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Exam: Maharashtra HSC Class 12
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Set · 5 questions
Find
d
y
d
x
\frac{dy}{dx}
d
x
d
y
if
Q701
#701
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Moderate
Add
x
e
y
+
y
e
x
=
1
xe^y + ye^x = 1
x
e
y
+
y
e
x
=
1
Show model answer
[QEx 1.3 Q.3 (vi) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q702
#702
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Moderate
Add
e
x
+
y
=
cos
(
x
−
y
)
e^{x+y} = \cos(x-y)
e
x
+
y
=
cos
(
x
−
y
)
Show model answer
[QEx 1.3 Q.3 (vii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q703
#703
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Moderate
Add
cos
(
x
y
)
=
x
+
y
\cos(xy) = x + y
cos
(
x
y
)
=
x
+
y
Show model answer
[QEx 1.3 Q.3 (viii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q704
#704
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Hard
Add
e
e
x
−
y
=
x
y
e^{e^{x-y}} = \frac{x}{y}
e
e
x
−
y
=
y
x
Show model answer
[QEx 1.3 Q.3 (ix) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q705
#705
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Moderate
Add
x
+
sin
(
x
+
y
)
=
y
−
cos
(
x
−
y
)
x + \sin(x+y) = y - \cos(x-y)
x
+
sin
(
x
+
y
)
=
y
−
cos
(
x
−
y
)
Show model answer
[QEx 1.3 Q.3 (x) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Set · 8 questions
Show that
d
y
d
x
=
y
x
\frac{dy}{dx} = \frac{y}{x}
d
x
d
y
=
x
y
in the following, where
a
a
a
and
p
p
p
are constants.
Q706
#706
Mathematics → Differentiation → Logarithmic Differentiation
·
Moderate
Add
x
7
y
5
=
(
x
+
y
)
12
x^7 y^5 = (x+y)^{12}
x
7
y
5
=
(
x
+
y
)
12
Show model answer
[QEx 1.3 Q.4 (i) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q707
#707
Mathematics → Differentiation → Logarithmic Differentiation
·
Moderate
Add
x
p
y
4
=
(
x
+
y
)
p
+
4
,
p
∈
N
x^p y^4 = (x+y)^{p+4},\ p \in N
x
p
y
4
=
(
x
+
y
)
p
+
4
,
p
∈
N
Show model answer
[QEx 1.3 Q.4 (ii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q708
#708
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Hard
Add
sec
(
x
5
+
y
5
x
5
−
y
5
)
=
a
2
\sec\left(\frac{x^5+y^5}{x^5-y^5}\right) = a^2
sec
(
x
5
−
y
5
x
5
+
y
5
)
=
a
2
Show model answer
[QEx 1.3 Q.4 (iii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q709
#709
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Hard
Add
tan
−
1
(
3
x
2
−
4
y
2
3
x
2
+
4
y
2
)
=
a
2
\tan^{-1}\left(\frac{3x^2-4y^2}{3x^2+4y^2}\right) = a^2
tan
−
1
(
3
x
2
+
4
y
2
3
x
2
−
4
y
2
)
=
a
2
Show model answer
[QEx 1.3 Q.4 (iv) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q710
#710
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Hard
Add
cos
−
1
(
7
x
4
+
5
y
4
7
x
4
−
5
y
4
)
=
tan
−
1
a
\cos^{-1}\left(\frac{7x^4+5y^4}{7x^4-5y^4}\right) = \tan^{-1} a
cos
−
1
(
7
x
4
−
5
y
4
7
x
4
+
5
y
4
)
=
tan
−
1
a
Show model answer
[QEx 1.3 Q.4 (v) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q711
#711
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Moderate
Add
log
(
x
20
−
y
20
x
20
+
y
20
)
=
20
\log\left(\frac{x^{20}-y^{20}}{x^{20}+y^{20}}\right) = 20
lo
g
(
x
20
+
y
20
x
20
−
y
20
)
=
20
Show model answer
[QEx 1.3 Q.4 (vi) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q712
#712
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Hard
Add
e
x
7
−
y
7
x
7
+
y
7
=
a
e^{\frac{x^7-y^7}{x^7+y^7}} = a
e
x
7
+
y
7
x
7
−
y
7
=
a
Show model answer
[QEx 1.3 Q.4 (vii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q713
#713
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Hard
Add
sin
(
x
3
−
y
3
x
3
+
y
3
)
=
a
3
\sin\left(\frac{x^3-y^3}{x^3+y^3}\right) = a^3
sin
(
x
3
+
y
3
x
3
−
y
3
)
=
a
3
Show model answer
[QEx 1.3 Q.4 (viii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Set · 10 questions
Q714
#714
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Moderate
Add
If
log
(
x
+
y
)
=
log
(
x
y
)
+
p
\log(x+y) = \log(xy) + p
lo
g
(
x
+
y
)
=
lo
g
(
x
y
)
+
p
, where
p
p
p
is constant then prove that
d
y
d
x
=
−
y
2
x
2
\frac{dy}{dx} = -\frac{y^2}{x^2}
d
x
d
y
=
−
x
2
y
2
.
Show model answer
[QEx 1.3 Q.5 (i) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q715
#715
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Moderate
Add
If
log
10
(
x
3
−
y
3
x
3
+
y
3
)
=
2
\log_{10}\left(\frac{x^3-y^3}{x^3+y^3}\right) = 2
lo
g
10
(
x
3
+
y
3
x
3
−
y
3
)
=
2
, show that
d
y
d
x
=
−
99
x
2
101
y
2
\frac{dy}{dx} = -\frac{99x^2}{101y^2}
d
x
d
y
=
−
101
y
2
99
x
2
.
