#### Chapters

Chapter 1.2: Matrices

Chapter 1.3: Differentiation

Chapter 1.4: Applications of Derivatives

Chapter 1.5: Integration

Chapter 1.6: Definite Integration

Chapter 1.7: Application of Definite Integration

Chapter 1.8: Differential Equation and Applications

Chapter 2.1: Commission, Brokerage and Discount

Chapter 2.2: Insurance and Annuity

Chapter 2.3: Linear Regression

Chapter 2.4: Time Series

Chapter 2.5: Index Numbers

Chapter 2.6: Linear Programming

Chapter 2.7: Assignment Problem and Sequencing

Chapter 2.8: Probability Distributions

## Chapter 1: Applications of Derivatives

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 Applications of Derivatives Q.1

#### MCQ [1 Mark]

**Choose the correct alternative:**

The slope of the tangent to the curve y = x^{3} – x^{2} – 1 at the point whose abscissa is – 2, is

– 8

8

16

– 16

**Choose the correct alternative:**

Slope of the normal to the curve 2x^{2} + 3y^{2} = 5 at the point (1, 1) on it is

`-2/3`

`2/3`

`3/2`

`-3/2`

**Choose the correct alternative:**

The function f(x) = x^{3} – 3x^{2} + 3x – 100, x ∈ R is

increasing for all x ∈ R, x ≠ 1

decreasing

neither increasing nor decreasing

decreasing for all x ∈ R, x ≠ 1

**Choose the correct alternative:**

If the marginal revenue is 28 and elasticity of demand is 3, then the price is

24

32

36

42

**Choose the correct alternative:**

The price P for the demand D is given as P = 183 + 120D − 3D^{2}, then the value of D for which price is increasing, is

D < 60

D > 60

D < 20

D > 20

**Choose the correct alternative:**

If the elasticity of the demand η = 1, then demand is

constant

inelastic

unitary elastic

elastic

**Choose the correct alternative:**

If 0 < η < 1, then the demand is

constant

inelastic

unitary elastic

elastic

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 Applications of Derivatives Q.2

#### Fill in the blanks [1 Mark]

The slope of tangent at any point (a, b) is also called as ______

If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______

The slope of the tangent to the curve x = `1/"t"`, y = `"t" - 1/"t"`, at t = 2 is ______

If the average revenue is 45 and elasticity of demand is 5, then marginal revenue is ______

The total cost function for production of articles is given as C = 100 + 600x – 3x^{2}, then the values of x for which the total cost is decreasing is ______

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 Applications of Derivatives Q.3

#### [1 Mark]

**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

**State whether the following statement is True or False:**

The function f(x) = `3/x` + 10, x ≠ 0 is decreasing

True

False

**State whether the following statement is True or False:**

The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing

True

False

**State whether the following statement is True or False:**

The equation of tangent to the curve y = x^{2} + 4x + 1 at (– 1, – 2) is 2x – y = 0

True

False

**State whether the following statement is True or False: **

If the function f(x) = x^{2} + 2x – 5 is an increasing function, then x < – 1

True

False

**State whether the following statement is True or False: **

If the marginal revenue is 50 and the price is ₹ 75, then elasticity of demand is 4

True

False

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 Applications of Derivatives Q.4

#### Solve the following: [3 Marks]

Find the equations of tangent and normal to the curve y = 3x^{2} – x + 1 at the point (1, 3) on it

Find the values of x such that f(x) = 2x^{3} – 15x^{2} + 36x + 1 is increasing function

Find the values of x such that f(x) = 2x^{3} – 15x^{2} – 144x – 7 is decreasing function

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

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

If the demand function is D = 50 – 3p – p^{2}. Find the elasticity of demand at p = 5 comment on the result

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

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

The total cost of manufacturing x articles is C = 47x + 300x^{2} - x^{4}. Find x, for which average cost is increasing

The total cost of manufacturing x articles C = 47x + 300x^{2} – x^{4} . Find x, for which average cost is decreasing

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 Applications of Derivatives Q.5

#### Solve the following: [4 Marks]

Determine the maximum and minimum value of the following function.

f(x) = 2x^{3} – 21x^{2} + 36x – 20

A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum

Find MPC, MPS, APC and APS, if the expenditure E_{c} of a person with income I is given as E_{c} = (0.0003) I^{2} + (0.075) I When I = 1000

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

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 profit is increasing

If x + y = 3 show that the maximum value of x^{2}y is 4.

