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Exam: Maharashtra HSC Class 12
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Set · 1 question
Find the approximate value of
Q326
#326
Mathematics → Application of Derivatives → Approximations
·
Moderate
Add
tan
−
1
(
1.001
)
\tan^{-1}(1.001)
tan
−
1
(
1.001
)
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[QEx 2.2 Q.3 (iii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Set · 3 questions
Find the approximate value of
Q327
#327
Mathematics → Application of Derivatives → Approximations
·
Easy
Add
e
0.995
e^{0.995}
e
0.995
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[QEx 2.2 Q.4 (i) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q328
#328
Mathematics → Application of Derivatives → Approximations
·
Easy
Add
e
2.1
e^{2.1}
e
2.1
given that
e
2
=
7.389
e^2 = 7.389
e
2
=
7.389
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[QEx 2.2 Q.4 (ii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q329
#329
Mathematics → Application of Derivatives → Approximations
·
Moderate
Add
3
2.01
3^{2.01}
3
2.01
given that
log
3
=
1.0986
\log 3 = 1.0986
lo
g
3
=
1.0986
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[QEx 2.2 Q.4 (iii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Set · 3 questions
Find the approximate value of
Q330
#330
Mathematics → Application of Derivatives → Approximations
·
Moderate
Add
log
e
(
101
)
\log_e(101)
lo
g
e
(
101
)
given that
log
e
10
=
2.3026
\log_e 10 = 2.3026
lo
g
e
10
=
2.3026
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[QEx 2.2 Q.5 (i) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q331
#331
Mathematics → Application of Derivatives → Approximations
·
Moderate
Add
log
e
(
9.01
)
\log_e(9.01)
lo
g
e
(
9.01
)
given that
log
3
=
1.0986
\log 3 = 1.0986
lo
g
3
=
1.0986
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[QEx 2.2 Q.5 (ii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q332
#332
Mathematics → Application of Derivatives → Approximations
·
Moderate
Add
log
10
(
1016
)
\log_{10}(1016)
lo
g
10
(
1016
)
given that
log
10
e
=
0.4343
\log_{10} e = 0.4343
lo
g
10
e
=
0.4343
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[QEx 2.2 Q.5 (iii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Set · 2 questions
Find the approximate value of
Q333
#333
Mathematics → Application of Derivatives → Approximations
·
Easy
Add
f
(
x
)
=
x
3
−
3
x
+
5
f(x) = x^3 - 3x + 5
f
(
x
)
=
x
3
−
3
x
+
5
at
x
=
1.99
x = 1.99
x
=
1.99
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[QEx 2.2 Q.6 (i) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q334
#334
Mathematics → Application of Derivatives → Approximations
·
Easy
Add
f
(
x
)
=
x
3
+
5
x
2
−
7
x
+
10
f(x) = x^3 + 5x^2 - 7x + 10
f
(
x
)
=
x
3
+
5
x
2
−
7
x
+
10
at
x
=
1.12
x = 1.12
x
=
1.12
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[QEx 2.2 Q.6 (ii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Set · 6 questions
Check the validity of the Rolle's theorem for the following functions.
Q335
#335
Mathematics → Application of Derivatives → Rolle's Theorem
·
Easy
Add
f
(
x
)
=
x
2
−
4
x
+
3
f(x) = x^2 - 4x + 3
f
(
x
)
=
x
2
−
4
x
+
3
,
x
∈
[
1
,
3
]
x \in [1, 3]
x
∈
[
1
,
3
]
Show model answer
[QEx 2.3 Q.1 (i) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q336
#336
Mathematics → Application of Derivatives → Rolle's Theorem
·
Moderate
Add
f
(
x
)
=
e
−
x
sin
x
f(x) = e^{-x} \sin x
f
(
x
)
=
e
−
x
sin
x
,
x
∈
[
0
,
π
]
x \in [0, \pi]
x
∈
[
0
,
π
]
Show model answer
[QEx 2.3 Q.1 (ii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q337
#337
Mathematics → Application of Derivatives → Rolle's Theorem
·
Easy
Add
f
(
x
)
=
2
x
2
−
5
x
+
3
f(x) = 2x^2 - 5x + 3
f
(
x
)
=
2
x
2
−
5
x
+
3
,
x
∈
[
1
,
3
]
x \in [1, 3]
x
∈
[
1
,
3
]
Show model answer
[QEx 2.3 Q.1 (iii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q338
#338
Mathematics → Application of Derivatives → Rolle's Theorem
·
Moderate
Add
f
(
x
)
=
sin
x
−
cos
x
+
3
f(x) = \sin x - \cos x + 3
f
(
x
)
=
sin
x
−
cos
x
+
3
,
x
∈
[
0
,
2
π
]
x \in [0, 2\pi]
x
∈
[
0
,
2
π
]
Show model answer
[QEx 2.3 Q.1 (iv) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q339
#339
Mathematics → Application of Derivatives → Rolle's Theorem
·
Hard
Add
f
(
x
)
=
x
2
f(x) = x^2
f
(
x
)
=
x
2
if
0
≤
x
≤
2
0 \leq x \leq 2
0
≤
x
≤
2
=
6
−
x
= 6 - x
=
6
−
x
if
2
≤
x
≤
6
2 \leq x \leq 6
2
≤
x
≤
6
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[QEx 2.3 Q.1 (v) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q340
#340
Mathematics → Application of Derivatives → Rolle's Theorem
·
Moderate
Add
f
(
x
)
=
x
2
3
f(x) = x^{\frac{2}{3}}
f
(
x
)
=
x
3
2
,
x
∈
[
−
1
,
1
]
x \in [-1, 1]
x
∈
[
−
1
,
1
]
Show model answer
[QEx 2.3 Q.1 (vi) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q341
#341
Mathematics → Application of Derivatives → Rolle's Theorem
·
Moderate
Add
Given an interval
[
a
,
b
]
[a, b]
[
a
,
b
]
that satisfies hypothesis of Rolle's thorem for the function
f
(
x
)
=
x
4
+
x
2
−
2
f(x) = x^4 + x^2 - 2
f
(
x
)
=
x
4
+
x
2
−
2
. It is known that
a
=
−
1
a = -1
a
=
−
1
. Find the value of
b
b
b
.
