# Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 2 - Applications of Derivatives [Latest edition]

## Chapter 2: Applications of Derivatives

Exercise 2.1Exercise 2.2Exercise 2.3Exercise 2.4Miscellaneous Exercise 2
Exercise 2.1 [Page 72]

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 2 Applications of Derivatives Exercise 2.1 [Page 72]

Exercise 2.1 | Q 1.1 | Page 72

Find the equations of tangents and normals to the following curves at the indicated points on them : y = x2 + 2ex 2 at (0, 4)

Exercise 2.1 | Q 1.2 | Page 72

Find the equations of tangents and normals to the following curves at the indicated points on them : x3 + y3 – 9xy = 0 at (2, 4)

Exercise 2.1 | Q 1.3 | Page 72

Find the equations of tangents and normals to the following curves at the indicated points on them : x^2 - sqrt(3)xy + 2y^2 = 5 "at"  (sqrt(3),2)

Exercise 2.1 | Q 1.4 | Page 72

Find the equations of tangents and normals to the following curves at the indicated points on them : 2xy + π sin y = 2pi  "at" (1, pi/2)

Exercise 2.1 | Q 1.5 | Page 72

Find the equations of tangents and normals to the following curves at the indicated points on them : x sin 2y = y cos 2x at (pi/4, pi/2)

Exercise 2.1 | Q 1.6 | Page 72

Find the equations of tangents and normals to the following curves at the indicated points on them : x = sin θ and y = cos 2θ at θ = pi/(6)

Exercise 2.1 | Q 1.7 | Page 72

Find the equations of tangents and normals to the following curves at the indicated points on them : x = sqrt(t), y = t  - (1)/sqrt(t) at = 4.

Exercise 2.1 | Q 2 | Page 72

Find the point of the curve y = sqrt(x - 3) where the tangent is perpendicular to the line 6x + 3y – 5 = 0.

Exercise 2.1 | Q 3 | Page 72

Find the points on the curve y = x3 – 2x2 – x where the tangents are parllel to 3x – y + 1 = 0.

Exercise 2.1 | Q 4 | Page 72

Find the equation of the tangents to the curve x2 + y2 – 2x – 4y + 1 =0 which a parallel to the X-axis.

Exercise 2.1 | Q 5 | Page 72

Find the equations of the normals to the curve 3x2 – y2 = 8, which are parallel to the line x + 3y = 4.

Exercise 2.1 | Q 6 | Page 72

If the line y = 4x – 5 touches the curves y2 = ax3 + b at the point (2, 3), find a and b.

Exercise 2.1 | Q 7 | Page 72

A particle moves along the curve 6y = x3 + 2. Find the points on the curve at which y-coordinate is changing 8 times as fast as the x-coordinate.

Exercise 2.1 | Q 8 | Page 72

A spherical soap bubble is expanding so that its radius is increasing at the rate of 0.02 cm/sec. At what rate is the surface area increasing, when its radius is 5 cm?

Exercise 2.1 | Q 9 | Page 72

The surface area of a spherical balloon is increasing at the rate of (2"cm"^2)/sec. At what rate is the volume of the balloon is increasing, when the radius of the balloon is 6 cm?

Exercise 2.1 | Q 10 | Page 72

If each side of an equilateral triangle increases at the rate of (sqrt(2)"cm")/sec, find the rate of increase of its area when its side of length 3 cm.

Exercise 2.1 | Q 11 | Page 72

The volume of a sphere increases at the rate of 20 cm3/sec. Find the rate of change of its surface area, when its radius is 5 cm

Exercise 2.1 | Q 12 | Page 72

The edge of a cube is decreasing at the rate of( 0.6"cm")/sec. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.

Exercise 2.1 | Q 13 | Page 72

A man of height 2 metres walks at a uniform speed of 6 km/hr away from a lamp post of 6 metres high. Find the rate at which the length of the shadow is increasing

Exercise 2.1 | Q 14 | Page 72

A man of height 1.5 metres walks towards a lamp post of height 4.5 metres, at the rate of (3/4)"metre"/sec.
Find the rate at which
(ii) the tip of shadow is moving.

