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Exam: Maharashtra HSC Class 12
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Set · 2 questions
Differentiate the following
w
.
r
.
t
.
x
w.\ r.\ t.\ x
w
.
r
.
t
.
x
.
Q501
#501
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Moderate
Add
y
=
(
x
3
+
2
x
−
3
)
4
(
x
+
cos
x
)
3
y = (x^3 + 2x - 3)^4 (x + \cos x)^3
y
=
(
x
3
+
2
x
−
3
)
4
(
x
+
cos
x
)
3
Show model answer
[Q1.1.3 SolvedEx.2 (iv) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q502
#502
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Hard
Add
y
=
(
1
+
cos
2
x
)
4
×
x
+
tan
x
y = (1 + \cos^2 x)^4 \times \sqrt{x + \sqrt{\tan x}}
y
=
(
1
+
cos
2
x
)
4
×
x
+
tan
x
Show model answer
[Q1.1.3 SolvedEx.2 (v) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Set · 6 questions
Differentiate the following
w
.
r
.
t
.
x
w.\ r.\ t.\ x
w
.
r
.
t
.
x
.
Q503
#503
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Moderate
Add
y
=
log
3
(
log
5
x
)
y = \log_3(\log_5 x)
y
=
lo
g
3
(
lo
g
5
x
)
Show model answer
[Q1.1.3 SolvedEx.3 (i) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q504
#504
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Moderate
Add
y
=
log
[
e
3
x
⋅
(
3
x
−
4
)
2
3
2
x
+
5
3
]
y = \log\left[e^{3x} \cdot \frac{(3x - 4)^{\frac{2}{3}}}{\sqrt[3]{2x + 5}}\right]
y
=
lo
g
[
e
3
x
⋅
3
2
x
+
5
(
3
x
−
4
)
3
2
]
Show model answer
[Q1.1.3 SolvedEx.3 (ii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q505
#505
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Hard
Add
y
=
log
[
1
−
cos
(
3
x
2
)
1
+
cos
(
3
x
2
)
]
y = \log\left[\sqrt{\frac{1 - \cos\left(\frac{3x}{2}\right)}{1 + \cos\left(\frac{3x}{2}\right)}}\right]
y
=
lo
g
[
1
+
c
o
s
(
2
3
x
)
1
−
c
o
s
(
2
3
x
)
]
Show model answer
[Q1.1.3 SolvedEx.3 (iii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q506
#506
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Hard
Add
y
=
log
[
x
+
x
2
+
a
2
x
2
+
a
2
−
x
]
y = \log\left[\frac{x + \sqrt{x^2 + a^2}}{\sqrt{x^2 + a^2} - x}\right]
y
=
lo
g
[
x
2
+
a
2
−
x
x
+
x
2
+
a
2
]
Show model answer
[Q1.1.3 SolvedEx.3 (iv) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q507
#507
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Hard
Add
y
=
(
4
)
log
2
(
sin
x
)
+
(
9
)
log
3
(
cos
x
)
y = (4)^{\log_2(\sin x)} + (9)^{\log_3(\cos x)}
y
=
(
4
)
l
o
g
2
(
s
i
n
x
)
+
(
9
)
l
o
g
3
(
c
o
s
x
)
Show model answer
[Q1.1.3 SolvedEx.3 (v) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q508
#508
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Hard
Add
y
=
a
a
log
a
(
cot
x
)
y = a^{a^{\log_a(\cot x)}}
y
=
a
a
l
o
g
a
(
c
o
t
x
)
Show model answer
[Q1.1.3 SolvedEx.3 (vi) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q509
#509
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Moderate
Add
If
f
(
x
)
=
7
g
(
x
)
−
3
f(x) = \sqrt{7g(x) - 3}
f
(
x
)
=
7
g
(
x
)
−
3
,
g
(
3
)
=
4
g(3) = 4
g
(
3
)
=
4
and
g
′
(
3
)
=
5
g'(3) = 5
g
′
(
3
)
=
5
, find
f
′
(
3
)
f'(3)
f
′
(
3
)
.
