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Exam: Maharashtra HSC Class 12
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Set · 3 questions
Find
d
y
d
x
\frac{dy}{dx}
d
x
d
y
if
Q726
#726
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
x
=
cos
(
log
t
)
x = \cos(\log t)
x
=
cos
(
lo
g
t
)
,
y
=
log
(
cos
t
)
y = \log(\cos t)
y
=
lo
g
(
cos
t
)
Show model answer
[Q1.4.2 SolvedEx.1 (iii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q727
#727
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Hard
Add
x
=
a
(
θ
+
sin
θ
)
x = a(\theta + \sin\theta)
x
=
a
(
θ
+
sin
θ
)
,
y
=
a
(
1
−
cos
θ
)
y = a(1 - \cos\theta)
y
=
a
(
1
−
cos
θ
)
Show model answer
[Q1.4.2 SolvedEx.1 (iv) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q728
#728
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
x
=
1
−
t
2
x = \sqrt{1 - t^2}
x
=
1
−
t
2
,
y
=
sin
−
1
t
y = \sin^{-1} t
y
=
sin
−
1
t
Show model answer
[Q1.4.2 SolvedEx.1 (v) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Set · 3 questions
Find
d
y
d
x
\frac{dy}{dx}
d
x
d
y
if
Q729
#729
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
x
=
sec
2
θ
x = \sec^2\theta
x
=
sec
2
θ
,
y
=
tan
3
θ
y = \tan^3\theta
y
=
tan
3
θ
, at
θ
=
π
3
\theta = \frac{\pi}{3}
θ
=
3
π
Show model answer
[Q1.4.2 SolvedEx.2 (i) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q730
#730
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
x
=
t
+
1
t
x = t + \frac{1}{t}
x
=
t
+
t
1
,
y
=
1
t
2
y = \frac{1}{t^2}
y
=
t
2
1
, at
t
=
1
2
t = \frac{1}{2}
t
=
2
1
Show model answer
[Q1.4.2 SolvedEx.2 (ii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q731
#731
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Hard
Add
x
=
3
cos
t
−
2
cos
3
t
x = 3\cos t - 2\cos^3 t
x
=
3
cos
t
−
2
cos
3
t
,
y
=
3
sin
t
−
2
sin
3
t
y = 3\sin t - 2\sin^3 t
y
=
3
sin
t
−
2
sin
3
t
, at
t
=
π
6
t = \frac{\pi}{6}
t
=
6
π
Show model answer
[Q1.4.2 SolvedEx.2 (iii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q732
#732
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Hard
Add
If
x
2
+
y
2
=
t
+
1
t
x^2 + y^2 = t + \frac{1}{t}
x
2
+
y
2
=
t
+
t
1
and
x
4
+
y
4
=
t
2
+
1
t
2
x^4 + y^4 = t^2 + \frac{1}{t^2}
x
4
+
y
4
=
t
2
+
t
2
1
, then show that
x
3
y
d
y
d
x
=
−
1
x^3 y \frac{dy}{dx} = -1
x
3
y
d
x
d
y
=
−
1
.
Show model answer
[Q1.4.2 SolvedEx.3 · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q733
#733
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Hard
Add
If
x
=
a
(
t
−
1
t
)
x = a\left(t - \frac{1}{t}\right)
x
=
a
(
t
−
t
1
)
and
y
=
b
(
t
+
1
t
)
y = b\left(t + \frac{1}{t}\right)
y
=
b
(
t
+
t
1
)
, then show that
d
y
d
x
=
b
2
x
a
2
y
\frac{dy}{dx} = \frac{b^2 x}{a^2 y}
d
x
d
y
=
a
2
y
b
2
x
.
Show model answer
[Q1.4.2 SolvedEx.4 · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q734
#734
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Hard
Add
If
x
=
a
sin
−
1
t
x = \sqrt{a^{\sin^{-1} t}}
x
=
a
s
i
n
−
1
t
and
y
=
a
cos
−
1
t
y = \sqrt{a^{\cos^{-1} t}}
y
=
a
c
o
s
−
1
t
, then show that
d
y
d
x
=
−
y
x
\frac{dy}{dx} = -\frac{y}{x}
d
x
d
y
=
−
x
y
.
Show model answer
[Q1.4.2 SolvedEx.5 · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q735
#735
Mathematics → Differentiation → Differentiation of One Function with respect to Another
·
Easy
Add
Find the derivative of
7
x
7^x
7
x
w.r.t.
x
7
x^7
x
7
.
Show model answer
[Q1.4.3 SolvedEx.1 · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q736
#736
Mathematics → Differentiation → Differentiation of One Function with respect to Another
·
Moderate
Add
Find the derivative of
cos
−
1
x
\cos^{-1} x
cos
−
1
x
w.r.t.
1
−
x
2
\sqrt{1-x^2}
1
−
x
2
.
Show model answer
[Q1.4.3 SolvedEx.2 · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q737
#737
Mathematics → Differentiation → Differentiation of One Function with respect to Another
·
Hard
Add
Find the derivative of
tan
−
1
(
1
+
x
2
−
1
x
)
\tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)
tan
−
1
(
x
1
+
x
2
−
1
)
w.r.t.
sin
−
1
(
2
x
1
+
x
2
)
\sin^{-1}\left(\frac{2x}{1+x^2}\right)
sin
−
1
(
1
+
x
2
2
x
)
.