Exercise 2.1 | Q 15 | Page 72

A ladder 10 metres long is leaning against a vertical wall. If the bottom of the ladder is pulled horizontally away from the wall at the rate of 1.2 metres per second, find how fast the top of the ladder is sliding down the wall, when the bottom is 6 metres away from the wall.

Exercise 2.1 | Q 16 | Page 72

If water is poured into an inverted hollow cone whose semi-vertical angle is 30°, so that its depth (measured along the axis) increases at the rate of( 1"cm")/sec. Find the rate at which the volume of water increasing when the depth is 2 cm.

Exercise 2.2 [Page 75]

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 2 Applications of Derivatives Exercise 2.2 [Page 75]

Exercise 2.2 | Q 1.1 | Page 75

Find the approximate values of : sqrt(8.95)

Exercise 2.2 | Q 1.2 | Page 75

Find the approximate values of : root(3)(28)

Exercise 2.2 | Q 1.3 | Page 75

Find the approximate values of : root(5)(31.98)

Exercise 2.2 | Q 1.4 | Page 75

Find the approximate values of : (3.97)4

Exercise 2.2 | Q 1.5 | Page 75

Find the approximate values of : (4.01)3

Exercise 2.2 | Q 2.1 | Page 75

Find the approximate values of : sin 61° , given that 1° = 0.0174°, sqrt(3) = 1.732

Exercise 2.2 | Q 2.2 | Page 75

Find the approximate values of : sin (29° 30'), given that 1°= 0.0175°, sqrt(3) = 1.732

Exercise 2.2 | Q 2.3 | Page 75

Find the approximate values of : cos(60° 30°), given that 1° = 0.0175°, sqrt(3) = 1.732

Exercise 2.2 | Q 2.4 | Page 75

Find the approximate values of : tan (45° 40°), given that 1° = 0.0175°.

Exercise 2.2 | Q 3.1 | Page 75

Find the approximate values of : tan–1(0.999)

Exercise 2.2 | Q 3.2 | Page 75

Find the approximate values of : cot–1 (0.999)

Exercise 2.2 | Q 3.3 | Page 75

Find the approximate values of : tan–1 (1.001)

Exercise 2.2 | Q 4.1 | Page 75

Find the approximate values of : e0.995, given that e = 2.7183.

Exercise 2.2 | Q 4.2 | Page 75

Find the approximate values of : e2.1, given that e2 = 7.389

Exercise 2.2 | Q 4.3 | Page 75

Find the approximate values of : 32.01, given that log 3 = 1.0986

Exercise 2.2 | Q 5.1 | Page 75

Find the approximate values of : loge(101), given that loge10 = 2.3026.

Exercise 2.2 | Q 5.2 | Page 75

Find the approximate values of : loge(9.01), given that log 3 = 1.0986.

Exercise 2.2 | Q 5.3 | Page 75

Find the approximate values of : log10(1016), given that log10e = 0.4343.

Exercise 2.2 | Q 6.1 | Page 75

Find the approximate values of : f(x) = x3 – 3x + 5 at x = 1.99.

Exercise 2.2 | Q 6.2 | Page 75

Find the approximate values of : f(x) = x3 + 5x2 – 7x + 10 at x = 1.12.

Exercise 2.3 [Page 80]

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 2 Applications of Derivatives Exercise 2.3 [Page 80]

Exercise 2.3 | Q 1.1 | Page 80

Check the validity of the Rolle’s theorem for the following functions : f(x) = x2 – 4x + 3, x ∈ [1, 3]

Exercise 2.3 | Q 1.2 | Page 80

Check the validity of the Rolle’s theorem for the following functions : f(x) = e–x sin x, x ∈ [0, π].

