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Exam: Maharashtra HSC Class 12
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Q426
#426
Mathematics → Application of Derivatives → Derivative as a Rate Measure
·
Moderate
Add
A man of height 180 cm is moving away from a lamp post at the rate of 1.2 meters per second. If the height of the lamp post is 4.5 meters, find the rate at which (i) his shadow is lengthening. (ii) the tip of the shadow is moving.
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[Q2.1.3 SolvedEx.4 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q427
#427
Mathematics → Application of Derivatives → Velocity, Acceleration and Jerk
·
Easy
Add
A car is moving in such a way that the distance it covers, is given by the equation
s
=
4
t
2
+
3
t
s = 4t^2 + 3t
s
=
4
t
2
+
3
t
where
s
s
s
is in meters and
t
t
t
is in seconds. What would be the velocity and the acceleration of the car at time
t
=
20
t = 20
t
=
20
second ?
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[Q2.1.4 SolvedEx.1 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q428
#428
Mathematics → Application of Derivatives → Velocity, Acceleration and Jerk
·
Moderate
Add
The displacement of a particle at time
t
t
t
is given by
s
=
2
t
3
−
5
t
2
+
4
t
−
3
s = 2t^3 - 5t^2 + 4t - 3
s
=
2
t
3
−
5
t
2
+
4
t
−
3
. Find the time when the acceleration is 14 ft/ sec
2
^2
2
, the velocity and the displacement at that time.
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[Q2.1.4 SolvedEx.2 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q429
#429
Mathematics → Application of Derivatives → Approximations
·
Easy
Add
Find the approximate value of
64.1
\sqrt{64.1}
64.1
.
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[Q2.2.1 SolvedEx.1 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q430
#430
Mathematics → Application of Derivatives → Approximations
·
Easy
Add
Find the approximate value of
(
3.98
)
3
(3.98)^3
(
3.98
)
3
.
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[Q2.2.1 SolvedEx.2 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q431
#431
Mathematics → Application of Derivatives → Approximations
·
Moderate
Add
Find the approximate value of
sin
(
30
∘
30
′
)
\sin(30^\circ\, 30')
sin
(
3
0
∘
3
0
′
)
. Given that
1
∘
=
0.0175
c
1^\circ = 0.0175^c
1
∘
=
0.017
5
c
and
cos
30
∘
=
0.866
\cos 30^\circ = 0.866
cos
3
0
∘
=
0.866
.
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[Q2.2.1 SolvedEx.3 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q432
#432
Mathematics → Application of Derivatives → Approximations
·
Moderate
Add
Find the approximate value of
tan
−
1
(
0.99
)
\tan^{-1}(0.99)
tan
−
1
(
0.99
)
, Given that
π
≈
3.1416
\pi \approx 3.1416
π
≈
3.1416
.
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[Q2.2.1 SolvedEx.4 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q433
#433
Mathematics → Application of Derivatives → Approximations
·
Easy
Add
Find the approximate value of
e
1.005
e^{1.005}
e
1.005
. Given that
e
=
2.7183
e = 2.7183
e
=
2.7183
.
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[Q2.2.1 SolvedEx.5 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q434
#434
Mathematics → Application of Derivatives → Approximations
·
Moderate
Add
Find the approximate value of
log
10
(
998
)
\log_{10}(998)
lo
g
10
(
998
)
. Given that
log
10
e
=
0.4343
\log_{10} e = 0.4343
lo
g
10
e
=
0.4343
.
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[Q2.2.1 SolvedEx.6 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q435
#435
Mathematics → Application of Derivatives → Approximations
·
Easy
Add
Find the approximate value of
f
(
x
)
=
x
3
+
5
x
2
−
2
x
+
3
f(x) = x^3 + 5x^2 - 2x + 3
f
(
x
)
=
x
3
+
5
x
2
−
2
x
+
3
at
x
=
1.98
x = 1.98
x
=
1.98
.
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[Q2.2.1 SolvedEx.7 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Set · 2 questions
Check whether conditions of Rolle's theorem are satisfied by the following functions.
Q436
#436
Mathematics → Application of Derivatives → Rolle's Theorem
·
Easy
Add
f
(
x
)
=
2
x
3
−
5
x
2
+
3
x
+
2
f(x) = 2x^3 - 5x^2 + 3x + 2
f
(
x
)
=
2
x
3
−
5
x
2
+
3
x
+
2
,
x
∈
[
0
,
3
2
]
x \in \left[0, \dfrac{3}{2}\right]
x
∈
[
0
,
2
3
]
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[Q2.3.1 SolvedEx.1 (i) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q437
#437
Mathematics → Application of Derivatives → Rolle's Theorem
·
Easy
Add
f
(
x
)
=
x
2
−
2
x
+
3
f(x) = x^2 - 2x + 3
f
(
x
)
=
x
2
−
2
x
+
3
,
x
∈
[
1
,
4
]
x \in [1, 4]
x
∈
[
1
,
4
]
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[Q2.3.1 SolvedEx.1 (ii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q438
#438
Mathematics → Application of Derivatives → Rolle's Theorem
·
Moderate
Add
Verify Rolle's theorem for the function
f
(
x
)
=
x
2
−
4
x
+
10
f(x) = x^2 - 4x + 10
f
(
x
)
=
x
2
−
4
x
+
10
on
[
0
,
4
]
[0, 4]
[
0
,
4
]
.
