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Exam: Maharashtra HSC Class 12
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Q751
#751
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
If
x
=
a
sec
θ
−
tan
θ
x = a\sqrt{\sec \theta - \tan \theta}
x
=
a
sec
θ
−
tan
θ
,
y
=
a
sec
θ
+
tan
θ
y = a\sqrt{\sec \theta + \tan \theta}
y
=
a
sec
θ
+
tan
θ
, then show that
d
y
d
x
=
−
y
x
\frac{dy}{dx} = -\frac{y}{x}
d
x
d
y
=
−
x
y
.
Show model answer
[QEx 1.4 Q.3 (i) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q752
#752
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
If
x
=
e
sin
3
t
x = e^{\sin 3t}
x
=
e
s
i
n
3
t
,
y
=
e
cos
3
t
y = e^{\cos 3t}
y
=
e
c
o
s
3
t
, then show that
d
y
d
x
=
−
y
log
x
x
log
y
\frac{dy}{dx} = -\frac{y \log x}{x \log y}
d
x
d
y
=
−
x
l
o
g
y
y
l
o
g
x
.
Show model answer
[QEx 1.4 Q.3 (ii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q753
#753
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
If
x
=
t
+
1
t
−
1
x = \frac{t+1}{t-1}
x
=
t
−
1
t
+
1
,
y
=
t
−
1
t
+
1
y = \frac{t-1}{t+1}
y
=
t
+
1
t
−
1
, then show that
y
2
+
d
y
d
x
=
0
y^2 + \frac{dy}{dx} = 0
y
2
+
d
x
d
y
=
0
.
Show model answer
[QEx 1.4 Q.3 (iii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q754
#754
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
If
x
=
a
cos
3
t
x = a \cos^3 t
x
=
a
cos
3
t
,
y
=
a
sin
3
t
y = a \sin^3 t
y
=
a
sin
3
t
, then show that
d
y
d
x
=
−
(
y
x
)
1
3
\frac{dy}{dx} = -\left(\frac{y}{x}\right)^{\frac{1}{3}}
d
x
d
y
=
−
(
x
y
)
3
1
.
Show model answer
[QEx 1.4 Q.3 (iv) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q755
#755
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
If
x
=
2
cos
4
(
t
+
3
)
x = 2 \cos^4 (t+3)
x
=
2
cos
4
(
t
+
3
)
,
y
=
3
sin
4
(
t
+
3
)
y = 3 \sin^4 (t+3)
y
=
3
sin
4
(
t
+
3
)
, show that
d
y
d
x
=
−
3
y
2
x
\frac{dy}{dx} = -\sqrt{\frac{3y}{2x}}
d
x
d
y
=
−
2
x
3
y
.
Show model answer
[QEx 1.4 Q.3 (v) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q756
#756
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
If
x
=
log
(
1
+
t
2
)
x = \log (1 + t^2)
x
=
lo
g
(
1
+
t
2
)
,
y
=
t
−
tan
−
1
t
y = t - \tan^{-1} t
y
=
t
−
tan
−
1
t
, show that
d
y
d
x
=
e
x
−
1
2
\frac{dy}{dx} = \frac{\sqrt{e^x - 1}}{2}
d
x
d
y
=
2
e
x
−
1
.
Show model answer
[QEx 1.4 Q.3 (vi) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q757
#757
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
If
x
=
sin
−
1
(
e
t
)
x = \sin^{-1} (e^t)
x
=
sin
−
1
(
e
t
)
,
y
=
1
−
e
2
t
y = \sqrt{1 - e^{2t}}
y
=
1
−
e
2
t
, show that
sin
x
+
d
y
d
x
=
0
\sin x + \frac{dy}{dx} = 0
sin
x
+
d
x
d
y
=
0
.