Show model answer
[Q1.4.3 SolvedEx.3 · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Set · 8 questions
Find
d
y
d
x
\frac{dy}{dx}
d
x
d
y
if
Q738
#738
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Easy
Add
x
=
a
t
2
x = at^2
x
=
a
t
2
,
y
=
2
a
t
y = 2at
y
=
2
a
t
Show model answer
[QEx 1.4 Q.1 (i) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q739
#739
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Easy
Add
x
=
a
cot
θ
x = a \cot \theta
x
=
a
cot
θ
,
y
=
b
cosec
θ
y = b \operatorname{cosec} \theta
y
=
b
cosec
θ
Show model answer
[QEx 1.4 Q.1 (ii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q740
#740
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Easy
Add
x
=
a
2
+
m
2
x = \sqrt{a^2 + m^2}
x
=
a
2
+
m
2
,
y
=
log
(
a
2
+
m
2
)
y = \log (a^2 + m^2)
y
=
lo
g
(
a
2
+
m
2
)
Show model answer
[QEx 1.4 Q.1 (iii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q741
#741
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Easy
Add
x
=
sin
θ
x = \sin \theta
x
=
sin
θ
,
y
=
tan
θ
y = \tan \theta
y
=
tan
θ
Show model answer
[QEx 1.4 Q.1 (iv) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q742
#742
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Easy
Add
x
=
a
(
1
−
cos
θ
)
x = a(1 - \cos \theta)
x
=
a
(
1
−
cos
θ
)
,
y
=
b
(
θ
−
sin
θ
)
y = b(\theta - \sin \theta)
y
=
b
(
θ
−
sin
θ
)
Show model answer
[QEx 1.4 Q.1 (v) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q743
#743
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
x
=
(
t
+
1
t
)
a
x = \left(t + \frac{1}{t}\right)^a
x
=
(
t
+
t
1
)
a
,
y
=
a
t
+
1
t
y = a^{t + \frac{1}{t}}
y
=
a
t
+
t
1
, where
a
>
0
a > 0
a
>
0
,
a
≠
1
a \neq 1
a
=
1
and
t
≠
0
t \neq 0
t
=
0
.
Show model answer
[QEx 1.4 Q.1 (vi) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q744
#744
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
x
=
cos
−
1
(
2
t
1
+
t
2
)
x = \cos^{-1}\left(\frac{2t}{1+t^2}\right)
x
=
cos
−
1
(
1
+
t
2
2
t
)
,
y
=
sec
−
1
(
1
+
t
2
)
y = \sec^{-1}\left(\sqrt{1+t^2}\right)
y
=
sec
−
1
(
1
+
t
2
)
Show model answer
[QEx 1.4 Q.1 (vii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q745
#745
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Hard
Add
x
=
cos
−
1
(
4
t
3
−
3
t
)
x = \cos^{-1}(4t^3 - 3t)
x
=
cos
−
1
(
4
t
3
−
3
t
)
,
y
=
tan
−
1
(
1
−
t
2
t
)
y = \tan^{-1}\left(\frac{\sqrt{1-t^2}}{t}\right)
y
=
tan
−
1
(
t
1
−
t
2
)
Show model answer
[QEx 1.4 Q.1 (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
Q746
#746
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
x
=
cosec
2
θ
x = \operatorname{cosec}^2 \theta
x
=
cosec
2
θ
,
y
=
cot
3
θ
y = \cot^3 \theta
y
=
cot
3
θ
, at
θ
=
π
6
\theta = \frac{\pi}{6}
θ
=
6
π
Show model answer
[QEx 1.4 Q.2 (i) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q747
#747
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
x
=
a
cos
3
θ
x = a \cos^3 \theta
x
=
a
cos
3
θ
,
y
=
a
sin
3
θ
y = a \sin^3 \theta
y
=
a
sin
3
θ
, at
θ
=
π
3
\theta = \frac{\pi}{3}
θ
=
3
π
Show model answer
[QEx 1.4 Q.2 (ii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q748
#748
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
x
=
t
2
+
t
+
1
x = t^2 + t + 1
x
=
t
2
+
t
+
1
,
y
=
sin
(
π
t
2
)
+
cos
(
π
t
2
)
y = \sin\left(\frac{\pi t}{2}\right) + \cos\left(\frac{\pi t}{2}\right)
y
=
sin
(
2
π
t
)
+
cos
(
2
π
t
)
, at
t
=
1
t = 1
t
=
1
Show model answer
[QEx 1.4 Q.2 (iii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q749
#749
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
x
=
2
cos
t
+
cos
2
t
x = 2 \cos t + \cos 2t
x
=
2
cos
t
+
cos
2
t
,
y
=
2
sin
t
−
sin
2
t
y = 2 \sin t - \sin 2t
y
=
2
sin
t
−
sin
2
t
, at
t
=
π
4
t = \frac{\pi}{4}
t
=
4
π
Show model answer
[QEx 1.4 Q.2 (iv) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q750
#750
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
x
=
t
+
2
sin
(
π
t
)
x = t + 2 \sin(\pi t)
x
=
t
+
2
sin
(
π
t
)
,
y
=
3
t
−
cos
(
π
t
)
y = 3t - \cos(\pi t)
y
=
3
t
−
cos
(
π
t
)
, at
t
=
1
2
t = \frac{1}{2}
t
=
2
1
Show model answer
[QEx 1.4 Q.2 (v) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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