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Exam: Maharashtra HSC Class 12
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Q401
#401
Mathematics → Application of Derivatives → Derivative as a Rate Measure
·
Moderate
Add
A water tank in the form of an inverted cone is being emptied at the rate of 2 cubic feet per second. The height of the cone is 8 feet and the radius is 4 feet. Find the rate of change of the water level when the depth is 6 feet.
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[QMisc II Q.4 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q402
#402
Mathematics → Application of Derivatives → Derivative as a Rate Measure
·
Moderate
Add
Find all points on the ellipse
9
x
2
+
16
y
2
=
400
9x^2 + 16y^2 = 400
9
x
2
+
16
y
2
=
400
, at which the y-coordinate is decreasing and the x-coordinate is increasing at the same rate.
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[QMisc II Q.5 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q403
#403
Mathematics → Application of Derivatives → Rolle's Theorem
·
Moderate
Add
Verify Rolle's theorem for the function
f
(
x
)
=
2
e
x
+
e
−
x
f(x) = \frac{2}{e^x + e^{-x}}
f
(
x
)
=
e
x
+
e
−
x
2
on
[
−
1
,
1
]
[-1, 1]
[
−
1
,
1
]
.
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[QMisc II Q.6 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q404
#404
Mathematics → Application of Derivatives → Lagrange's Mean Value Theorem
·
Moderate
Add
The position of a particle is given by the function
s
(
t
)
=
2
t
2
+
3
t
−
4
s(t) = 2t^2 + 3t - 4
s
(
t
)
=
2
t
2
+
3
t
−
4
. Find the time
t
=
c
t = c
t
=
c
in the interval
0
≤
t
≤
4
0 \leq t \leq 4
0
≤
t
≤
4
when the instantaneous velocity of the particle equals to its average velocity in this interval.
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[QMisc II Q.7 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q405
#405
Mathematics → Application of Derivatives → Approximations
·
Easy
Add
Find the approximate value of the function
f
(
x
)
=
x
2
+
3
x
f(x) = \sqrt{x^2 + 3x}
f
(
x
)
=
x
2
+
3
x
at
x
=
1.02
x = 1.02
x
=
1.02
.
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[QMisc II Q.8 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q406
#406
Mathematics → Application of Derivatives → Approximations
·
Moderate
Add
Find the approximate value of
cos
−
1
(
0.51
)
\cos^{-1}(0.51)
cos
−
1
(
0.51
)
given
π
=
3.1416
\pi = 3.1416
π
=
3.1416
,
2
3
=
1.1547
\frac{2}{\sqrt{3}} = 1.1547
3
2
=
1.1547
.
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[QMisc II Q.9 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q407
#407
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Hard
Add
Find the intervals on which the function
y
=
x
x
y = x^x
y
=
x
x
,
(
x
>
0
)
(x > 0)
(
x
>
0
)
is increasing and decreasing.
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[QMisc II Q.10 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q408
#408
Mathematics → Application of Derivatives → Increasing and Decreasing Functions
·
Moderate
Add
Find the intervals on the which the function
f
(
x
)
=
x
log
x
f(x) = \frac{x}{\log x}
f
(
x
)
=
l
o
g
x
x
, is increasing and decreasing.
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[QMisc II Q.11 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q409
#409
Mathematics → Application of Derivatives → Maxima and Minima
·
Hard
Add
An open box with a square base is to be made out of a given quantity of sheet of area
a
2
a^2
a
2
, Show the maximum volume of the box is
a
3
6
3
\frac{a^3}{6\sqrt{3}}
6
3
a
3
.
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[QMisc II Q.12 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q410
#410
Mathematics → Application of Derivatives → Maxima and Minima
·
Moderate
Add
Show that of all rectangles inscribed in a given circle, the square has the maximum area.
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[QMisc II Q.13 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q411
#411
Mathematics → Application of Derivatives → Maxima and Minima
·
Moderate
Add
Show that a closed right circular cyclinder of given surface area has maximum volume if its height equals the diameter of its base.
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[QMisc II Q.14 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q412
#412
Mathematics → Application of Derivatives → Maxima and Minima
·
Hard
Add
A window is in the form of a rectangle surmounted by a semi-circle. If the perimeter be 30 m, find the dimensions so that the greatest possible amount of light may be admitted.
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[QMisc II Q.15 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q413
#413
Mathematics → Application of Derivatives → Maxima and Minima
·
Hard
Add
Show that the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.