Exercise 2.3 | Q 1.3 | Page 80

Check the validity of the Rolle’s theorem for the following functions : f(x) = 2x2 – 5x + 3, x ∈ [1, 3].

Exercise 2.3 | Q 1.4 | Page 80

Check the validity of the Rolle’s theorem for the following functions : f(x) = sin x – cos x + 3, x ∈ [0, 2π].

Exercise 2.3 | Q 1.5 | Page 80

Check the validity of the Rolle’s theorem for the following functions : f(x = x2, if 0 ≤ x ≤ 2
= 6 – x, if 2 < x ≤ 6.

Exercise 2.3 | Q 1.6 | Page 80

Check the validity of the Rolle’s theorem for the following functions : f(x) = x^(2/3), x ∈ [ - 1, 1].

Exercise 2.3 | Q 2 | Page 80

Given an interval [a, b] that satisfies hypothesis of Rolle's theorem for the function f(x) = x4 + x2 – 2. It is known that a = – 1. Find the value of b.

Exercise 2.3 | Q 3.1 | Page 80

Verify Rolle’s theorem for the following functions : f(x) = sin x + cos x + 7, x ∈ [0, 2π]

Exercise 2.3 | Q 3.2 | Page 80

Verify Rolle’s theorem for the following functions  : f(x) = sin(x/2), x ∈ [0, 2pi]

Exercise 2.3 | Q 3.3 | Page 80

Verify Rolle’s theorem for the following functions : f(x) = x2 – 5x + 9, x ∈ 1, 4].

Exercise 2.3 | Q 4 | Page 80

If Rolle's theorem holds for the function f(x) = x3 + px2 + qx + 5, x ∈ [1, 3] with c = 2 + (1)/sqrt(3), find the values of p and q.

Exercise 2.3 | Q 5 | Page 80

If Rolle’s theorem holds for the function f(x) = (x –2) log x, x ∈ [1, 2], show that the equation x log x = 2 – x is satisfied by at least one value of x in (1, 2).

Exercise 2.3 | Q 6 | Page 80

The function f(x) = x(x + 3)e^(-(x)/2) satisfies all the conditions of Rolle's theorem on [– 3, 0]. Find the value of c such that f'(c) = 0.

Exercise 2.3 | Q 7.1 | Page 80

Verify Lagrange’s mean value theorem for the following functions : f(x) = log x on [1, e].

Exercise 2.3 | Q 7.2 | Page 80

Verify Lagrange’s mean value theorem for the following functions : f(x) = (x – 1)(x – 2)(x – 3) on [0, 4].

Exercise 2.3 | Q 7.3 | Page 80

Verify Lagrange’s mean value theorem for the following functions : x^2 - 3x - 1, x ∈ [(-11)/7 , 13/7].

Exercise 2.3 | Q 7.4 | Page 80

Verify Lagrange’s mean value theorem for the following functions : f(x) = 2x – x2, x ∈ [0, 1].

Exercise 2.3 | Q 7.5 | Page 80

Verify Lagrange’s mean value theorem for the following functions : f(x) = (x - 1)/(x - 3) on [4, 5].

Exercise 2.4 [Pages 89 - 90]

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 2 Applications of Derivatives Exercise 2.4 [Pages 89 - 90]

Exercise 2.4 | Q 1.1 | Page 89

Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.

Exercise 2.4 | Q 1.2 | Page 89

Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.

Exercise 2.4 | Q 1.3 | Page 89

Test whether the following functions are increasing or decreasing : f(x) = (1)/x, x ∈ R , x ≠ 0.

