{\mathrm{d} x}=a_{n-1}
and
\frac{\mathrm{d} (a_{n})}{\mathrm{d} x}=0
Hence, option A is the correct option.
Equation
x^n-1=0, n > 1, n\epsilon N
, has roots
1, a_2, ....a_n
.
The value of
\displaystyle \sum_{r=2}^{n}\frac {1}{2-a_r}
, is
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0%
\displaystyle \frac {2^{n-1}(n-2)+1}{2^n-1}
0%
\displaystyle \frac {2^{n}(n-2)+1}{2^n-1}
0%
\displaystyle \frac {2^{n-1}(n-1)-1}{2^n-1}
0%
none of these
Explanation
Equation
x^n-1=0, n > 1, n\epsilon N
, has roots
1, a_2, \dots a_n
.
We have to find the value of
\displaystyle \sum_{r=2}^{n}\frac {1}{2-a_r}
{ x }^{ n }-1=\left( x-1 \right) \left( x-a_{ 2 } \right)\dots\left( x-{ a }_{ n } \right)\rightarrow
{i}
Taking log on both the sides
\log { \left( { x }^{ n }-1 \right) =\log { \left( x-1 \right)} +\log { \left( x-{ a }_{ 2 } \right) +\dots+ } \log { \left( x-{ a }_{ n } \right) } }
Differentiating both the sides, we get
\displaystyle \frac { n{ x }^{ n-1 } }{ { x }^{ n }-1 } =\displaystyle \frac { 1 }{ x-1 } +\displaystyle \frac { 1 }{ x-{ a }_{ 2 } } +\dots+\displaystyle \frac { 1 }{ x-{ a }_{ n } } \rightarrow
{ii}
Putting
x=2
in eqn {ii}
\displaystyle \frac { n{ 2 }^{ n-1 } }{ { 2 }^{ n }-1 } =1+\displaystyle \frac { 1 }{ 2-{ a }_{ 2 } } +\dots+\displaystyle \frac { 1 }{ 2-{ a }_{ n } }
\displaystyle \frac { 1 }{ 2-{ a }_{ 2 } } +\dots+\displaystyle \frac { 1 }{ 2-{ a }_{ n } } =\displaystyle \frac { n{ 2 }^{ n-1 } }{ { 2 }^{ n }-1 } -1=\displaystyle \frac { n{ 2 }^{ n-1 }-{ 2 }^{ n }+1 }{ { 2 }^{ n }-1 }
\displaystyle \frac { 1 }{ 2-{ a }_{ 2 } } +\dots+\displaystyle \frac { 1 }{ 2-{ a }_{ n } } =\displaystyle \frac { { 2 }^{ n-1 }\left( n-2 \right) +1 }{ { 2 }^{ n }-1 }
Let
f(x) = \sqrt{x - 1} + \sqrt{x + 24 - 10\sqrt{x - 1}}, 1 \le x \le 26
be a real valued function, then
f'(x)
for
1 < x < 26
is
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0
0%
\displaystyle\frac{1}{\sqrt{x - 1}}
0%
2\sqrt{x - 1}
0%
1
Explanation
We have,
f(x) = \sqrt{x - 1} + \sqrt{x + 24 - 10\sqrt{x - 1}}, 1 < x < 26
Rearrange the terms so as to get
{ \left( a-b \right) }^{ 2 }=a^2-2ab+b^2
f(x) = \sqrt{x - 1} + \sqrt{(x - 1) + 25 - 10\sqrt{x - 1}}
\therefore f(x) = \sqrt{x - 1} + \sqrt{(\sqrt {x - 1} - 5)^2}
f(x) = \sqrt{x - 1} + \left|\sqrt{x - 1} - 5\right|
\therefore f(x) = \sqrt{x - 1} - \left(\sqrt{x - 1} - 5\right) \quad\quad [\because\sqrt{x - 1} - 5 < 0 \mbox{ for } 1 < x < 26]
Now, differentiating w.r. to
x
, we get
\therefore f'(x) = 0
for all
x \in (1, 26)
f:R\rightarrow R
and
f(x)=2ax+sin 2x
, then the set of values of a for which
f(x)
is one-one and onto is
