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Exam: Maharashtra HSC Class 12
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Q676
#676
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Hard
Add
If
sin
(
p
x
m
−
q
y
m
p
x
m
+
q
y
m
)
=
r
\sin\left(\frac{px^m - qy^m}{px^m + qy^m}\right) = r
sin
(
p
x
m
+
q
y
m
p
x
m
−
q
y
m
)
=
r
, then show that
d
y
d
x
=
y
x
\frac{dy}{dx} = \frac{y}{x}
d
x
d
y
=
x
y
, where
r
r
r
is a constant.
Show model answer
[Q1.3.3 SolvedEx.3 · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q677
#677
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Hard
Add
If
sec
−
1
(
x
3
+
y
3
x
3
−
y
3
)
=
2
a
\sec^{-1}\left(\frac{x^3 + y^3}{x^3 - y^3}\right) = 2a
sec
−
1
(
x
3
−
y
3
x
3
+
y
3
)
=
2
a
, then show that
d
y
d
x
=
x
2
tan
2
a
y
2
\frac{dy}{dx} = \frac{x^2\tan^2 a}{y^2}
d
x
d
y
=
y
2
x
2
t
a
n
2
a
, where
a
a
a
is a constant.
Show model answer
[Q1.3.3 SolvedEx.4 · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q678
#678
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Hard
Add
If
y
=
tan
x
+
tan
x
+
tan
x
+
…
∞
y = \sqrt{\tan x + \sqrt{\tan x + \sqrt{\tan x + \ldots \infty}}}
y
=
tan
x
+
tan
x
+
tan
x
+
…
∞
, then show that
d
y
d
x
=
sec
2
x
2
y
−
1
\frac{dy}{dx} = \frac{\sec^2 x}{2y - 1}
d
x
d
y
=
2
y
−
1
s
e
c
2
x
.
Show model answer
[Q1.3.3 SolvedEx.5 · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q679
#679
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Hard
Add
If
1
−
x
2
+
1
−
y
2
=
a
(
x
−
y
)
\sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y)
1
−
x
2
+
1
−
y
2
=
a
(
x
−
y
)
, then show that
d
y
d
x
=
1
−
y
2
1
−
x
2
\frac{dy}{dx} = \sqrt{\frac{1 - y^2}{1 - x^2}}
d
x
d
y
=
1
−
x
2
1
−
y
2
.
Show model answer
[Q1.3.3 SolvedEx.6 · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Set · 8 questions
Differentiate the following w. r. t. x
Q680
#680
Mathematics → Differentiation → Logarithmic Differentiation
·
Moderate
Add
(
x
+
1
)
2
(
x
+
2
)
3
(
x
+
3
)
4
\frac{(x+1)^2}{(x+2)^3\,(x+3)^4}
(
x
+
2
)
3
(
x
+
3
)
4
(
x
+
1
)
2
Show model answer
[QEx 1.3 Q.1 (i) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q681
#681
Mathematics → Differentiation → Logarithmic Differentiation
·
Moderate
Add
4
x
−
1
(
2
x
+
3
)
(
5
−
2
x
)
2
3
\sqrt[3]{\frac{4x-1}{(2x+3)\,(5-2x)^2}}
3
(
2
x
+
3
)
(
5
−
2
x
)
2
4
x
−
1
Show model answer
[QEx 1.3 Q.1 (ii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q682
#682
Mathematics → Differentiation → Logarithmic Differentiation
·
Moderate
Add
(
x
2
+
3
)
3
2
⋅
sin
3
2
x
⋅
2
x
2
(x^2+3)^{\frac{3}{2}}\cdot\sin^3 2x\cdot 2^{x^2}
(
x
2
+
3
)
2
3
⋅
sin
3
2
x
⋅
2
x
2
Show model answer
[QEx 1.3 Q.1 (iii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q683
#683
Mathematics → Differentiation → Logarithmic Differentiation
·
Hard
Add
(
x
2
+
2
x
+
2
)
3
2
(
x
+
3
)
3
(
cos
x
)
x
\frac{(x^2+2x+2)^{\frac{3}{2}}}{(\sqrt{x}+3)^3\,(\cos x)^x}
(
x
+
3
)
3
(
c
o
s
x
)
x
(
x
2
+
2
x
+
2
)
2
3
Show model answer
[QEx 1.3 Q.1 (iv) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q684
#684
Mathematics → Differentiation → Logarithmic Differentiation
·
Moderate
Add
x
5
⋅
tan
3
4
x
sin
2
3
x
\frac{x^5\cdot\tan^3 4x}{\sin^2 3x}
s
i
n
2
3
x
x
5
⋅
t
a
n
3
4
x
Show model answer
[QEx 1.3 Q.1 (v) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q685
#685
Mathematics → Differentiation → Logarithmic Differentiation
·
Moderate
Add
x
tan
−
1
x
x^{\tan^{-1} x}
x
t
a
n
−
1
x
Show model answer
[QEx 1.3 Q.1 (vi) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q686
#686
Mathematics → Differentiation → Logarithmic Differentiation
·
Easy
Add
(
sin
x
)
x
(\sin x)^x
(
sin
x
)
x
Show model answer
[QEx 1.3 Q.1 (vii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q687
#687
Mathematics → Differentiation → Logarithmic Differentiation
·
Moderate
Add
sin
x
x
\sin x^x
sin
x
x
Show model answer
[QEx 1.3 Q.1 (viii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Set · 8 questions
Differentiate the following w. r. t. x.
