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

## 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.

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

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?

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.

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

## Chapter 4: Applications of Derivatives

Exercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Miscellaneous Exercise 4

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

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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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