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[Q2.3.1 SolvedEx.2 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q439
#439
Mathematics → Application of Derivatives → Rolle's Theorem
·
Moderate
Add
Given an interval
[
a
,
b
]
[a, b]
[
a
,
b
]
that satisfies hypothesis of Rolle's theorem for the function
f
(
x
)
=
x
3
−
2
x
2
+
3
f(x) = x^3 - 2x^2 + 3
f
(
x
)
=
x
3
−
2
x
2
+
3
. It is known that
a
=
0
a = 0
a
=
0
. Find the value of
b
b
b
.
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[Q2.3.1 SolvedEx.3 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q440
#440
Mathematics → Application of Derivatives → Rolle's Theorem
·
Hard
Add
Verify Rolle's theorem for the function
f
(
x
)
=
e
x
(
sin
x
−
cos
x
)
f(x) = e^x(\sin x - \cos x)
f
(
x
)
=
e
x
(
sin
x
−
cos
x
)
on
[
π
4
,
5
π
4
]
\left[\dfrac{\pi}{4}, \dfrac{5\pi}{4}\right]
[
4
π
,
4
5
π
]
.
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[Q2.3.1 SolvedEx.4 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q441
#441
Mathematics → Application of Derivatives → Lagrange's Mean Value Theorem
·
Moderate
Add
Verify Lagrange's mean value theorem for the function
f
(
x
)
=
x
+
4
f(x) = \sqrt{x + 4}
f
(
x
)
=
x
+
4
on the interval
[
0
,
5
]
[0, 5]
[
0
,
5
]
.
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[Q2.3.2 SolvedEx.1 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q442
#442
Mathematics → Application of Derivatives → Lagrange's Mean Value Theorem
·
Moderate
Add
Verify Lagrange's mean value theorem for the function
f
(
x
)
=
x
+
1
x
f(x) = x + \dfrac{1}{x}
f
(
x
)
=
x
+
x
1
on the interval
[
1
,
3
]
[1, 3]
[
1
,
3
]
.
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[Q2.3.2 SolvedEx.2 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q443
#443
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Easy
Add
Show that the function
f
(
x
)
=
x
3
+
10
x
+
7
f(x) = x^3 + 10x + 7
f
(
x
)
=
x
3
+
10
x
+
7
for
x
∈
R
x \in R
x
∈
R
is strictly increasing.
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[Q2.4.1 SolvedEx.1 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q444
#444
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Moderate
Add
Test whether the function
f
(
x
)
=
x
3
+
6
x
2
+
12
x
−
5
f(x) = x^3 + 6x^2 + 12x - 5
f
(
x
)
=
x
3
+
6
x
2
+
12
x
−
5
is increasing or decreasing for all
x
∈
R
x \in R
x
∈
R
.
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[Q2.4.1 SolvedEx.2 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q445
#445
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Moderate
Add
Find the values of
x
x
x
, for which the funciton
f
(
x
)
=
x
3
+
12
x
2
+
36
x
+
6
f(x) = x^3 + 12x^2 + 36x + 6
f
(
x
)
=
x
3
+
12
x
2
+
36
x
+
6
is (i) monotonically increasing. (ii) monotonically decreasing.
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[Q2.4.1 SolvedEx.3 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q446
#446
Mathematics → Application of Derivatives → Maxima and Minima
·
Moderate
Add
Find the local maxima or local minima of
f
(
x
)
=
x
3
−
3
x
f(x) = x^3 - 3x
f
(
x
)
=
x
3
−
3
x
.
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[Q2.4.3 SolvedEx.1 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q447
#447
Mathematics → Application of Derivatives → Maxima and Minima
·
Moderate
Add
Find the local maximum and local minimum value of
f
(
x
)
=
x
3
−
3
x
2
−
24
x
+
5
f(x) = x^3 - 3x^2 - 24x + 5
f
(
x
)
=
x
3
−
3
x
2
−
24
x
+
5
.
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[Q2.4.4 SolvedEx.1 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q448
#448
Mathematics → Application of Derivatives → Maxima and Minima
·
Easy
Add
A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.
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[Q2.4.4 SolvedEx.2 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q449
#449
Mathematics → Application of Derivatives → Maxima and Minima
·
Hard
Add
A Rectangular sheet of paper has it area 24 sq. meters. The margin at the top and the bottom are 75 cm each and at the sides 50 cm each. What are the dimensions of the paper, if the area of the printed space is maximum ?
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[Q2.4.4 SolvedEx.3 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q450
#450
Mathematics → Application of Derivatives → Maxima and Minima
·
Moderate
Add
An open box is to be cut out of piece of square card of side 18 cm by cutting of equal squares from the corners and turning up the sides. Find the maximum volume of the box.
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[Q2.4.4 SolvedEx.4 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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