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a\in \left (-\dfrac {1}{2}, \dfrac {1}{2}\right )
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a\in (-1, 1)
0%
a\in R-\left (-\dfrac {1}{2}, \dfrac {1}{2}\right )
0%
a\in R-(-1, 1)
Explanation
f(x)=2ax+\sin 2x
Now
f'(x)=2a+2\cos 2x
=2(a+\cos 2x)
Now
1\leq \cos 2x\leq -1
Hence if
a\in 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\in 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 }={ \lambda }^{ 2 }
, then
\displaystyle y\frac { d }{ dx } \left( \frac { y }{ \lambda } \right) +\frac { d }{ dx } \left( \frac { { z }^{ 2 } }{ \lambda } \right)
is equal to
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\displaystyle \frac { z }{ \lambda } \frac { dz }{ dx }
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\displaystyle \frac { z }{ \lambda } \frac { dx }{ dz }
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\displaystyle \frac { \lambda }{ z } \frac { dz }{ dx }
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None of these
Explanation
We have,
\lambda =\sqrt { { y }^{ 2 }+{ z }^{ 2 } }
Now,
\displaystyle y\dfrac { d }{ dx } \left( \dfrac { y }{ \lambda } \right) +\dfrac { d }{ dx } \left( \dfrac { { z }^{ 2 } }{ \lambda } \right)
\displaystyle =y\dfrac { d }{ dx } \left( \dfrac { y }{ \sqrt { { y }^{ 2 }+{ z }^{ 2 } } } \right) +\dfrac { d }{ dx } \left( \dfrac { { z }^{ 2 } }{ \sqrt { { y }^{ 2 }+{ z }^{ 2 } } } \right)
\displaystyle =y\left[ \dfrac { \sqrt { { y }^{ 2 }+{ z }^{ 2 } } \dfrac { d }{ dx } -\dfrac { y\left( y\dfrac { dy }{ dx } +z\dfrac { dz }{ dx } \right) }{ \sqrt { { y }^{ 2 }+{ z }^{ 2 } } } }{ { y }^{ 2 }+{ z }^{ 2 } } \right] +\left[ \dfrac { \left( \sqrt { { y }^{ 2 }+{ z }^{ 2 } } \right) 2z\dfrac { dz }{ dx } -\dfrac { { z }^{ 2 }\left( y\dfrac { dy }{ dx } +z\dfrac { dz }{ dx } \right) }{ \sqrt { { y }^{ 2 }+{ z }^{ 2 } } } }{ { y }^{ 2 }+{ z }^{ 2 } } \right]
\displaystyle =\dfrac { 1 }{ { \left( { y }^{ 2 }+{ z }^{ 2 } \right) }^{ \dfrac { 3 }{ 2 } } }\times
\displaystyle \left[ y\dfrac { dy }{ dx } \left( { y }^{ 2 }+{ z }^{ 2 } \right) -{ y }^{ 2 }\left( y\dfrac { dy }{ dx } +z\dfrac { dz }{ dx } \right) +2z\dfrac { dz }{ dx } \left( { y }^{ 2 }+{ z }^{ 2 } \right) -{ z }^{ 2 }\left( y\dfrac { dy }{ dx } +z\dfrac { dz }{ dx } \right) \right]
\displaystyle =\dfrac { 1 }{ { \left( { y }^{ 2 }+{ z }^{ 2 } \right) }^{ \dfrac { 3 }{ 2 } } } \left[ z\left( { y }^{ 2 }+{ z }^{ 2 } \right) \dfrac { dz }{ dx } \right]
\displaystyle =\dfrac { z }{ \sqrt { { y }^{ 2 }+{ z }^{ 2 } } } \dfrac { dz }{ dx } =\dfrac { z }{ \lambda } \dfrac { dz }{ dx } .