Exercise 2.4 | Q 2.1 | Page 89

Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6

Exercise 2.4 | Q 2.2 | Page 89

Find the values of x for which the following functions are strictly increasing : f(x) = 3 + 3x – 3x2 + x3

Exercise 2.4 | Q 2.3 | Page 89

Find the values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7

Exercise 2.4 | Q 3.1 | Page 89

Find the values of x for which the following functions are strictly decreasing : f(x) = 2x3 – 3x2 – 12x + 6

Exercise 2.4 | Q 3.2 | Page 89

Find the values of x for which the following functions are strictly decreasing : f(x) = x + (25)/x

Exercise 2.4 | Q 3.3 | Page 89

Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12

Exercise 2.4 | Q 4 | Page 90

Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing

Exercise 2.4 | Q 5 | Page 90

Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is

(a) Strictly increasing
(b) strictly decreasing

Exercise 2.4 | Q 6 | Page 90

Find the values of x for which f(x) = x/(x^2 + 1) is (a) strictly increasing (b) decreasing.

Exercise 2.4 | Q 7 | Page 90

show that f(x) = 3x + (1)/(3x) is increasing in (1/3, 1) and decreasing in (1/9, 1/3).

Exercise 2.4 | Q 8 | Page 90

Show that f(x) = x – cos x is increasing for all x.

Exercise 2.4 | Q 9.1 | Page 90

Find the maximum and minimum of the following functions : y = 5x3 + 2x2 – 3x

Exercise 2.4 | Q 9.2 | Page 90

Find the maximum and minimum of the following functions : f(x) = 2x3 – 21x2 + 36x – 20

Exercise 2.4 | Q 9.3 | Page 90

Find the maximum and minimum of the following functions : f(x) = x3 – 9x2 + 24x

Exercise 2.4 | Q 9.4 | Page 90

Find the maximum and minimum of the following functions : f(x) = x^2 + (16)/x^2

Exercise 2.4 | Q 9.5 | Page 90

Find the maximum and minimum of the following functions : f(x) = x log x

Exercise 2.4 | Q 9.6 | Page 90

Find the maximum and minimum of the following functions : f(x) = logx/x

Exercise 2.4 | Q 10 | Page 90

Divide the number 30 into two parts such that their product is maximum.

Exercise 2.4 | Q 11 | Page 90

Divide the number 20 into two parts such that sum of their squares is minimum.

Exercise 2.4 | Q 12 | Page 90

A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.

Exercise 2.4 | Q 13 | Page 90

A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t2 . Find the maximum height it can reach.

Exercise 2.4 | Q 14 | Page 90

Find the largest size of a rectangle that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter.

Exercise 2.4 | Q 15 | Page 90

An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of pia^3cu cm of water. Find the dimensions so that the quantity of the metal sheet required is minimum.

Exercise 2.4 | Q 16 | Page 90

The perimeter of a triangle is 10 cm. If one of the side is 4 cm. What are the other two sides of the triangle for its maximum area?

Exercise 2.4 | Q 17 | Page 90

A box with a square base is to have an open top. The surface area of the box is 192 sq cm. What should be its dimensions in order that the volume is largest?

Exercise 2.4 | Q 18 | Page 90

The profit function P(x) of a firm, selling x items per day is given by P(x) = (150 – x)x – 1625 . Find the number of items the firm should manufacture to get maximum profit. Find the maximum profit.

Exercise 2.4 | Q 20 | Page 90

Show that among rectangles of given area, the square has least perimeter.

Exercise 2.4 | Q 21 | Page 90

Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.

Exercise 2.4 | Q 22 | Page 90

Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.

Exercise 2.4 | Q 23 | Page 90

Show that y = log (1 + x) – (2x)/(2 + x), x > - 1 is an increasing function on its domain.

Exercise 2.4 | Q 24 | Page 90

Prove that y = (4sinθ)/(2 + cosθ) - θ is an increasing function if θ ∈[0, pi/2]

Miscellaneous Exercise 2 [Page 92]

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 2 Applications of Derivatives Miscellaneous Exercise 2 [Page 92]

Miscellaneous Exercise 2 | Q 1 | Page 92

Choose the correct option from the given alternatives :

If the function f(x) = ax3 + bx2 + 11x – 6 satisfies conditions of Rolle's theoreem in [1, 3] and  f'(2 + 1/sqrt(3)) = 0, then values of a and b are respectively