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[QEx 2.3 Q.2 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Set · 3 questions
Verify Rolle's theorem for the following functions.
Q342
#342
Mathematics → Application of Derivatives → Rolle's Theorem
·
Moderate
Add
f
(
x
)
=
sin
x
+
cos
x
+
7
f(x) = \sin x + \cos x + 7
f
(
x
)
=
sin
x
+
cos
x
+
7
,
x
∈
[
0
,
2
π
]
x \in [0, 2\pi]
x
∈
[
0
,
2
π
]
Show model answer
[QEx 2.3 Q.3 (i) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q343
#343
Mathematics → Application of Derivatives → Rolle's Theorem
·
Moderate
Add
f
(
x
)
=
sin
(
x
2
)
f(x) = \sin \left( \frac{x}{2} \right)
f
(
x
)
=
sin
(
2
x
)
,
x
∈
[
0
,
2
π
]
x \in [0, 2\pi]
x
∈
[
0
,
2
π
]
Show model answer
[QEx 2.3 Q.3 (ii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q344
#344
Mathematics → Application of Derivatives → Rolle's Theorem
·
Moderate
Add
f
(
x
)
=
x
2
−
5
x
+
9
f(x) = x^2 - 5x + 9
f
(
x
)
=
x
2
−
5
x
+
9
,
x
∈
[
1
,
4
]
x \in [1, 4]
x
∈
[
1
,
4
]
Show model answer
[QEx 2.3 Q.3 (iii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q345
#345
Mathematics → Application of Derivatives → Rolle's Theorem
·
Hard
Add
If Rolle's theorem holds for the function
f
(
x
)
=
x
3
+
p
x
2
+
q
x
+
5
f(x) = x^3 + px^2 + qx + 5
f
(
x
)
=
x
3
+
p
x
2
+
q
x
+
5
,
x
∈
[
1
,
3
]
x \in [1, 3]
x
∈
[
1
,
3
]
with
c
=
2
+
1
3
c = 2 + \frac{1}{\sqrt{3}}
c
=
2
+
3
1
, find the values of
p
p
p
and
q
q
q
.
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[QEx 2.3 Q.4 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q346
#346
Mathematics → Application of Derivatives → Rolle's Theorem
·
Moderate
Add
Rolle's theorem holds for the function
f
(
x
)
=
(
x
−
2
)
log
x
f(x) = (x - 2) \log x
f
(
x
)
=
(
x
−
2
)
lo
g
x
,
x
∈
[
1
,
2
]
x \in [1, 2]
x
∈
[
1
,
2
]
, show that the equation
x
log
x
=
2
−
x
x \log x = 2 - x
x
lo
g
x
=
2
−
x
is satisfied by at least one value of
x
x
x
in
(
1
,
2
)
(1, 2)
(
1
,
2
)
.
Show model answer
[QEx 2.3 Q.5 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q347
#347
Mathematics → Application of Derivatives → Rolle's Theorem
·
Moderate
Add
The function
f
(
x
)
=
x
(
x
+
3
)
e
−
x
2
f(x) = x (x + 3) e^{-\frac{x}{2}}
f
(
x
)
=
x
(
x
+
3
)
e
−
2
x
satisfies all the conditions of Rolle's theorem on
[
−
3
,
0
]
[-3, 0]
[
−
3
,
0
]
. Find the value of
c
c
c
such that
f
′
(
c
)
=
0
f'(c) = 0
f
′
(
c
)
=
0
.
Show model answer
[QEx 2.3 Q.6 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Set · 3 questions
Verify Lagrange's mean value theorem for the following functions.
Q348
#348
Mathematics → Application of Derivatives → Lagrange's Mean Value Theorem
·
Moderate
Add
f
(
x
)
=
log
x
f(x) = \log x
f
(
x
)
=
lo
g
x
, on
[
1
,
e
]
[1, e]
[
1
,
e
]
Show model answer
[QEx 2.3 Q.7 (i) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q349
#349
Mathematics → Application of Derivatives → Lagrange's Mean Value Theorem
·
Moderate
Add
f
(
x
)
=
(
x
−
1
)
(
x
−
2
)
(
x
−
3
)
f(x) = (x - 1) (x - 2) (x - 3)
f
(
x
)
=
(
x
−
1
)
(
x
−
2
)
(
x
−
3
)
on
[
0
,
4
]
[0, 4]
[
0
,
4
]
Show model answer
[QEx 2.3 Q.7 (ii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q350
#350
Mathematics → Application of Derivatives → Lagrange's Mean Value Theorem
·
Moderate
Add
f
(
x
)
=
x
2
−
3
x
−
1
f(x) = x^2 - 3x - 1
f
(
x
)
=
x
2
−
3
x
−
1
,
x
∈
[
−
11
7
,
13
7
]
x \in \left[ -\frac{11}{7}, \frac{13}{7} \right]
x
∈
[
−
7
11
,
7
13
]
Show model answer
[QEx 2.3 Q.7 (iii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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