Let
h(x)
be differentiable for all
x
and let
f(x)=(kx+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)=\left( kx+{ e }^{ x } \right) .h(x)
h(0)=5,h'(0)=-2,f'(0)=18
f'(x)=\displaystyle \dfrac { d }{ dx } \left[ \left( kx+{ e }^{ x } \right) h(x) \right]
\Rightarrow f'(x)=\left( kx+{ e }^{ x } \right) \displaystyle \frac { d }{ dx } h(x)+h(x)\frac { d }{ dx } \left( kx+{ e }^{ x } \right)
\Rightarrow f'(x)=\left( kx+{ e }^{ x } \right) h'(x)+h(x)\left( k+{ e }^{ x } \right)
put
x=0
, we get
f'(0)=\left( 0+1 \right) h'(0)+h(0)\left( k+{ e }^{ 0 } \right)
f'(0)=h'(0)+h(0)\left( k+1 \right)
18=-2+5\left( k+1 \right)
20=5\left( k+1 \right)
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,\space y,\space z \in R
Then,
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f'(x) < 0
for all
x \in R
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f'(x) = 0
for all
x \in R
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f'(x) > 0
for all
x \in R
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none of these
Explanation
We have,
f(x) + f(y) + f(z) + f(x)f(y)f(z) = 14
for all
x,\space y,\space z \in R \quad\quad ...(i)
Putting
x = y = z = 0
, we get,
3f(0) + \left\{f(0)\right\}^3 = 14
\left\{f(0)\right\}^3 + 3f(0) - 14 = 0
f(0) = 2.
Now, putting
y = z = x
in
(i)
, we get
3f'(x) + 3\left\{f(x)\right\}^2f'(x) = 0 \quad
for all
x \in R
\left\{\left\{f(x)\right\}^2 + 1\right\}f'(x) = 0 \quad
for all
x \in R
f'(x) = 0 \quad
for all
x \in R
Let
f(x)=x^{2}+xg^{'}(1)+g^{"}(2)
and
g(x)=f(1).x^{2}+xf^{'}(x)+f^"(x)
then
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f^{'}(1)+f^{'}(2)=0
0%
g^{'}(2)=g^{'}(1)
0%
g^{''}(2)+f^{''}(3)=6
0%
none of these
Explanation
Let
g'\left( 1 \right) =a
and
g'\left( 2 \right) =b
--------(1)
Then
f\left( x \right) ={ x }^{ 2 }+ax+b,f\left( 1 \right) =1+a+b\\ f'\left( x \right) =2x+a,f''\left( x \right) =2\\ \therefore g\left( x \right) =\left( 1+a+b \right) { x }^{ 2 }+\left( 2x+a \right) x+2\\ ={ x }^{ 2 }\left( 3+a+b \right) +ax+2\\ \Rightarrow g'\left( x \right) =2x\left( 3+a+b \right) +a
Hence
g'\left( 1 \right) =2\left( 3+a+b \right) +a
--------(2)
g'\left( 2 \right) =4\left( 3+a+b \right) +a
--------(3)
From (1),(2) and (3)
a=2\left( 3+a+b \right) +a
and
b=2\left( 3+a+b \right)
\Rightarrow 3+a+b=0
and
b+2a+6=0
\therefore f\left( x \right) ={ x }^{ 2 }-3x
and
g\left( x \right) =-3x+2
\therefore f'\left( 1 \right) +f'\left( 2 \right) =0\\ \\ g''\left( 2 \right) +f''\left( 3 \right) =6
If
\sqrt {y+x}+\sqrt {y-x}=c
(where
c\neq 0
), then
\displaystyle \frac {dy}{dx}
has the value equal to
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\displaystyle \frac {2x}{c^2}
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\displaystyle \frac {x}{y+\sqrt {y^2-x^2}}
