Maharashtra State BoardHSC Commerce 12th Board Exam
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Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 4 - Applications of Derivatives [Latest edition]

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Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board - Shaalaa.com

Chapter 4: Applications of Derivatives

Exercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Miscellaneous Exercise 4
Exercise 4.1 [Page 105]

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 4 Applications of Derivatives Exercise 4.1 [Page 105]

Exercise 4.1 | Q 1.1 | Page 105

Find the equation of tangent and normal to the curve at the given points on it.

y = 3x2 - x + 1 at (1, 3)

Exercise 4.1 | Q 1.2 | Page 105

Find the equation of tangent and normal to the curve at the given points on it.

2x2 + 3y2 = 5 at (1, 1)

Exercise 4.1 | Q 1.3 | Page 105

Find the equation of tangent and normal to the curve at the given points on it.

x2 + y2 + xy = 3 at (1, 1)

Exercise 4.1 | Q 2 | Page 105

Find the equations of tangent and normal to the curve y = x2 + 5 where the tangent is parallel to the line 4x − y + 1 = 0.

Exercise 4.1 | Q 3 | Page 105

Find the equations of tangent and normal to the curve y = 3x2 - 3x - 5 where the tangent is parallel to the line 3x − y + 1 = 0.

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Exercise 4.2 [Page 106]

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 4 Applications of Derivatives Exercise 4.2 [Page 106]

Exercise 4.2 | Q 1.1 | Page 106

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

Exercise 4.2 | Q 1.2 | Page 106

Test whether the function is increasing or decreasing. 

f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0, 

Exercise 4.2 | Q 1.3 | Page 106

Test whether the following function is increasing or decreasing.

f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0

Exercise 4.2 | Q 2.1 | Page 106

Find the value of x, such that f(x) is increasing function.

f(x) = 2x3 - 15x2 + 36x + 1 

Exercise 4.2 | Q 2.2 | Page 106

Find the value of x, such that f(x) is increasing function.

f(x) = x2 + 2x - 5 

Exercise 4.2 | Q 2.3 | Page 106

Find the value of x, such that f(x) is increasing function.

f(x) = 2x3 - 15x2 - 144x - 7 

Exercise 4.2 | Q 3.1 | Page 106

Find the value of x, such that f(x) is decreasing function.

f(x) = 2x3 - 15x2 - 144x - 7 

Exercise 4.2 | Q 3.2 | Page 106

Find the value of x such that f(x) is decreasing function.

f(x) = x4 − 2x3 + 1

Exercise 4.2 | Q 3.3 | Page 106

Find the value of x, such that f(x) is decreasing function.

f(x) = 2x3 - 15x2 - 84x - 7 

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Exercise 4.3 [Page 109]

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 4 Applications of Derivatives Exercise 4.3 [Page 109]

Exercise 4.3 | Q 1.1 | Page 109

Determine the maximum and minimum value of the following function.

f(x) = 2x3 - 21x2 + 36x - 20

Exercise 4.3 | Q 1.2 | Page 109

Determine the maximum and minimum value of the following function.

f(x) = x log x

Exercise 4.3 | Q 1.3 | Page 109

Determine the maximum and minimum value of the following function.

f(x) = `"x"^2 + 16/"x"`

Exercise 4.3 | Q 2 | Page 109

Divide the number 20 into two parts such that their product is maximum.

Exercise 4.3 | Q 3 | Page 109

A metal wire of 36cm long is bent to form a rectangle. Find it's dimensions when it's area is maximum.

Exercise 4.3 | Q 4 | Page 109

The total cost of producing x units is ₹ (x2 + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?

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Exercise 4.4 [Pages 112 - 113]

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 4 Applications of Derivatives Exercise 4.4 [Pages 112 - 113]

Exercise 4.4 | Q 1 | Page 112

The demand function of a commodity at price P is given as, D = `40 - "5P"/8`. Check whether it is increasing or decreasing function.

Exercise 4.4 | Q 2 | Page 112

Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing

Exercise 4.4 | Q 3 | Page 112

The total cost function for production of x articles is given as C = 100 + 600x – 3x2 . Find the values of x for which total cost is decreasing.

Exercise 4.4 | Q 4.1 | Page 112

The manufacturing company produces x items at the total cost of ₹ 180 + 4x. The demand function for this product is P = (240 – x). Find x for which revenue is increasing.

