Tamil Nadu Board of Secondary EducationHSC Commerce Class 12th

# Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide chapter 3 - Integral Calculus – 2 [Latest edition]

## Chapter 3: Integral Calculus – 2

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Miscellaneous problems
Exercise 3.1 [Pages 64 - 65]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 3 Integral Calculus – 2Exercise 3.1 [Pages 64 - 65]

Exercise 3.1 | Q 1 | Page 64

Using Integration, find the area of the region bounded the line 2y + x = 8, the x-axis and the lines x = 2, x = 4

Exercise 3.1 | Q 2 | Page 65

Find the area bounded by the lines y – 2x – 4 = 0, y = 0, y = 3 and the y-axis

Exercise 3.1 | Q 3 | Page 65

Calculate the area bounded by the parabola y2 = 4ax and its latus rectum

Exercise 3.1 | Q 4 | Page 65

Find the area bounded by the line y = x and x-axis and the ordinates x = 1, x = 2

Exercise 3.1 | Q 5 | Page 65

Using integration, find the area of the region bounded by the line y – 1 = x, the x-axis and the ordinates x = – 2, x = 3

Exercise 3.1 | Q 6 | Page 65

Find the area of the region lying in the first quadrant bounded by the region y = 4x2, x = 0, y = 0 and y = 4

Exercise 3.1 | Q 7 | Page 65

Find the area bounded by the curve y = x2 and the line y = 4

Exercise 3.2 [Pages 72 - 73]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 3 Integral Calculus – 2Exercise 3.2 [Pages 72 - 73]

Exercise 3.2 | Q 1 | Page 72

The cost of an overhaul of an engine is ₹ 10,000 The operating cost per hour is at the rate of 2x – 240 where the engine has run x km. Find out the total cost if the engine runs for 300 hours after overhaul

Exercise 3.2 | Q 2 | Page 72

Elasticity of a function ("E"y)/("E"x) is given by ("E"y)/("E"x) = (-7x)/((1 - 2x)(2 + 3x)). Find the function when x = 2, y = 3/8

Exercise 3.2 | Q 3 | Page 72

The elasticity of demand with respect to price for a commodity is given by ((4 - x))/x, where p is the price when demand is x. Find the demand function when the price is 4 and the demand is 2. Also, find the revenue function

Exercise 3.2 | Q 4 | Page 72

A company receives a shipment of 500 scooters every 30 days. From experience, it is known that the inventory on hand is related to the number of days x. Since the shipment, I(x) = 500 – 0.03x2, the daily holding cost per scooter is ₹ 0.3. Determine the total cost for maintaining inventory for 30 days

Exercise 3.2 | Q 5 | Page 72

An account fetches interest at the rate of 5% per annum compounded continuously. An individual deposits ₹ 1,000 each year in his account. How much will be in the account after 5 years. (e0.25 = 1.284)

Exercise 3.2 | Q 6 | Page 72

The marginal cost function of a product is given by "dc"/("d"x) = 100 – 10x + 0.1x2 where x is the output. Obtain the total and the average cost function of the firm under the assumption, that its fixed cost is ₹ 500

Exercise 3.2 | Q 7 | Page 72

The marginal cost function is MC = 300  x^(2/5) and fixed cost is zero. Find out the total cost and average cost functions

Exercise 3.2 | Q 8 | Page 72

If the marginal cost function of x units of output is "a"/sqrt("a"x + "b") and if the cost of output is zero. Find the total cost as a function of x

Exercise 3.2 | Q 9 | Page 72

Determine the cost of producing 200 air conditioners if the marginal cost (is per unit) is C'(x) = x^2/200 + 4

Exercise 3.2 | Q 10 | Page 72

The marginal revenue (in thousands of Rupees) functions for a particular commodity is 5 + 3"e"^(- 003x) where x denotes the number of units sold. Determine the total revenue from the sale of 100 units. (Given e–3 = 0.05 approximately)

Exercise 3.2 | Q 11 | Page 72

If the marginal revenue function for a commodity is MR = 9 – 4x2. Find the demand function

Exercise 3.2 | Q 12 | Page 72

Given the marginal revenue function 4/(2x + 3)^2 - 1 show that the average revenue function is P = 4/(6x + 9) - 1