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\displaystyle \frac {y-\sqrt {y^2-x^2}}{x}
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\displaystyle \frac {c^2}{2y}
Explanation
\sqrt {y+x}+\sqrt {y-x}=c
Squaring both sides, we get
{ \left( \sqrt { y+x } +\sqrt { y-x } \right) }^{ 2 }={ c }^{ 2 }
2y+2\sqrt { { y }^{ 2 }-{ x }^{ 2 } } ={ c }^{ 2 }
y+\sqrt { { y }^{ 2 }-{ x }^{ 2 } } =\displaystyle \frac { { c }^{ 2 } }{ 2 }
...........(1)
Differentiating bothe the sides, we get
\displaystyle \frac { dy }{ dx } +\displaystyle \frac { 1 }{ 2\sqrt { { y }^{ 2 }-{ x }^{ 2 } } } \left( 2y\displaystyle \frac { dy }{ dx } -2x \right) =0
\displaystyle \frac { dy }{ dx } +\displaystyle \frac { y }{ \sqrt { { y }^{ 2 }-{ x }^{ 2 } } } \displaystyle \frac { dy }{ dx } -\displaystyle \frac { x }{ \sqrt { { y }^{ 2 }-{ x }^{ 2 } } } =0
\displaystyle \frac { dy }{ dx } \left( 1+\displaystyle \frac { y }{ \sqrt { { y }^{ 2 }-{ x }^{ 2 } } } \right) =\displaystyle \frac { x }{ \sqrt { { y }^{ 2 }-{ x }^{ 2 } } }
\displaystyle \frac { dy }{ dx } =\displaystyle \frac { x }{ y+\sqrt { { y }^{ 2 }-{ x }^{ 2 } } }
..................Option (b)
On rationalizing we get,
\displaystyle \frac { dy }{ dx } =\displaystyle \frac { x\left( y-\sqrt { { y }^{ 2 }-{ x }^{ 2 } } \right) }{ { x }^{ 2 } }
=\displaystyle \frac { y-\sqrt { { y }^{ 2 }-{ x }^{ 2 } } }{ x }
........Option (c)
Putting
y+\sqrt { { y }^{ 2 }-{ x }^{ 2 } } ={ c }^{ 2 }
. we get
\displaystyle \frac { dy }{ dx } =\displaystyle \frac { x }{ \displaystyle \frac { { c }^{ 2 } }{ 2 } } =\displaystyle \frac { 2x }{ { c }^{ 2 } }
......... Option (a)
Equation
x^n-1=0, n > 1, n\epsilon N
, has roots
1, a_2, ....a_n
.
The value of
\displaystyle \sum_{r=2}^{n}\frac {1}{1-a_r}
, is
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\displaystyle \frac {n}{4}
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\displaystyle \frac {n(n-1)}{2}
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\displaystyle \frac {n-1}{2}
0%
none of these
Explanation
Equation
x^n-1=0, n > 1, n\epsilon N
, has roots
1, a_1, a_2, \dots a_n
We have to find value of
\displaystyle \sum_{r=2}^{n}\frac {1}{1-a_r}
{ x }^{ n }-1=\left( x-1 \right) \left( x-a_{ 2 } \right)\dots\left( x-{ a }_{ n } \right)\rightarrow
{i}
Taking log on both the sides
\log { \left( { x }^{ n }-1 \right) =\log { \left( x-1 \right) + } \log { \left( x-{ a }_{ 2 } \right) +\dots+ } \log { \left( x-{ a }_{ n } \right) } }
Differentiating both the sides, we get
\displaystyle \frac { n{ x }^{ n-1 } }{ { x }^{ n }-1 } =\displaystyle \frac { 1 }{ x-1 } +\displaystyle \frac { 1 }{ x-{ a }_{ 2 } } +\cdots+\displaystyle \frac { 1 }{ x-{ a }_{ n } }\rightarrow
{ii}
\displaystyle \frac { n{ x }^{ n-1 } }{ { x }^{ n }-1 } -\displaystyle \frac { 1 }{ x-1 } =\displaystyle \frac { 1 }{ x-{ a }_{ 2 } } +\dots\displaystyle \frac { 1 }{ x-{ a }_{ n } }