Show model answer
[QEx 1.3 Q.5 (ii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q716
#716
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Moderate
Add
If
log
5
(
x
4
+
y
4
x
4
−
y
4
)
=
2
\log_5\left(\frac{x^4+y^4}{x^4-y^4}\right) = 2
lo
g
5
(
x
4
−
y
4
x
4
+
y
4
)
=
2
, show that
d
y
d
x
=
−
12
x
3
13
y
3
\frac{dy}{dx} = -\frac{12x^3}{13y^3}
d
x
d
y
=
−
13
y
3
12
x
3
.
Show model answer
[QEx 1.3 Q.5 (iii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q717
#717
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Moderate
Add
If
e
x
+
e
y
=
e
x
+
y
e^x + e^y = e^{x+y}
e
x
+
e
y
=
e
x
+
y
, then show that
d
y
d
x
=
−
e
y
−
x
\frac{dy}{dx} = -e^{y-x}
d
x
d
y
=
−
e
y
−
x
.
Show model answer
[QEx 1.3 Q.5 (iv) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q718
#718
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Hard
Add
If
sin
−
1
(
x
5
−
y
5
x
5
+
y
5
)
=
π
6
\sin^{-1}\left(\frac{x^5-y^5}{x^5+y^5}\right) = \frac{\pi}{6}
sin
−
1
(
x
5
+
y
5
x
5
−
y
5
)
=
6
π
, show that
d
y
d
x
=
x
4
3
y
4
\frac{dy}{dx} = \frac{x^4}{3y^4}
d
x
d
y
=
3
y
4
x
4
.
Show model answer
[QEx 1.3 Q.5 (v) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q719
#719
Mathematics → Differentiation → Logarithmic Differentiation
·
Moderate
Add
If
x
y
=
e
x
−
y
x^y = e^{x-y}
x
y
=
e
x
−
y
, then show that
d
y
d
x
=
log
x
(
1
+
log
x
)
2
\frac{dy}{dx} = \frac{\log x}{(1+\log x)^2}
d
x
d
y
=
(
1
+
l
o
g
x
)
2
l
o
g
x
.
Show model answer
[QEx 1.3 Q.5 (vi) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q720
#720
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Hard
Add
If
y
=
cos
x
+
cos
x
+
cos
x
+
.
.
.
∞
y = \sqrt{\cos x + \sqrt{\cos x + \sqrt{\cos x + ...\infty}}}
y
=
cos
x
+
cos
x
+
cos
x
+
...∞
, then show that
d
y
d
x
=
sin
x
1
−
2
y
\frac{dy}{dx} = \frac{\sin x}{1-2y}
d
x
d
y
=
1
−
2
y
s
i
n
x
.
Show model answer
[QEx 1.3 Q.5 (vii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q721
#721
Mathematics → Differentiation → Derivatives of Implicit Functions
·
Hard
Add
If
y
=
log
x
+
log
x
+
log
x
+
.
.
.
∞
y = \sqrt{\log x + \sqrt{\log x + \sqrt{\log x + ...\infty}}}
y
=
lo
g
x
+
lo
g
x
+
lo
g
x
+
...∞
, then show that
d
y
d
x
=
1
x
(
2
y
−
1
)
\frac{dy}{dx} = \frac{1}{x(2y-1)}
d
x
d
y
=
x
(
2
y
−
1
)
1
.
Show model answer
[QEx 1.3 Q.5 (viii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q722
#722
Mathematics → Differentiation → Logarithmic Differentiation
·
Hard
Add
If
y
=
x
x
x
⋅
⋅
⋅
∞
y = x^{x^{x^{\cdot^{\cdot^{\cdot\infty}}}}}
y
=
x
x
x
⋅
⋅
⋅
∞
, then show that
d
y
d
x
=
y
2
x
(
1
−
log
y
)
\frac{dy}{dx} = \frac{y^2}{x(1-\log y)}
d
x
d
y
=
x
(
1
−
l
o
g
y
)
y
2
.
Show model answer
[QEx 1.3 Q.5 (ix) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q723
#723
Mathematics → Differentiation → Logarithmic Differentiation
·
Hard
Add
If
e
y
=
y
x
e^y = y^x
e
y
=
y
x
, then show that
d
y
d
x
=
(
log
y
)
2
log
y
−
1
\frac{dy}{dx} = \frac{(\log y)^2}{\log y - 1}
d
x
d
y
=
l
o
g
y
−
1
(
l
o
g
y
)
2
.
Show model answer
[QEx 1.3 Q.5 (x) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Set · 2 questions
Find
d
y
d
x
\frac{dy}{dx}
d
x
d
y
if
Q724
#724
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Easy
Add
x
=
a
t
4
x = at^4
x
=
a
t
4
,
y
=
2
a
t
2
y = 2at^2
y
=
2
a
t
2
Show model answer
[Q1.4.2 SolvedEx.1 (i) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q725
#725
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
x
=
t
−
t
x = t - \sqrt{t}
x
=
t
−
t
,
y
=
t
+
t
y = t + \sqrt{t}
y
=
t
+
t
Show model answer
[Q1.4.2 SolvedEx.1 (ii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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