Q688
#688
Mathematics → Differentiation → Logarithmic Differentiation
·
Moderate
Add
x
e
+
x
x
+
e
x
+
e
e
x^e + x^x + e^x + e^e
x
e
+
x
x
+
e
x
+
e
e
Show model answer
[QEx 1.3 Q.2 (i) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q689
#689
Mathematics → Differentiation → Logarithmic Differentiation
·
Hard
Add
x
x
x
+
e
x
x
x^{x^x} + e^{x^x}
x
x
x
+
e
x
x
Show model answer
[QEx 1.3 Q.2 (ii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q690
#690
Mathematics → Differentiation → Logarithmic Differentiation
·
Moderate
Add
(
log
x
)
x
−
(
cos
x
)
cot
x
(\log x)^x - (\cos x)^{\cot x}
(
lo
g
x
)
x
−
(
cos
x
)
c
o
t
x
Show model answer
[QEx 1.3 Q.2 (iii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q691
#691
Mathematics → Differentiation → Logarithmic Differentiation
·
Moderate
Add
x
e
x
+
(
log
x
)
sin
x
x^{e^x} + (\log x)^{\sin x}
x
e
x
+
(
lo
g
x
)
s
i
n
x
Show model answer
[QEx 1.3 Q.2 (iv) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q692
#692
Mathematics → Differentiation → Logarithmic Differentiation
·
Moderate
Add
e
tan
x
+
(
log
x
)
tan
x
e^{\tan x} + (\log x)^{\tan x}
e
t
a
n
x
+
(
lo
g
x
)
t
a
n
x
Show model answer
[QEx 1.3 Q.2 (v) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q693
#693
Mathematics → Differentiation → Logarithmic Differentiation
·
Moderate
Add
(
sin
x
)
tan
x
+
(
cos
x
)
cot
x
(\sin x)^{\tan x} + (\cos x)^{\cot x}
(
sin
x
)
t
a
n
x
+
(
cos
x
)
c
o
t
x
Show model answer
[QEx 1.3 Q.2 (vi) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q694
#694
Mathematics → Differentiation → Logarithmic Differentiation
·
Hard
Add
10
x
x
+
x
x
10
+
x
10
x
10^{x^x} + x^{x^{10}} + x^{10^x}
1
0
x
x
+
x
x
10
+
x
1
0
x
Show model answer
[QEx 1.3 Q.2 (vii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q695
#695
Mathematics → Differentiation → Logarithmic Differentiation
·
Hard
Add
[
(
tan
x
)
tan
x
]
tan
x
\left[(\tan x)^{\tan x}\right]^{\tan x}
[
(
tan
x
)
t
a
n
x
]
t
a
n
x
at
x
=
π
4
x = \frac{\pi}{4}
x
=
4
π
Show model answer
[QEx 1.3 Q.2 (viii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Set · 5 questions
Find
d
y
d
x
\frac{dy}{dx}
d
x
d
y
if
Q696
#696
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Easy
Add
x
+
y
=
a
\sqrt{x} + \sqrt{y} = \sqrt{a}
x
+
y
=
a
Show model answer
[QEx 1.3 Q.3 (i) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q697
#697
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Easy
Add
x
x
+
y
y
=
a
a
x\sqrt{x} + y\sqrt{y} = a\sqrt{a}
x
x
+
y
y
=
a
a
Show model answer
[QEx 1.3 Q.3 (ii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q698
#698
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Moderate
Add
x
+
x
y
+
y
=
1
x + \sqrt{xy} + y = 1
x
+
x
y
+
y
=
1
Show model answer
[QEx 1.3 Q.3 (iii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q699
#699
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Moderate
Add
x
3
+
x
2
y
+
x
y
2
+
y
3
=
81
x^3 + x^2 y + xy^2 + y^3 = 81
x
3
+
x
2
y
+
x
y
2
+
y
3
=
81
Show model answer
[QEx 1.3 Q.3 (iv) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q700
#700
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Hard
Add
x
2
y
2
−
tan
−
1
x
2
+
y
2
=
cot
−
1
x
2
+
y
2
x^2 y^2 - \tan^{-1}\sqrt{x^2+y^2} = \cot^{-1}\sqrt{x^2+y^2}
x
2
y
2
−
tan
−
1
x
2
+
y
2
=
cot
−
1
x
2
+
y
2
Show model answer
[QEx 1.3 Q.3 (v) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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