\because (x^n-1)=(x-1)(1+x+x^2+x^3+x^4+\dots+x^{n-1})
\therefore \displaystyle \frac { n{ x }^{ n-1 }-1\left( 1+x+{ x }^{ 2 }+\dots+{ x }^{ n-1 } \right) }{ { x }^{ n}-1 } =\displaystyle \frac { 1 }{ x-{ a }_{ 2 } } +\dots\displaystyle \frac { 1 }{ x-{ a }_{ n } }
\displaystyle\lim _{ x\rightarrow 1 }{ \left( \displaystyle \frac { n{ x }^{ n-1 }-1\left( 1+x+{ x }^{ 2 }+\dots+{ x }^{ n-1 } \right) }{ { x }^{ n}-1 } \right) } =\lim _{ x\rightarrow 1 }{ \left(\displaystyle \frac { 1 }{ x-{ a }_{ 2 } } +\dots\displaystyle \frac { 1 }{ x-{ a }_{ n } } \right) }
Applying L'Hospital's rule on L.H.S
\displaystyle\lim _{ x\rightarrow 1 }{\displaystyle \frac { n\left( n-1 \right) { x }^{ n-2 }-\left\{ 1+2x+\cdots+\left( n-1 \right) { x }^{ n-2 } \right\} }{ n{ x }^{ n-1 } } =\displaystyle \frac { 1 }{ 1-{ a }_{ 2 } } +\dots\displaystyle \frac { 1 }{ 1-{ a }_{ n } } }
\Rightarrow \displaystyle \frac { n\left( n-1 \right) -\left[ 1+2+\dots+\left( n-1 \right) \right] }{ n } =\displaystyle \frac { 1 }{ 1-{ a }_{ 2 } } +\dots\displaystyle \frac { 1 }{ 1-{ a }_{ n } }
\therefore\displaystyle \frac { 1 }{ 1-{ a }_{ 2 } } +\dots\displaystyle \frac { 1 }{ 1-{ a }_{ n } } =\displaystyle \frac { n-1 }{ 2 }
Suppose,
A=\displaystyle \frac {dy}{dx}
of
x^2+y^2=4
at
(\sqrt 2, \sqrt 2), B=\displaystyle \frac {dy}{dx}
of
sin y+sin x=sin x\cdot sin y
at
(\pi, \pi)
and
C=\displaystyle \frac {dy}{dx}
of
2e^{xy}+e^xe^y-e^x=e^{xy+1}
at
(1, 1)
, then
(A-B-C)
has the value equal to .....
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\displaystyle \frac { 1 }{ 2 }
0%
\displaystyle \frac { 1 }{ 3 }
0%
1
0%
2
Explanation
A: \displaystyle \frac {d}{dx} (x^2+y^2=4)
at
(\sqrt 2, \sqrt 2))
2x+2y\displaystyle \frac { dy }{ dx } =0
\displaystyle \frac { dy }{ dx } =-\displaystyle \frac { x }{ y } =-\displaystyle \frac { \sqrt { 2 } }{ \sqrt { 2 } } =-1
B: \displaystyle \frac {d}{dx}(\sin y+\sin x=\sin x\cdot \sin y)
at
(\pi, \pi)
\cos { y\displaystyle \frac { dy }{ dx } } +\cos { x } =\sin { x } \cos { y\displaystyle \frac { dy }{ dx } +\sin { y } \cos { x } }
\displaystyle \frac { dy }{ dx } \left( \cos { y } -\sin { x } \cos { y } \right) =\sin { y\cos { x-\cos { x } } }
\displaystyle \frac { dy }{ dx } =\displaystyle \frac { \cos { x\left( \sin { y-1 } \right) } }{ \cos { y\left( 1-\sin { x } \right) } }
B=\displaystyle \frac { -1\left( 0-1 \right) }{ -1\left( 1-0 \right) } =-1
C: \displaystyle \frac {d}{dx}(2e^{xy}+e^xe^y-e^x=e^{xy+1})
at
(1, 1)
\left[ 2{ e }^{ xy }\left( x\displaystyle \frac { dy }{ dx } +y \right) \right] +{ e }^{ x }{ e }^{ y }\displaystyle \frac { dy }{ dx } +{ e }^{ y }{ e }^{ x }-{ e }^{ x }={ e }^{ xy+1 }\left( x\displaystyle \frac { dy }{ dx } +y \right)