Show model answer
[Q1.1.3 SolvedEx.4 · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q510
#510
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Hard
Add
If
F
(
x
)
=
G
{
3
G
[
5
G
(
x
)
]
}
F(x) = G\{3G[5G(x)]\}
F
(
x
)
=
G
{
3
G
[
5
G
(
x
)]}
,
G
(
0
)
=
0
G(0) = 0
G
(
0
)
=
0
and
G
′
(
0
)
=
3
G'(0) = 3
G
′
(
0
)
=
3
, find
F
′
(
0
)
F'(0)
F
′
(
0
)
.
Show model answer
[Q1.1.3 SolvedEx.5 · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q511
#511
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Moderate
Add
Select the appropriate hint from the hint basket and fill in the blank spaces in the following paragraph. [Activity] "Let
f
(
x
)
=
sin
x
f(x) = \sin x
f
(
x
)
=
sin
x
and
g
(
x
)
=
log
x
g(x) = \log x
g
(
x
)
=
lo
g
x
then
f
[
g
(
x
)
]
=
f[g(x)] =
f
[
g
(
x
)]
=
______ and
g
[
f
(
x
)
]
=
g[f(x)] =
g
[
f
(
x
)]
=
______. Now
f
′
(
x
)
=
f'(x) =
f
′
(
x
)
=
______ and
g
′
(
x
)
=
g'(x) =
g
′
(
x
)
=
______. The derivative of
f
[
g
(
x
)
]
f[g(x)]
f
[
g
(
x
)]
w
.
r
.
t
.
x
w.\ r.\ t.\ x
w
.
r
.
t
.
x
in terms of
f
f
f
and
g
g
g
is ______. Therefore
d
d
x
[
f
[
g
(
x
)
]
]
=
\frac{d}{dx}\left[f[g(x)]\right] =
d
x
d
[
f
[
g
(
x
)]
]
=
______ and
[
d
d
x
[
f
[
g
(
x
)
]
]
]
x
=
1
=
\left[\frac{d}{dx}[f[g(x)]]\right]_{x=1} =
[
d
x
d
[
f
[
g
(
x
)]]
]
x
=
1
=
______. The derivative of
g
[
f
(
x
)
]
g[f(x)]
g
[
f
(
x
)]
w
.
r
.
t
.
x
w.\ r.\ t.\ x
w
.
r
.
t
.
x
in terms of
f
f
f
and
g
g
g
is ______. Therefore
d
d
x
[
g
[
f
(
x
)
]
]
=
\frac{d}{dx}\left[g[f(x)]\right] =
d
x
d
[
g
[
f
(
x
)]
]
=
______ and
[
d
d
x
[
g
[
f
(
x
)
]
]
]
x
=
π
3
=
\left[\frac{d}{dx}[g[f(x)]]\right]_{x=\frac{\pi}{3}} =
[
d
x
d
[
g
[
f
(
x
)]]
]
x
=
3
π
=
______." Hint basket :
{
f
′
[
g
(
x
)
]
⋅
g
′
(
x
)
,
cos
(
log
x
)
x
,
1
,
g
′
[
f
(
x
)
]
⋅
f
′
(
x
)
,
cot
x
,
3
,
sin
(
log
x
)
,
log
(
sin
x
)
,
cos
x
,
1
x
}
\left\{f'[g(x)] \cdot g'(x),\ \frac{\cos(\log x)}{x},\ 1,\ g'[f(x)] \cdot f'(x),\ \cot x,\ \sqrt{3},\ \sin(\log x),\ \log(\sin x),\ \cos x,\ \frac{1}{x}\right\}
{
f
′
[
g
(
x
)]
⋅
g
′
(
x
)
,
x
c
o
s
(
l
o
g
x
)
,
1
,
g
′
[
f
(
x
)]
⋅
f
′
(
x
)
,
cot
x
,
3
,
sin
(
lo
g
x
)
,
lo
g
(
sin
x
)
,
cos
x
,
x
1
}
Show model answer
[Q1.1.3 SolvedEx.6 · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Set · 6 questions
Differentiate w. r. t.
x
x
x
.