Show model answer
[QEx 1.4 Q.3 (vii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q758
#758
Mathematics → Differentiation → Derivatives of Parametric Functions
·
Moderate
Add
If
x
=
2
b
t
1
+
t
2
x = \frac{2bt}{1+t^2}
x
=
1
+
t
2
2
b
t
,
y
=
a
(
1
−
t
2
1
+
t
2
)
y = a\left(\frac{1-t^2}{1+t^2}\right)
y
=
a
(
1
+
t
2
1
−
t
2
)
, show that
d
x
d
y
=
−
b
2
y
a
2
x
\frac{dx}{dy} = -\frac{b^2 y}{a^2 x}
d
y
d
x
=
−
a
2
x
b
2
y
.
Show model answer
[QEx 1.4 Q.3 (viii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q759
#759
Mathematics → Differentiation → Differentiation of One Function with respect to Another
·
Easy
Add
Differentiate
x
sin
x
x \sin x
x
sin
x
w.r.t.
tan
x
\tan x
tan
x
.
Show model answer
[QEx 1.4 Q.4 (i) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q760
#760
Mathematics → Differentiation → Differentiation of One Function with respect to Another
·
Moderate
Add
Differentiate
sin
−
1
(
2
x
1
+
x
2
)
\sin^{-1}\left(\frac{2x}{1+x^2}\right)
sin
−
1
(
1
+
x
2
2
x
)
w.r.t.
cos
−
1
(
1
−
x
2
1
+
x
2
)
\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)
cos
−
1
(
1
+
x
2
1
−
x
2
)
.
Show model answer
[QEx 1.4 Q.4 (ii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q761
#761
Mathematics → Differentiation → Differentiation of One Function with respect to Another
·
Hard
Add
Differentiate
tan
−
1
(
x
1
−
x
2
)
\tan^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)
tan
−
1
(
1
−
x
2
x
)
w.r.t.
sec
−
1
(
1
2
x
2
−
1
)
\sec^{-1}\left(\frac{1}{2x^2-1}\right)
sec
−
1
(
2
x
2
−
1
1
)
.
Show model answer
[QEx 1.4 Q.4 (iii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q762
#762
Mathematics → Differentiation → Differentiation of One Function with respect to Another
·
Moderate
Add
Differentiate
cos
−
1
(
1
−
x
2
1
+
x
2
)
\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)
cos
−
1
(
1
+
x
2
1
−
x
2
)
w.r.t.
tan
−
1
x
\tan^{-1} x
tan
−
1
x
.
Show model answer
[QEx 1.4 Q.4 (iv) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q763
#763
Mathematics → Differentiation → Differentiation of One Function with respect to Another
·
Moderate
Add
Differentiate
3
x
3^x
3
x
w.r.t.
log
x
3
\log_x 3
lo
g
x
3
.
Show model answer
[QEx 1.4 Q.4 (v) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q764
#764
Mathematics → Differentiation → Differentiation of One Function with respect to Another
·
Hard
Add
Differentiate
tan
−
1
(
cos
x
1
+
sin
x
)
\tan^{-1}\left(\frac{\cos x}{1+\sin x}\right)
tan
−
1
(
1
+
s
i
n
x
c
o
s
x
)
w.r.t.
sec
−
1
x
\sec^{-1} x
sec
−
1
x
.
Show model answer
[QEx 1.4 Q.4 (vi) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q765
#765
Mathematics → Differentiation → Differentiation of One Function with respect to Another
·
Moderate
Add
Differentiate
x
x
x^x
x
x
w.r.t.
x
sin
x
x^{\sin x}
x
s
i
n
x
.
Show model answer
[QEx 1.4 Q.4 (vii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q766
#766
Mathematics → Differentiation → Differentiation of One Function with respect to Another
·
Hard
Add
Differentiate
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.
tan
−
1
(
2
x
1
−
x
2
1
−
2
x
2
)
\tan^{-1}\left(\frac{2x\sqrt{1-x^2}}{1-2x^2}\right)
tan
−
1
(
1
−
2
x
2
2
x
1
−
x
2
)
.