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[QMisc II Q.16 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q414
#414
Mathematics → Application of Derivatives → Maxima and Minima
·
Hard
Add
A wire of length
l
l
l
is cut in to two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is least, if the radius of the circle is half the side of the square.
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[QMisc II Q.17 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q415
#415
Mathematics → Application of Derivatives → Maxima and Minima
·
Hard
Add
A rectangular sheet of paper of fixed perimeter with the sides having their length in the ratio
8
:
15
8 : 15
8
:
15
converted in to an open rectangular box by folding after removing the squares of equal area from all corners. If the total area of the removed squares is 100, the resulting box has maximum valume. Find the lengths of the sides of rectangular sheet of paper.
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[QMisc II Q.18 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q416
#416
Mathematics → Application of Derivatives → Maxima and Minima
·
Hard
Add
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a shpere of radius
r
r
r
is
4
r
3
\frac{4r}{3}
3
4
r
.
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[QMisc II Q.19 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q417
#417
Mathematics → Application of Derivatives → Maxima and Minima
·
Hard
Add
Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is
2
R
3
\frac{2R}{\sqrt{3}}
3
2
R
. Also find the maximum volume.
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[QMisc II Q.20 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q418
#418
Mathematics → Application of Derivatives → Maxima and Minima
·
Moderate
Add
Find the maximum and minimum values of the function
f
(
x
)
=
cos
2
x
+
sin
x
f(x) = \cos^2 x + \sin x
f
(
x
)
=
cos
2
x
+
sin
x
.
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[QMisc II Q.21 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Set · 3 questions
Find the equations of tangent and normal to the curve at the given point on it.
Q419
#419
Mathematics → Application of Derivatives → Tangents and Normals
·
Easy
Add
y
=
2
x
3
−
x
2
+
2
y = 2x^3 - x^2 + 2
y
=
2
x
3
−
x
2
+
2
at
(
1
2
,
2
)
\left(\frac{1}{2}, 2\right)
(
2
1
,
2
)
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[Q2.1.2 SolvedEx.1 (i) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q420
#420
Mathematics → Application of Derivatives → Tangents and Normals
·
Moderate
Add
x
3
+
2
x
2
y
−
9
x
y
=
−
2
x^3 + 2x^2 y - 9xy = -2
x
3
+
2
x
2
y
−
9
x
y
=
−
2
at
(
2
,
1
)
(2, 1)
(
2
,
1
)
Show model answer
[Q2.1.2 SolvedEx.1 (ii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q421
#421
Mathematics → Application of Derivatives → Tangents and Normals
·
Hard
Add
x
=
2
sin
3
θ
x = 2 \sin^3 \theta
x
=
2
sin
3
θ
,
y
=
3
cos
3
θ
y = 3 \cos^3 \theta
y
=
3
cos
3
θ
at
θ
=
π
4
\theta = \frac{\pi}{4}
θ
=
4
π
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[Q2.1.2 SolvedEx.1 (iii) · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q422
#422
Mathematics → Application of Derivatives → Tangents and Normals
·
Moderate
Add
Find points on the curve given by
y
=
x
3
−
6
x
2
+
x
+
3
y = x^3 - 6x^2 + x + 3
y
=
x
3
−
6
x
2
+
x
+
3
where the tangents are parallel to the line
y
=
x
+
5
y = x + 5
y
=
x
+
5
.
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[Q2.1.2 SolvedEx.2 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q423
#423
Mathematics → Application of Derivatives → Derivative as a Rate Measure
·
Easy
Add
A stone is dropped in to a quiet lake and waves in the form of circles are generated, radius of the circular wave increases at the rate of 5 cm/ sec. At the instant when the radius of the circular wave is 8 cm, how fast the area enclosed is increasing ?
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[Q2.1.3 SolvedEx.1 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q424
#424
Mathematics → Application of Derivatives → Derivative as a Rate Measure
·
Moderate
Add
The volume of the spherical ball is increasing at the rate of
4
π
4\pi
4
π
cc/sec. Find the rate at which the radius and the surface area are changing when the volume is
288
π
288\pi
288
π
cc.
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[Q2.1.3 SolvedEx.2 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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Q425
#425
Mathematics → Application of Derivatives → Derivative as a Rate Measure
·
Easy
Add
Water is being poured at the rate of 36 m
3
^{3}
3
/sec in to a cylindrical vessel of base radius 3 meters. Find the rate at which water level is rising.
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[Q2.1.3 SolvedEx.3 · Maharashtra State Board (Class 12) — Application of Derivatives (Balbharati textbook)]
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