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Exam: Maharashtra HSC Class 12
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Set · 2 questions
Verify Lagrange's mean value theorem for the following functions.
Q351
#351
Mathematics → Application of Derivatives → Lagrange's Mean Value Theorem
·
Easy
Add
f
(
x
)
=
2
x
−
x
2
f(x) = 2x - x^2
f
(
x
)
=
2
x
−
x
2
,
x
∈
[
0
,
1
]
x \in [0, 1]
x
∈
[
0
,
1
]
Show model answer
[QEx 2.3 Q.7 (iv) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q352
#352
Mathematics → Application of Derivatives → Lagrange's Mean Value Theorem
·
Moderate
Add
f
(
x
)
=
x
−
1
x
−
3
f(x) = \frac{x - 1}{x - 3}
f
(
x
)
=
x
−
3
x
−
1
on
[
4
,
5
]
[4, 5]
[
4
,
5
]
Show model answer
[QEx 2.3 Q.7 (v) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Set · 3 questions
Test whether the following functions are increasing or decreasing.
Q353
#353
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Easy
Add
f
(
x
)
=
x
3
−
6
x
2
+
12
x
−
16
f(x) = x^3 - 6x^2 + 12x - 16
f
(
x
)
=
x
3
−
6
x
2
+
12
x
−
16
,
x
∈
R
x \in \mathbb{R}
x
∈
R
Show model answer
[QEx 2.4 Q.1 (i) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q354
#354
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Easy
Add
f
(
x
)
=
2
−
3
x
+
3
x
2
−
x
3
f(x) = 2 - 3x + 3x^2 - x^3
f
(
x
)
=
2
−
3
x
+
3
x
2
−
x
3
,
x
∈
R
x \in \mathbb{R}
x
∈
R
Show model answer
[QEx 2.4 Q.1 (ii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q355
#355
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Moderate
Add
f
(
x
)
=
x
−
1
x
f(x) = x - \dfrac{1}{x}
f
(
x
)
=
x
−
x
1
,
x
∈
R
x \in \mathbb{R}
x
∈
R
and
x
≠
0
x \neq 0
x
=
0
Show model answer
[QEx 2.4 Q.1 (iii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Set · 3 questions
Find the values of
x
x
x
for which the following functions are strictly increasing -
Q356
#356
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Easy
Add
f
(
x
)
=
2
x
3
−
3
x
2
−
12
x
+
6
f(x) = 2x^3 - 3x^2 - 12x + 6
f
(
x
)
=
2
x
3
−
3
x
2
−
12
x
+
6
Show model answer
[QEx 2.4 Q.2 (i) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q357
#357
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Easy
Add
f
(
x
)
=
3
+
3
x
−
3
x
2
+
x
3
f(x) = 3 + 3x - 3x^2 + x^3
f
(
x
)
=
3
+
3
x
−
3
x
2
+
x
3
Show model answer
[QEx 2.4 Q.2 (ii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q358
#358
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Easy
Add
f
(
x
)
=
x
3
−
6
x
2
−
36
x
+
7
f(x) = x^3 - 6x^2 - 36x + 7
f
(
x
)
=
x
3
−
6
x
2
−
36
x
+
7
Show model answer
[QEx 2.4 Q.2 (iii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Set · 3 questions
Find the values of
x
x
x
for which the following functions are strictly decreasing -
Q359
#359
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Easy
Add
f
(
x
)
=
2
x
3
−
3
x
2
−
12
x
+
6
f(x) = 2x^3 - 3x^2 - 12x + 6
f
(
x
)
=
2
x
3
−
3
x
2
−
12
x
+
6
Show model answer
[QEx 2.4 Q.3 (i) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q360
#360
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Easy
Add
f
(
x
)
=
x
+
25
x
f(x) = x + \dfrac{25}{x}
f
(
x
)
=
x
+
x
25
Show model answer
[QEx 2.4 Q.3 (ii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q361
#361
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Easy
Add
f
(
x
)
=
x
3
−
9
x
2
+
24
x
+
12
f(x) = x^3 - 9x^2 + 24x + 12
f
(
x
)
=
x
3
−
9
x
2
+
24
x
+
12
Show model answer
[QEx 2.4 Q.3 (iii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q362
#362
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Easy
Add
Find the values of
x
x
x
for which the function
f
(
x
)
=
x
3
−
12
x
2
−
144
x
+
13
f(x) = x^3 - 12x^2 - 144x + 13
f
(
x
)
=
x
3
−
12
x
2
−
144
x
+
13
(a) Increasing (b) Decreasing
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[QEx 2.4 Q.4 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q363