\displaystyle \frac { dy }{ dx } =\displaystyle \frac { y{ e }^{ xy+1 }-2y{ e }^{ xy }-{ e }^{ x+y }+{ e }^{ x } }{ 2x{ e }^{ xy }+{ e }^{ x+y }-x{ e }^{ xy+1 } }
.......... since
{ e }^{ x }{ e }^{ y }={ e }^{ x+y }
C=\displaystyle \frac { { e }^{ 2 }-2{ e }^{ 1 }-{ e }^{ 2 }+{ e }^{ 1 } }{ 2{ e }^{ 1 }+{ e }^{ 2 }-{ e }^{ 2 } }
=\displaystyle \frac { -{ e }^{ 1 } }{ 2{ e }^{ 1 } } =-\displaystyle \frac { 1 }{ 2 }
Therefore
A-B-C=\displaystyle \frac { 1 }{ 2 }
If
f(x)=\sqrt {x+2\sqrt {2x-4}}+\sqrt {x-2\sqrt {2x-4}}
, then the value of
10 f'(102^+)
is
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-1
0%
0
0%
1
0%
Does not exist
Explanation
f(x)=\sqrt {x+2\sqrt {2x-4}}+\sqrt {x-2\sqrt {2x-4}}
f(x)=\sqrt { { \left( \sqrt { x-2 } +\sqrt { 2 } \right) }^{ 2 } } +\sqrt { { \left( \sqrt { x-2 } -\sqrt { 2 } \right) }^{ 2 } }
f(x)=\left| \sqrt { x-2 } +\sqrt { 2 } \right| +\left| \sqrt { x-2 } -\sqrt { 2 } \right| \\
For
\sqrt { x-2 }
to exist,
x\ge 2
Also
\sqrt { x-1 } +\sqrt { 2 } >0
.............(always true)
But
\sqrt { x-1 } -\sqrt { 2 } \ge 0\quad
only if
x\ge 4
<0
only if
x<4
Now
f(x)
becomes
f(x)=\sqrt { x-2 } +\sqrt { 2 } -\sqrt { x-2 } +\sqrt { 2 }
for
2\le x<4
f(x)=\sqrt { x-2 } +\sqrt { 2 } +\sqrt { x-2 } -\sqrt { 2 }
for
x\ge 4
f(x)=2\sqrt { 2, }
for
2\le x<4
f(x)=2\sqrt { x-2 }
for
4\le x<\infty
f is continous
\left[ 2,4 \right] \cup \left[ 4,\infty \right]
f'(x)=0
for
2\le x<4
f'(x)=\displaystyle \frac { 1 }{ \sqrt { x-2 } }
for
\le x<\infty
f'({ 102 }^{ + })=\displaystyle \frac { 1 }{ \sqrt { 102-2 } } =\displaystyle \frac { 1 }{ 10 } \\ 10f'({ 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
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G.P.
0%
H.P.
0%
A.G.P
0%
A.P.
Explanation
Let
f\left( x \right) ={ px }^{ 2 }+qx+r
Differentiatin g w.r. to
x
, we get
f^{ ' }\left( x \right) =2px+q
Now,
f\left( -1 \right) =f\left( 1 \right)
\therefore p-q+r=p+q+r
\therefore q=0
\therefore f^{ ' }\left( x \right) =2px
f^{ ' }\left( a \right) =2pa.f^{ ' }\left( b \right) =2pb,f^{ ' }\left( c \right) =2pc
Since
a,b,c
are in
A.P.
, therefore
2b=a+c
4pb=2pa+2pc
2f^{ ' }\left( b \right) =f^{ ' }\left( a \right) +f^{ ' }\left( c \right)
\therefore f^{ ' }\left( a \right) ,f^{ ' }\left( b \right) ,f^{ ' }\left( c \right)
are in A.P
For the curve represented implicitly as
3^x-2^y=1
, the value of
\displaystyle\lim_{x\rightarrow \infty }\left ( \dfrac{dy}{dx} \right )
is
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0%
equal to
1
0%
equal to
0
0%
equal to
\log _2{3}
0%
non existent
Explanation
\displaystyle y=\frac{log(3^{x}-1)}{log2}
\displaystyle y'=\frac{3^{x}log3}{(3^{x}-1)log2}
At
x\rightarrow \infty, \displaystyle\frac{3^{x}}{3^{x}-1} \rightarrow 1
So,
\displaystyle y'=\frac{log3}{log2}
Hence, option C is the correct option.