Exercise 3.2 | Q 13 | Page 72

A firm’s marginal revenue function is MR = 20"e"^((-x)/10) (1 - x/10). Find the corresponding demand function

Exercise 3.2 | Q 14 | Page 72

The marginal cost of production of a firm is given by C'(x) = 5 + 0.13x, the marginal revenue is given by R'(x) = 18 and the fixed cost is ₹ 120. Find the profit function

Exercise 3.2 | Q 15 | Page 72

If the marginal revenue function is R'(x) = 1500 – 4x – 3x2. Find the revenue function and average revenue function

Exercise 3.2 | Q 16 | Page 72

Find the revenue function and the demand function if the marginal revenue for x units is MR = 10 + 3x – x2

Exercise 3.2 | Q 17 | Page 73

The marginal cost function of a commodity is given by MC = 14000/sqrt(7x + 4) and the fixed cost is ₹ 18,000. Find the total cost and average cost

Exercise 3.2 | Q 18 | Page 73

If the marginal cost (MC) of production of the company is directly proportional to the number of units (x) produced, then find the total cost function, when the fixed cost is ₹ 5,000 and the cost of producing 50 units is ₹ 5,625

Exercise 3.2 | Q 19 | Page 73

If MR = 20 – 5x + 3x2, Find total revenue function

Exercise 3.2 | Q 20 | Page 73

If MR = 14 – 6x + 9x2, Find the demand function

Exercise 3.3 [Page 75]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 3 Integral Calculus – 2Exercise 3.3 [Page 75]

Exercise 3.3 | Q 1 | Page 75

Calculate consumer’s surplus if the demand function p = 50 – 2x and x = 20

Exercise 3.3 | Q 2 | Page 75

Calculate consumer’s surplus if the demand function p = 122 – 5x – 2x2, and x = 6

Exercise 3.3 | Q 3 | Page 75

The demand function p = 85 – 5x and supply function p = 3x – 35. Calculate the equilibrium price and quantity demanded. Also, calculate consumer’s surplus

Exercise 3.3 | Q 4 | Page 75

The demand function for a commodity is p = e–x .Find the consumer’s surplus when p = 0.5

Exercise 3.3 | Q 5 | Page 75

Calculate the producer’s surplus at x = 5 for the supply function p = 7 + x

Exercise 3.3 | Q 6 | Page 75

If the supply function for a product is p = 3x + 5x2. Find the producer’s surplus when x = 4

Exercise 3.3 | Q 7 | Page 75

The demand function for a commodity is p =36/(x + 4). Find the consumer’s surplus when the prevailing market price is ₹ 6

Exercise 3.3 | Q 8 | Page 75

The demand and supply functions under perfect competition are pd = 1600 – x2 and ps = 2x2 + 400 respectively. Find the producer’s surplus

Exercise 3.3 | Q 9 | Page 75

Under perfect competition for a commodity the demand and supply laws are Pd = 8/(x + 1) - 2 and Ps = (x + 3)/2 respectively. Find the consumer’s and producer’s surplus

Exercise 3.3 | Q 10 | Page 75

The demand equation for a products is x = sqrt(100 - "p") and the supply equation is x = "P"/2 - 10. Determine the consumer’s surplus and producer’s surplus, under market equilibrium

Exercise 3.3 | Q 11 | Page 75

Find the consumer’s surplus and producer’s surplus for the demand function pd = 25 – 3x and supply function ps = 5 + 2x

Exercise 3.4 [Pages 75 - 77]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 3 Integral Calculus – 2Exercise 3.4 [Pages 75 - 77]

#### MCQ

Exercise 3.4 | Q 1 | Page 75

Choose the correct alternative:

Area bounded by the curve y = x(4 – x) between the limits 0 and 4 with x-axis is

• 30/5 sq.units

• 31/2 sq.units

• 32/3 sq.units

• 15/2 sq.units

Exercise 3.4 | Q 2 | Page 75

Choose the correct alternative:

Area bounded by the curve y = e–2x between the limits 0 ≤ x ≤ oo is

• 1 sq.units

• 1/2 sq.unit

• 5 sq.units

• 2 sq.units

Exercise 3.4 | Q 3 | Page 75

Choose the correct alternative:

Area bounded by the curve y = 1/x between the limits 1 and 2 is

• log2 sq.units

• log5 sq.units

• log3 sq.unitslog3 sq.units

• log 4 sq.units

Exercise 3.4 | Q 4 | Page 75

Choose the correct alternative:

If the marginal revenue function of a firm is MR = "e"^((-x)/10), then revenue is

• - 10"e"^((-x)/10)

• 1 - "e"^((-x)/10)

• 10(1 - "e"^((-x)/10))

• "e"^((-x)/10) + 10

Exercise 3.4 | Q 5 | Page 75

Choose the correct alternative:

If MR and MC denotes the marginal revenue and marginal cost functions, then the profit functions is

• P = int ("MR" - "MC")  "d"x + "k"

• P = int ("MR" + "MC")  "d"x + "k"

• P = int ("MR") ("MC")  "d"x + "k"

• P = int ("R" - "C")  "d"x + "k"

Exercise 3.4 | Q 6 | Page 76

Choose the correct alternative:

The demand and supply functions are given by D(x) = 16 – x2 and S(x) = 2x2 + 4 are under perfect competition, then the equilibrium price x is

• 2

• 3

• 4

• 5

Exercise 3.4 | Q 7 | Page 76

Choose the correct alternative:

The marginal revenue and marginal cost functions of a company are MR = 30 – 6x and MC = – 24 + 3x where x is the product, then the profit function is

• 9x2 + 54x

• 9x2 – 54x

• 54x - (9x^2)/2

• 54x - (9x^2)/2 + "k"

Exercise 3.4 | Q 8 | Page 76

Choose the correct alternative:

The given demand and supply function are given by D(x) = 20 – 5x and S(x) = 4x + 8 if they are under perfect competition then the equilibrium demand is

• 40

• 41/2

• 40/3

• 41/5

Exercise 3.4 | Q 9 | Page 76

Choose the correct alternative:

If the marginal revenue MR = 35 + 7x – 3x2, then the average revenue AR is

• 35x + (7x^2)/2 - x^3

• 35x + (7x)/2 - x^2

• 35x + (7x)/2 + x^2

• 35 + 7x + x2

Exercise 3.4 | Q 10 | Page 76

Choose the correct alternative:

The profit of a function p(x) is maximum when

• MC – MR = 0

• MC = 0

• MR = 0

• MC + MR = 0

Exercise 3.4 | Q 11 | Page 76

Choose the correct alternative:

For the demand function p(x), the elasticity of demand with respect to price is unity then

• Revenue is constant

• Cost function is constant

• Profit is constant

• None of these

Exercise 3.4 | Q 12 | Page 76

Choose the correct alternative:

The demand function for the marginal function MR = 100 – 9x2 is

• 100 – 3x2

• 100x – 3x2

• 100x – 9x2

• 100 + 9x2

Exercise 3.4 | Q 13 | Page 76

Choose the correct alternative:

When x0 = 5 and p0 = 3 the consumer’s surplus for the demand function pd = 28 – x2 is

• 250 units

• 250/3 units

• 251/2 units

• 251/3 units

Exercise 3.4 | Q 14 | Page 76

Choose the correct alternative:

When x0 = 2 and P0 = 12 the producer’s surplus for the supply function Ps = 2x2 + 4 is

• 31/5 units

• 31/2 units

• 32/3 units

• 30/7 units

Exercise 3.4 | Q 15 | Page 76

Choose the correct alternative:

Area bounded by y = x between the lines y = 1, y = 2 with y-axis is

• 1/2 sq.units

• 5/2 sq.units

• 3/2 sq.units

• 1 sq.unit

Exercise 3.4 | Q 16 | Page 76

Choose the correct alternative:

The producer’s surplus when the supply function for a commodity is P = 3 + x and x0 = 3 is

• 5/2

• 9/2

• 3/2

• 7/2

Exercise 3.4 | Q 17 | Page 76

Choose the correct alternative:

The marginal cost function is MC = 100sqrt(x). find AC given that TC = 0 when the output is zero is

• 200/3 x^(1/2)

• 200/3 x^(3/2)

• 200/(3x^(3/2)

• 200/(3x^(1/2)

Exercise 3.4 | Q 18 | Page 76

Choose the correct alternative:

The demand and supply function of a commodity are P(x) = (x – 5)2 and S(x) = x2 + x + 3 then the equilibrium quantity x0 is

• 5

• 2

• 3

• 19

Exercise 3.4 | Q 19 | Page 76

Choose the correct alternative:

The demand and supply function of a commodity are D(x) = 25 – 2x and S(x) = (10 + x)/4 then the equilibrium price p0 is

• 5

• 2

• 3

• 10

Exercise 3.4 | Q 20 | Page 77

Choose the correct alternative:

If MR and MC denote the marginal revenue and marginal cost and MR – MC = 36x – 3x2 – 81, then the maximum profit at x is equal to

• 3

• 6

• 9

• 5

Exercise 3.4 | Q 21 | Page 77

Choose the correct alternative:

If the marginal revenue of a firm is constant, then the demand function is

• MR

• MC

• C(x)

• AC

Exercise 3.4 | Q 22 | Page 77

Choose the correct alternative:

For a demand function p, if int "dp"/"p" = "k" int ("d"x)/x then k is equal to

• ηd

• d

• (-1)/eta_"d"

• 1/eta_"d"

Exercise 3.4 | Q 23 | Page 77

Choose the correct alternative:

Area bounded by y = ex between the limits 0 to 1 is

• (e – 1) sq.units

• (e + 1) sq.units

• (1 - 1/"e") sq.units

• (1 + 1/"e") sq.units

Exercise 3.4 | Q 24 | Page 77

Choose the correct alternative:

The area bounded by the parabola y2 = 4x bounded by its latus rectum is

• 16/3 sq.units

• 8/3 sq.units

• 73/3 sq.units

• 1/3 sq.units

Exercise 3.4 | Q 25 | Page 77

Choose the correct alternative:

Area bounded by y = |x| between the limits 0 and 2 is

• 1 sq.units

• 3 sq.units

• 2 sq.units

• 4 sq.units

Miscellaneous problems [Page 77]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 3 Integral Calculus – 2Miscellaneous problems [Page 77]

Miscellaneous problems | Q 1 | Page 77

A manufacture’s marginal revenue function is given by MR = 275 – x – 0.3x2. Find the increase in the manufactures total revenue if the production is increased from 10 to 20 units

Miscellaneous problems | Q 2 | Page 77

A company has determined that marginal cost function for x product of a particular commodity is given by MC = 125 + 10x - x^2/9. Where C is the cost of producing x units of the commodity. If the fixed cost is ₹ 250 what is the cost of producing 15 units

Miscellaneous problems | Q 3 | Page 77

The marginal revenue function for a firm given by MR = 2/(x + 3) - (2x)/(x + 3)^2 + 5. Show that the demand function is P = (2x)/(x + 3)^2 + 5

Miscellaneous problems | Q 4 | Page 77

For the marginal revenue function MR = 6 – 3x2 – x3, Find the revenue function and demand function

Miscellaneous problems | Q 5 | Page 77

The marginal cost of production of a firm is given by C'(x) = 20 + x/20 the marginal revenue is given by R’(x) = 30 and the fixed cost is ₹ 100. Find the profit function

Miscellaneous problems | Q 6 | Page 77

The demand equation for a product is Pd = 20 – 5x and the supply equation is Ps = 4x + 8. Determine the consumers surplus and producer’s surplus under market equilibrium

Miscellaneous problems | Q 7 | Page 77

A company requires f(x) number of hours to produce 500 units. It is represented by f(x) = 1800x–0.4. Find out the number of hours required to produce additional 400 units. [(900)0.6 = 59.22, (500)0.6 = 41.63]

Miscellaneous problems | Q 8 | Page 77

The price elasticity of demand for a commodity is "p"/x^3. Find the demand function if the quantity of demand is 3 when the price is ₹ 2.

Miscellaneous problems | Q 9 | Page 77

Find the area of the region bounded by the curve between the parabola y = 8x2 – 4x + 6 the y-axis and the ordinate at x = 2

Miscellaneous problems | Q 10 | Page 77

Find the area of the region bounded by the curve y2 = 27x3 and the lines x = 0, y = 1 and y = 2

## Chapter 3: Integral Calculus – 2

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Miscellaneous problems

## Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide chapter 3 - Integral Calculus – 2

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