#363
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Easy
Add
Find the values of
x
x
x
for which
f
(
x
)
=
2
x
3
−
15
x
2
−
144
x
−
7
f(x) = 2x^3 - 15x^2 - 144x - 7
f
(
x
)
=
2
x
3
−
15
x
2
−
144
x
−
7
is (a) strictly increasing (b) strictly decreasing
Show model answer
[QEx 2.4 Q.5 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q364
#364
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Moderate
Add
Find the values of
x
x
x
for which
f
(
x
)
=
x
x
2
+
1
f(x) = \dfrac{x}{x^2 + 1}
f
(
x
)
=
x
2
+
1
x
is (a) strictly increasing (b) strictly decreasing
Show model answer
[QEx 2.4 Q.6 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q365
#365
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Moderate
Add
Show that
f
(
x
)
=
3
x
+
1
3
x
f(x) = 3x + \dfrac{1}{3x}
f
(
x
)
=
3
x
+
3
x
1
increasing in
(
1
3
,
1
)
\left(\dfrac{1}{3}, 1\right)
(
3
1
,
1
)
and decreasing in
(
1
9
,
1
3
)
\left(\dfrac{1}{9}, \dfrac{1}{3}\right)
(
9
1
,
3
1
)
.
Show model answer
[QEx 2.4 Q.7 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q366
#366
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Easy
Add
Show that
f
(
x
)
=
x
−
cos
x
f(x) = x - \cos x
f
(
x
)
=
x
−
cos
x
is increasing for all
x
x
x
.
Show model answer
[QEx 2.4 Q.8 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Set · 6 questions
Find the maximum and minimum of the following functions -
Q367
#367
Mathematics → Application of Derivatives → Maxima and Minima
·
Moderate
Add
y
=
5
x
3
+
2
x
2
−
3
x
y = 5x^3 + 2x^2 - 3x
y
=
5
x
3
+
2
x
2
−
3
x
Show model answer
[QEx 2.4 Q.9 (i) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q368
#368
Mathematics → Application of Derivatives → Maxima and Minima
·
Easy
Add
f
(
x
)
=
2
x
3
−
21
x
2
+
36
x
−
20
f(x) = 2x^3 - 21x^2 + 36x - 20
f
(
x
)
=
2
x
3
−
21
x
2
+
36
x
−
20
Show model answer
[QEx 2.4 Q.9 (ii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q369
#369
Mathematics → Application of Derivatives → Maxima and Minima
·
Easy
Add
f
(
x
)
=
x
3
−
9
x
2
+
24
x
f(x) = x^3 - 9x^2 + 24x
f
(
x
)
=
x
3
−
9
x
2
+
24
x
Show model answer
[QEx 2.4 Q.9 (iii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q370
#370
Mathematics → Application of Derivatives → Maxima and Minima
·
Moderate
Add
f
(
x
)
=
x
2
+
16
x
2
f(x) = x^2 + \dfrac{16}{x^2}
f
(
x
)
=
x
2
+
x
2
16
Show model answer
[QEx 2.4 Q.9 (iv) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q371
#371
Mathematics → Application of Derivatives → Maxima and Minima
·
Moderate
Add
f
(
x
)
=
x
log
x
f(x) = x \log x
f
(
x
)
=
x
lo
g
x
Show model answer
[QEx 2.4 Q.9 (v) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q372
#372
Mathematics → Application of Derivatives → Maxima and Minima
·
Moderate
Add
f
(
x
)
=
log
x
x
f(x) = \dfrac{\log x}{x}
f
(
x
)
=
x
lo
g
x
Show model answer
[QEx 2.4 Q.9 (vi) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q373
#373
Mathematics → Application of Derivatives → Maxima and Minima
·
Easy
Add
Divide the number 30 in to two parts such that their product is maximum.
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[QEx 2.4 Q.10 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q374
#374
Mathematics → Application of Derivatives → Maxima and Minima
·
Easy
Add
Divide the number 20 in to two parts such that sum of their squares is minimum.
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[QEx 2.4 Q.11 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q375
#375
Mathematics → Application of Derivatives → Maxima and Minima
·
Easy
Add
A wire of length 36 meter is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.
Show model answer
[QEx 2.4 Q.12 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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