Q512
#512
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Easy
Add
(
x
3
−
2
x
−
1
)
5
(x^3 - 2x - 1)^5
(
x
3
−
2
x
−
1
)
5
Show model answer
[QEx 1.1 Q.1 (i) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q513
#513
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Moderate
Add
(
2
x
3
2
−
3
x
4
3
−
5
)
5
2
\left(2x^{\frac{3}{2}} - 3x^{\frac{4}{3}} - 5\right)^{\frac{5}{2}}
(
2
x
2
3
−
3
x
3
4
−
5
)
2
5
Show model answer
[QEx 1.1 Q.1 (ii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q514
#514
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Easy
Add
x
2
+
4
x
−
7
\sqrt{x^2 + 4x - 7}
x
2
+
4
x
−
7
Show model answer
[QEx 1.1 Q.1 (iii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q515
#515
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Moderate
Add
x
2
+
x
2
+
1
\sqrt{x^2 + \sqrt{x^2 + 1}}
x
2
+
x
2
+
1
Show model answer
[QEx 1.1 Q.1 (iv) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q516
#516
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Moderate
Add
8
3
(
2
x
2
−
7
x
−
5
)
11
3
\frac{8}{3\sqrt[3]{\left(2x^2 - 7x - 5\right)^{11}}}
3
3
(
2
x
2
−
7
x
−
5
)
11
8
Show model answer
[QEx 1.1 Q.1 (v) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q517
#517
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Moderate
Add
(
3
x
−
5
−
1
3
x
−
5
)
5
\left(\sqrt{3x - 5} - \frac{1}{\sqrt{3x - 5}}\right)^5
(
3
x
−
5
−
3
x
−
5
1
)
5
Show model answer
[QEx 1.1 Q.1 (vi) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Set · 8 questions
Differentiate the following w.r.t.
x
x
x
Q518
#518
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Easy
Add
cos
(
x
2
+
a
2
)
\cos\left(x^2 + a^2\right)
cos
(
x
2
+
a
2
)
Show model answer
[QEx 1.1 Q.2 (i) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q519
#519
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Moderate
Add
e
(
3
x
+
2
)
+
5
\sqrt{e^{(3x + 2)} + 5}
e
(
3
x
+
2
)
+
5
Show model answer
[QEx 1.1 Q.2 (ii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q520
#520
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Moderate
Add
log
[
tan
(
x
2
)
]
\log\left[\tan\left(\frac{x}{2}\right)\right]
lo
g
[
tan
(
2
x
)
]
Show model answer
[QEx 1.1 Q.2 (iii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q521
#521
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Moderate
Add
tan
x
\sqrt{\tan \sqrt{x}}
tan
x
Show model answer
[QEx 1.1 Q.2 (iv) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q522
#522
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Moderate
Add
cot
3
[
log
(
x
3
)
]
\cot^3\left[\log\left(x^3\right)\right]
cot
3
[
lo
g
(
x
3
)
]
Show model answer
[QEx 1.1 Q.2 (v) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q523
#523
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Moderate
Add
5
sin
3
x
+
3
5^{\sin^3 x + 3}
5
s
i
n
3
x
+
3
Show model answer
[QEx 1.1 Q.2 (vi) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q524
#524
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Moderate
Add
cosec
(
cos
x
)
\operatorname{cosec}\left(\sqrt{\cos x}\right)
cosec
(
cos
x
)
Show model answer
[QEx 1.1 Q.2 (vii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q525
#525
Mathematics → Differentiation → Derivatives of Composite Functions (Chain Rule)
·
Easy
Add
log
[
cos
(
x
3
−
5
)
]
\log\left[\cos\left(x^3 - 5\right)\right]
lo
g
[
cos
(
x
3
−
5
)
]
Show model answer
[QEx 1.1 Q.2 (viii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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