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

#### Chapters ## Chapter 1: Applications of Derivatives

Q.1Q.2Q.3Q.4Q.5Q.6
Q.1

### 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]

Q.1 | Q 1

Choose the correct alternative:

The slope of the tangent to the curve y = x3 – x2 – 1 at the point whose abscissa is – 2, is

• – 8

• 8

• 16

• – 16

Q.1 | Q 2

Choose the correct alternative:

Slope of the normal to the curve 2x2 + 3y2 = 5 at the point (1, 1) on it is

• -2/3

• 2/3

• 3/2

• -3/2

Q.1 | Q 3

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

Q.1 | Q 4

Choose the correct alternative:

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

• 24

• 32

• 36

• 42

Q.1 | Q 5

Choose the correct alternative:

The price P for the demand D is given as P = 183 + 120D − 3D2, then the value of D for which price is increasing, is

• D < 60

• D > 60

• D < 20

• D > 20

Q.1 | Q 6

Choose the correct alternative:

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

• constant

• inelastic

• unitary elastic

• elastic

Q.1 | Q 7

Choose the correct alternative:

If 0 < η < 1, then the demand is

• constant

• inelastic

• unitary elastic

• elastic

Q.2

### 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]

Q.2 | Q 1

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

Q.2 | Q 2

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

Q.2 | Q 3

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

Q.2 | Q 4

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

Q.2 | Q 5

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

Q.3

### 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]

Q.3 | Q 1

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

Q.3 | Q 2

State whether the following statement is True or False:

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

• True

• False

Q.3 | Q 3

State whether the following statement is True or False:

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

• True

• False

Q.3 | Q 4

State whether the following statement is True or False:

The equation of tangent to the curve y = x2 + 4x + 1 at (– 1, – 2) is 2x – y = 0

• True

• False

Q.3 | Q 5

State whether the following statement is True or False:

If the function f(x) = x2 + 2x – 5 is an increasing function, then x < – 1

• True

• False

Q.3 | Q 6

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

Q.4

### 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]

Q.4 | Q 1

Find the equations of tangent and normal to the curve y = 3x2 – x + 1 at the point (1, 3) on it

Q.4 | Q 2

Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function

Q.4 | Q 3

Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function

Q.4 | Q 4

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

Q.4 | Q 5

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

Q.4 | Q 6. (i)

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

Q.4 | Q 6. (ii)

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

Q.4 | Q 7

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

Q.4 | Q 8. (i)

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

Q.4 | Q 8. (ii)

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

Q.5

### 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]

Q.5 | Q 1

Determine the maximum and minimum value of the following function.

f(x) = 2x3 – 21x2 + 36x – 20

Q.5 | Q 2

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

Q.5 | Q 3

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

Q.5 | Q 4. (i)

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

Q.5 | Q 4. (ii)

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

Q.5 | Q 5

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

Q.5 | Q 6

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

Q.5 | Q 7

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

Q.5 | Q 8

Find the equation of tangent to the curve y = x2 + 4x at the point whose ordinate is – 3

Q.6

### 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]

Q.6 | Q 1

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

Q.6 | Q 2

By completing the following activity, examine the function f(x) = x3 – 9x2 + 24x for maxima and minima

Solution: f(x) = x3 – 9x2 + 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

Q.6 | Q 3

By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.

Solution: f(x) = 2x3 – 15x2 – 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

Q.6 | Q 4. (i)

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

Q.6 | Q 4. (ii)

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

Q.6 | Q 4. (iii)

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

Q.1Q.2Q.3Q.4Q.5Q.6 ## 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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