• 1, – 6

• – 2, 1

• – 1, – 6

• – 1, 6

Miscellaneous Exercise 2 | Q 2 | Page 92

Choose the correct option from the given alternatives :

If f(x) = (x^2 - 1)/(x^2 + 1), for every real x, then the minimum ue of f is

• 1

• 0

• – 1

• 2

Miscellaneous Exercise 2 | Q 3 | Page 92

Choose the correct option from the given alternatives :

A ladder 5 m in length is resting against vertical wall. The bottom of the ladder is pulled along the ground away from the wall at the rate of (1.5 "m")/sec. The length of the higher point of ladder when the foot of the ladder is 4.0 m away from the wall decreases at the rate of

• 1

• 2

• 2.5

• – 1

Miscellaneous Exercise 2 | Q 4 | Page 92

Choose the correct option from the given alternatives :

Let f(x) and g(x) be differentiable for 0 x < 1 such that f(0) = 0, g(0), f(1) = 6. Let there exist a real number c in (0, 1) such that f'(c) = 2g'(c), then the value of g(1) must be

• 1

• 3

• 2.5

• – 1

Miscellaneous Exercise 2 | Q 5 | Page 92

Choose the correct option from the given alternatives :

Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in

• ( - oo, 1)

• [3, oo)

• ( - oo,] ∪ [3, oo)

• (1, 3)

Miscellaneous Exercise 2 | Q 6 | Page 92

Choose the correct option from the given alternatives :

If x = – 1 and x = 2 are the extreme points of y = oo log x + betax^2 + x, then

• oo = - 6, beta = (1)/(2)

• oo = - 6, beta = -(1)/(2)

• oo = 2, beta = -(1)/(2)

• oo = 2, beta = (1)/(2)

Miscellaneous Exercise 2 | Q 7 | Page 92

Choose the correct option from the given alternatives :

The normal to the curve x2 + 2xy – 3y2 = 0 at (1, 1)

• meets the curve again in second quadrant

• does not meet the curve again

• meets the curve again in third quadrant

• meets the curve again in fourth quadrant

Miscellaneous Exercise 2 | Q 8 | Page 92

Choose the correct option from the given alternatives :

The equation of the tangent to the curve y = 1 - e^(x/2) at the point of intersection with Y-axis is

• x + 2y = 0

• 2x + y = 0

• x – y = 2

• x + y = 2

Miscellaneous Exercise 2 | Q 9 | Page 92

Choose the correct option from the given alternatives :

If the tangent at (1, 1) on y2 = x(2 – x)2 meets the curve again at P, then P is

• (4, 4)

• (– 1, 2)

• (3, 6)

• (9/4, 3/8)

Miscellaneous Exercise 2 | Q 10 | Page 92

Choose the correct option from the given alternatives :

The approximate value of tan (44° 30°), given that 1° = 0.0175, is

• 0.8952

• 0.9528

• 0.9285

• 0.9825

Miscellaneous Exercise 2 [Pages 93 - 94]

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 2 Applications of Derivatives Miscellaneous Exercise 2 [Pages 93 - 94]

Miscellaneous Exercise 2 | Q 1 | Page 93

Solve the following : If the curves ax2 + by2 = 1 and a'x2 + b'y2 = 1, intersect orthogonally, then prove that (1)/a - (1)/b = (1)/a' - (1)/b'.

Miscellaneous Exercise 2 | Q 2 | Page 93

Solve the following : Determine the area of the triangle formed by the tangent to the graph of the function y = 3 – x2 drawn at the point (1, 2) and the coordinate axes.

Miscellaneous Exercise 2 | Q 3 | Page 93

Solve the following : Find the equation of the tangent and normal drawn to the curve y4 – 4x4 – 6xy = 0 at the point M (1, 2).

Miscellaneous Exercise 2 | Q 4 | Page 93

Solve the following : A water tank in the farm 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.