Show model answer
[QEx 1.4 Q.4 (viii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Set · 5 questions
Find the second order derivative of the following :
Q767
#767
Mathematics → Differentiation → Higher Order Derivatives
·
Easy
Add
x
3
+
7
x
2
−
2
x
−
9
x^3 + 7x^2 - 2x - 9
x
3
+
7
x
2
−
2
x
−
9
Show model answer
[Q1.5.1 SolvedEx.1 (i) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q768
#768
Mathematics → Differentiation → Higher Order Derivatives
·
Easy
Add
x
2
e
x
x^2 e^x
x
2
e
x
Show model answer
[Q1.5.1 SolvedEx.1 (ii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q769
#769
Mathematics → Differentiation → Higher Order Derivatives
·
Moderate
Add
e
2
x
sin
3
x
e^{2x} \sin 3x
e
2
x
sin
3
x
Show model answer
[Q1.5.1 SolvedEx.1 (iii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q770
#770
Mathematics → Differentiation → Higher Order Derivatives
·
Easy
Add
x
2
log
x
x^2 \log x
x
2
lo
g
x
Show model answer
[Q1.5.1 SolvedEx.1 (iv) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q771
#771
Mathematics → Differentiation → Higher Order Derivatives
·
Moderate
Add
sin
(
log
x
)
\sin (\log x)
sin
(
lo
g
x
)
Show model answer
[Q1.5.1 SolvedEx.1 (v) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Set · 2 questions
Find
d
2
y
d
x
2
\frac{d^2y}{dx^2}
d
x
2
d
2
y
if,
Q772
#772
Mathematics → Differentiation → Higher Order Derivatives
·
Hard
Add
x
=
cot
−
1
(
1
−
t
2
t
)
x = \cot^{-1}\left(\frac{\sqrt{1 - t^2}}{t}\right)
x
=
cot
−
1
(
t
1
−
t
2
)
and
x
=
cosec
−
1
(
1
+
t
2
2
t
)
x = \operatorname{cosec}^{-1}\left(\frac{1 + t^2}{2t}\right)
x
=
cosec
−
1
(
2
t
1
+
t
2
)
Show model answer
[Q1.5.1 SolvedEx.2 (i) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q773
#773
Mathematics → Differentiation → Higher Order Derivatives
·
Hard
Add
x
=
a
cos
3
θ
x = a \cos^3 \theta
x
=
a
cos
3
θ
,
y
=
b
sin
3
θ
y = b \sin^3 \theta
y
=
b
sin
3
θ
at
θ
=
π
4
\theta = \frac{\pi}{4}
θ
=
4
π
Show model answer
[Q1.5.1 SolvedEx.2 (ii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q774
#774
Mathematics → Differentiation → Higher Order Derivatives
·
Hard
Add
If
a
x
2
+
2
h
x
y
+
b
y
2
=
0
ax^2 + 2hxy + by^2 = 0
a
x
2
+
2
h
x
y
+
b
y
2
=
0
then show that
d
2
y
d
x
2
=
0
\frac{d^2y}{dx^2} = 0
d
x
2
d
2
y
=
0
.
Show model answer
[Q1.5.1 SolvedEx.3 · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q775
#775
Mathematics → Differentiation → Higher Order Derivatives
·
Hard
Add
If
y
=
cos
(
m
cos
−
1
x
)
y = \cos (m \cos^{-1} x)
y
=
cos
(
m
cos
−
1
x
)
then show that
(
1
−
x
2
)
d
2
y
d
x
2
−
x
d
y
d
x
+
m
2
y
=
0
(1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} + m^2 y = 0
(
1
−
x
2
)
d
x
2
d
2
y
−
x
d
x
d
y
+
m
2
y
=
0
.
Show model answer
[Q1.5.1 SolvedEx.4 · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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