Exercise 4.4 | Q 4.2 | Page 112

A manufacturing company produces x items at the total cost of Rs (180+4x). The demand function of this product is P=(240 - x). Find x for which profit is increasing. 

Exercise 4.4 | Q 5.1 | Page 112

For manufacturing x units, labour cost is 150 - 54x, processing cost is x2 and revenue R = 10800x - 4x. Find the value of x for which Total cost is decreasing.

Exercise 4.4 | Q 5.2 | Page 112

For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the values of x for which Revenue is increasing.

Exercise 4.4 | Q 6.1 | Page 112

The total cost of manufacturing x articles is C = (47x + 300x2 - x4).  Find x, for which average cost is increasing.

Exercise 4.4 | Q 6.2 | Page 112

The total cost of manufacturing x articles C = 47x + 300x2 – x4 . Find x, for which average cost is decreasing.

Exercise 4.4 | Q 7.1 | Page 112

Find the marginal revenue if the average revenue is 45 and elasticity of demand is 5.

Exercise 4.4 | Q 7.2 | Page 112

Find the price, if the marginal revenue is 28 and elasticity of demand is 3.

Exercise 4.4 | Q 7.3 | Page 112

Find the elasticity of demand, if the marginal revenue is 50 and price is Rs 75.

Exercise 4.4 | Q 8 | Page 112

If the demand function is D = `(("p" + 6)/("p" - 3))`, find the elasticity of demand at p = 4.

Exercise 4.4 | Q 9 | Page 113

Find the price for the demand function D = `((2"p" + 3)/(3"p" - 1))`, when elasticity of demand is `11/14`.

Exercise 4.4 | Q 10.1 | Page 113

If the demand function is D = 50 – 3p – p2. Find the elasticity of demand at p = 5 comment on the result.

Exercise 4.4 | Q 10.2 | Page 113

If the demand function is D = 50 – 3p – p2. Find the elasticity of demand at p = 2 comment on the result.

Exercise 4.4 | Q 11.1 | Page 113

For the demand function D = 100 – `"p"^2/2`. Find the elasticity of demand at p = 10 and comment on the results.

Exercise 4.4 | Q 11.2 | Page 113

For the demand function D = 100 – `"p"^2/2`. Find the elasticity of demand at p = 6 and comment on the results.

Exercise 4.4 | Q 12.1 | Page 113

A manufacturing company produces x items at a total cost of ₹ 40 + 2x. Their price is given as p = 120 – x. Find the value of x for which revenue is increasing.

Exercise 4.4 | Q 12.2 | Page 113

A manufacturing company produces x items at a total cost of ₹ 40 + 2x. Their price is given as p = 120 – x. Find the value of x for which profit is increasing.

Exercise 4.4 | Q 12.3 | Page 113

A manufacturing company produces x items at a total cost of ₹ 40 + 2x. Their price is given as p = 120 – x. Find the value of x for which also find an elasticity of demand for price 80.

Exercise 4.4 | Q 13 | Page 113

Find MPC, MPS, APC and APS, if the expenditure Ec of a person with income I is given as Ec = (0.0003) I2 + (0.075) I When I = 1000.

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Miscellaneous Exercise 4 [Pages 113 - 114]

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 4 Applications of Derivatives Miscellaneous Exercise 4 [Pages 113 - 114]

Miscellaneous Exercise 4 | Q 1.1 | Page 113

Choose the correct alternative.

The equation of tangent to the curve y = x2 + 4x + 1 at (-1, -2) is 

  • 2x - y = 0

  • 2x + y - 5 = 0

  • 2x - y - 1 = 0

  • x + y - 1 = 0

Miscellaneous Exercise 4 | Q 1.2 | Page 113

Choose the correct alternative.

The equation of tangent to the curve x2 + y2 = 5 where the tangent is parallel to the line 2x – y + 1 = 0 are

  • 2x – y + 5 = 0; 2x – y – 5 = 0

  • 2x + y + 5 = 0; 2x + y – 5 = 0

  • x – 2y + 5 = 0; x – 2y – 5 = 0

  • x + 2y + 5 = 0; x + 2y – 5 = 0

Miscellaneous Exercise 4 | Q 1.3 | Page 113

Choose the correct alternative.

