#### Online Mock Tests

#### 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

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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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