If
\displaystyle x^2+y^2=R^2
(R>0)
then
\displaystyle k= \frac{{y}''}{\sqrt{(1+y'^{2})^3}}
where
k
in terms of
R
alone is equal to
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0%
-\displaystyle \frac{1}{R^2}
0%
-\displaystyle \frac{1}{R}
0%
\displaystyle \frac{2}{R}
0%
-\displaystyle \frac{2}{R^2}
Explanation
\displaystyle x^{2}+y^{2}=R^{2}
\displaystyle x=-yy' => y'=\frac{-x}{y}
\displaystyle y''=\frac{-y+xy'}{y^{2}}=\frac{-y^{2}-x^{2}}{y^{3}}=-\frac{R^{2}}{y^{3}}
\displaystyle k=\frac{y''}{\sqrt{(1+y'^{2})^{3}}}=\frac{-\frac{R^{2}}{y^{3}}}{\sqrt{(1+\frac{x^{2}}{y^{2}})^{3}}}
\displaystyle =\frac{-\frac{R^{2}}{y^{3}}}{\sqrt{(\frac{R^{2}}{y^{2}})^{3}}}
\displaystyle =-\frac{R^{2}}{R^{3}}=\frac{-1}{R}
If
\displaystyle \frac { \cos ^{ 4 }{ \theta } }{ x } +\frac { \sin ^{ 4 }{ \theta } }{ y } =\frac { 1 }{ x+y }
then
\displaystyle \frac { dy }{ dx } =
Report Question
0%
xy
0%
\tan ^{ 2 }{ \theta }
0%
0
0%
\left( { x }^{ 2 }+{ y }^{ 2 } \right) \sec ^{ 2 }{ \theta }
Explanation
Given
\displaystyle \dfrac { \cos ^{ 4 }{ \theta } }{ x } +\dfrac { \sin ^{ 4 }{ \theta } }{ y } =\dfrac { 1 }{ x+y }
\displaystyle \Rightarrow \left( x+y \right) \left( \dfrac { \cos ^{ 4 }{ \theta } }{ x } +\dfrac { \sin ^{ 4 }{ \theta } }{ y } \right) ={ \left( \cos ^{ 2 }{ \theta } +\sin ^{ 2 }{ \theta } \right) }^{ 2 }
{ \left( \because \cos ^{ 2 }{ \theta } +\sin ^{ 2 }{ \theta } =1 \right) }
\displaystyle \therefore \dfrac { y }{ x } \cos ^{ 4 }{ \theta } +\dfrac { x }{ y } \sin ^{ 4 }{ \theta } -2\sin ^{ 2 }{ \theta } \cos ^{ 2 }{ \theta } =0
\displaystyle \Rightarrow { \left( \sqrt { \dfrac { y }{ x } } \cos ^{ 2 }{ \theta } -\sqrt { \dfrac { x }{ y } } \sin ^{ 2 }{ \theta } \right) }^{ 2 }=0
\displaystyle \therefore \tan ^{ 2 }{ \theta } =\dfrac { y }{ x } \Rightarrow y=x\tan ^{ 2 }{ \theta }
\displaystyle \Rightarrow \dfrac { dy }{ dx } =\tan ^{ 2 }{ \theta }
Two functions
f
and
g
have first and second derivatives at
x=0
and satisfy the relations,
f(0)=\displaystyle \frac{2}{g(0)},
f'(0)=2g'(0)=4g(0),
g''(0)=5f''(0)=6f(0)=3
then
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If
h(x)=\displaystyle \frac{f(x)}{g(x)}
then
h'(0)=\displaystyle \frac{15}{4}
0%
If
k(x)=f(x).g(x)\sin x
then
k'(0)=2
0%
\displaystyle\lim _{x\rightarrow{0}}\displaystyle \frac{g'(x)}{f'(x)}=\displaystyle \frac{1}{2}
0%
None of these
Explanation
6f(0) = 3; g(0) = \dfrac{2}{f(0)}, f'(0) = 4g(0), 2g'(0) = 4g(0)
\Rightarrow \displaystyle f(0)=\frac{1}{2}, g(0)=4, f'(0)=16, g'(0)=8
\displaystyle h(x)=\frac{f(x)}{g(x)}
\displaystyle h'(x)=\frac{g(x)f'(x)-f(x)g'(x)}{(g(x))^{2}}
So,
\displaystyle h'(0)=\frac{4.16-\dfrac{1}{2}.8}{4.4}=\frac{64-4}{16}=\frac{15}{4}
\displaystyle k(x)=f(x).g(x)\sin x
\displaystyle k'(x)=f(x).g(x)cosx+sinx(f(x)g'(x)+g(x)f'(x))
\displaystyle k'(0)=\frac{1}{2}.4.1+0=2
\displaystyle \lim_{x\rightarrow 0}=\frac{g'(x)}{f'(x)}=\frac{8}{16}=\frac{1}{2}
0:0:1
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Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers
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