If elasticity of demand η = 1, then demand is

  • constant

  • inelastic

  • unitary elastic

  • elastic

Miscellaneous Exercise 4 | Q 1.4 | Page 113

Choose the correct alternative.

If 0 < η < 1, then demand is

  • constant

  • inelastic

  • unitary elastic

  • elastic

Miscellaneous Exercise 4 | Q 1.5 | Page 113

Choose the correct alternative.

The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is

  • increasing for all x ∈ R, x ≠ 1

  • decreasing

  • neither, increasing nor decreasing

  • decreasing for all x ∈ R, x ≠ 1

Miscellaneous Exercise 4 | Q 1.6 | Page 113

Choose the correct alternative.

If f(x) = 3x3 - 9x2 - 27x + 15 then

  • f has maximum value 66

  • f has minimum value 30

  • f has maxima at x = –1

  • f has minima at x = –1

Miscellaneous Exercise 4 | Q 2.1 | Page 114

Fill in the blank:

The slope of tangent at any point (a, b) is called as _______.

Miscellaneous Exercise 4 | Q 2.2 | Page 114

Fill in the blank:

If f(x) = x - 3x2 + 3x - 100, x ∈ R then f''(x) is ______

Miscellaneous Exercise 4 | Q 2.3 | Page 114

Fill in the blank:

If f(x) = `7/"x" - 3`, x ∈ R x ≠ 0 then f ''(x) is ______

Miscellaneous Exercise 4 | Q 2.4 | Page 114

Fill in the blank:

A road of 108 m length is bent to form a rectangle. If the area of the rectangle is maximum, then its dimensions are _______.

Miscellaneous Exercise 4 | Q 2.5 | Page 114

Fill in the blank:

If f(x) = x log x, then its minimum value is______

Miscellaneous Exercise 4 | Q 3.1 | Page 114

State whether the following statement is True or False:

The equation of tangent to the curve y = 4xex at `(-1, (- 4)/"e")` is ye + 4 = 0

  • True

  • False

Miscellaneous Exercise 4 | Q 3.2 | Page 114

State whether the following statement is True or False:

x + 10y + 21 = 0 is the equation of normal to the curve y = 3x2 + 4x - 5 at (1, 2).

  • True

  • False

Miscellaneous Exercise 4 | Q 3.3 | Page 114

State whether the following statement is True or False:

An absolute maximum must occur at a critical point or at an end point.

  • True

  • False

Miscellaneous Exercise 4 | Q 3.4 | Page 114

State whether the following statement is True or False:

The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.

  • True

  • False

Miscellaneous Exercise 4 | Q 4.1 | Page 114

Find the equation of tangent and normal to the following curve.

xy = c2 at `("ct", "c"/"t")` where t is parameter.

Miscellaneous Exercise 4 | Q 4.1 | Page 114

Find the equation of tangent and normal to the following curve.

y = x2 + 4x at the point whose ordinate is -3.

Miscellaneous Exercise 4 | Q 4.1 | Page 114

Find the equation of tangent and normal to the following curve.

x = `1/"t",  "y" = "t" - 1/"t"`,  at t = 2

Miscellaneous Exercise 4 | Q 4.1 | Page 114

Find the equation of tangent and normal to the following curve.

y = x3 - x2 - 1 at the point whose abscissa is -2.

Miscellaneous Exercise 4 | Q 4.2 | Page 114

Find the equation of tangent to the curve y = `sqrt("x - 3")` which is perpendicular to the line 6x + 3y – 4 = 0.

Miscellaneous Exercise 4 | Q 4.3 | Page 114

Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.

Miscellaneous Exercise 4 | Q 4.4 | Page 114

Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.

Miscellaneous Exercise 4 | Q 4.5 | Page 114

If x + y = 3 show that the maximum value of x2y is 4.

Miscellaneous Exercise 4 | Q 4.6 | Page 114

6. Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x

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Chapter 4: Applications of Derivatives

Exercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Miscellaneous Exercise 4
Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board - Shaalaa.com

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 4 - Applications of Derivatives

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 4 (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 1 (Commerce) 12th Standard HSC Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 4 Applications of Derivatives are Introduction of Derivatives, Increasing and Decreasing Functions, Maxima and Minima, Application of Derivatives to Economics.

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