Miscellaneous Exercise 2 | Q 5 | Page 93

Solve the following : Find all points on the ellipse 9x2 + 16y2 = 400, at which the y-coordinate is decreasing and the coordinate is increasing at the same rate.

Miscellaneous Exercise 2 | Q 6 | Page 93

Solve the following : Verify Rolle’s theorem for the function f(x)  (2)/(e^x + e^-x) on [– 1, 1].

Miscellaneous Exercise 2 | Q 7 | Page 93

Solve the following : The position of a particle is given by the function s (t) = 2t2 + 3t – 4. Find the time t = c in the interval 0 ≤ t ≤ 4 when the instantaneous velocity of the particle equal to its average velocity in this interval.

Miscellaneous Exercise 2 | Q 8 | Page 93

Find the approximate value of the function f(x) = sqrt(x^2 + 3x) at x = 1.02.

Miscellaneous Exercise 2 | Q 9 | Page 93

Solve the following : Find the approximate value of cos–1 (0.51), given π = 3.1416, (2)/sqrt(3) = 1.1547.

Miscellaneous Exercise 2 | Q 10 | Page 93

Solve the following : Find the intervals on which the function y = xx, (x > 0) is increasing and decreasing.

Miscellaneous Exercise 2 | Q 11 | Page 93

Solve the following : Find the intervals on which the function f(x) = x/logx is increasing and decreasing.

Miscellaneous Exercise 2 | Q 12 | Page 93

Solve the following : An open box with a square base is to be made out of given quantity of sheet of area a2. Show that the maximum volume of the box is a^3/(6sqrt(3).

Miscellaneous Exercise 2 | Q 13 | Page 93

Solve the following : Show that of all rectangles inscribed in a given circle, the square has the maximum area.

Miscellaneous Exercise 2 | Q 14 | Page 93

Solve the following : Show that a closed right circular cylinder of given surface area has maximum volume if its height equals the diameter of its base.

Miscellaneous Exercise 2 | Q 15 | Page 93

Solve the following :  A window is in the form of a rectangle surmounted by a semicircle. If the perimeter be 30 m, find the dimensions so that the greatest possible amount of light may be admitted.

Miscellaneous Exercise 2 | Q 16 | Page 93

Solve the following : 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.

Miscellaneous Exercise 2 | Q 17 | Page 93

Solve the following:

A wire of length l is cut into 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 the least, if the radius of the circle is half of the side of the square.

Miscellaneous Exercise 2 | Q 18 | Page 94

Solve the following : A rectangular sheet of paper of fixed perimeter with the sides having their lengths in the ratio 8 : 15 converted into 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 volume. Find the lengths of the rectangular sheet of paper.

Miscellaneous Exercise 2 | Q 19 | Page 94

Solve the following : Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is  (4r)/(3).

Miscellaneous Exercise 2 | Q 20 | Page 94

Solve the following : Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is (2"R")/sqrt(3). Also, find the maximum volume.

Miscellaneous Exercise 2 | Q 21 | Page 94

Solve the following :  Find the maximum and minimum values of the function f(x) = cos2x + sinx.

## Chapter 2: Applications of Derivatives

Exercise 2.1Exercise 2.2Exercise 2.3Exercise 2.4Miscellaneous Exercise 2

## Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 2 - Applications of Derivatives

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 2 (Applications of Derivatives) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. Balbharati textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 2 Applications of Derivatives are Applications of Derivatives in Geometry, Derivatives as a Rate Measure, Approximations, Rolle's Theorem, Lagrange's Mean Value Theorem (Lmvt), Increasing and Decreasing Functions, Maxima and Minima.

Using Balbharati 12th Board Exam solutions Applications of Derivatives exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in Balbharati Solutions are important questions that can be asked in the final exam. Maximum students of Maharashtra State Board 12th Board Exam prefer Balbharati Textbook Solutions to score more in exam.

Get the free view of chapter 2 Applications of Derivatives 12th Board Exam extra questions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board and can use Shaalaa.com to keep it handy for your exam preparation