Find the equation of tangent to the curve x^{2} + y^{2} = 5, where the tangent is parallel to the line 2x – y + 1 = 0

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

Find the equation of tangent to the curve y = x^{2} + 4x at the point whose ordinate is – 3

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 Applications of Derivatives Q.6

#### Activity: [4 Marks]

A metal wire of 36 cm long is bent to form a rectangle. By completing the following activity, find it’s dimensions when it’s area is maximum.

**Solution:** Let the dimensions of the rectangle be x cm and y cm.

∴ 2x + 2y = 36

Let f(x) be the area of rectangle in terms of x, then

f(x) = `square`

∴ f'(x) = `square`

∴ f''(x) = `square`

For extreme value, f'(x) = 0, we get

x = `square`

∴ f''`(square)` = – 2 < 0

∴ Area is maximum when x = `square`, y = `square`

∴ Dimensions of rectangle are `square`

By completing the following activity, examine the function f(x) = x^{3} – 9x^{2} + 24x for maxima and minima

**Solution:** f(x) = x^{3} – 9x^{2} + 24x

∴ f'(x) = `square`

∴ f''(x) = `square`

For extreme values, f'(x) = 0, we get

x = `square` or `square`

∴ f''`(square)` = – 6 < 0

∴ f(x) is maximum at x = 2.

∴ Maximum value = `square`

∴ f''`(square)` = 6 > 0

∴ f(x) is maximum at x = 4.

∴ Minimum value = `square`

By completing the following activity, find the values of x such that f(x) = 2x^{3} – 15x^{2} – 84x – 7 is decreasing function.

**Solution: **f(x) = 2x^{3} – 15x^{2} – 84x – 7

∴ f'(x) = `square`

∴ f'(x) = 6`(square) (square)`

Since f(x) is decreasing function.

∴ f'(x) < 0

**Case 1:** `(square)` > 0 and (x + 2) < 0

∴ x ∈ `square`

**Case 2:** `(square)` < 0 and (x + 2) > 0

∴ x ∈ `square`

∴ f(x) is decreasing function if and only if x ∈ `square`

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

**Solution:** Total cost C = 40 + 2x and Price p = 120 – x

Revenue R = `square`

Differentiating w.r.t. x,

`("dR")/("d"x) = square`

Since Revenue is increasing,

`("dr")/("d"x)` > 0

∴ Revenue is increasing for `square`

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

**Solution:** Total cost C = 40 + 2x and Price p = 120 − x

Profit π = R – C

∴ π = `square`

Differentiating w.r.t. x,

`("d"pi)/("d"x)` = `square`

Since Profit is increasing,

`("d"pi)/("d"x)` > 0

∴ Profit is increasing for `square`

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

**Solution:** Total cost C = 40 + 2x and Price p = 120 – x

p = 120 – x

∴ x = 120 – p

Differentiating w.r.t. p,

`("d"x)/("dp")` = `square`

∴ Elasticity of demand is given by η = `- "P"/x*("d"x)/("dp")`

∴ η = `square`

When p = 80, then elasticity of demand η = `square`

## Chapter 1: Applications of Derivatives

## SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 chapter 1 - Applications of Derivatives

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 chapter 1 (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 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 chapter 1 Applications of Derivatives are Introduction of Derivatives, Increasing and Decreasing Functions, Maxima and Minima, Application of Derivatives to Economics.

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