#### Chapters

Chapter 2: Integral Calculus – 1

Chapter 3: Integral Calculus – 2

Chapter 4: Differential Equations

Chapter 5: Numerical Methods

Chapter 6: Random Variable and Mathematical expectation

Chapter 7: Probability Distributions

Chapter 8: Sampling techniques and Statistical Inference

Chapter 9: Applied Statistics

Chapter 10: Operations Research

## Chapter 3: Integral Calculus – 2

### 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]

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

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

Calculate the area bounded by the parabola y^{2} = 4ax and its latus rectum

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

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

Find the area of the region lying in the first quadrant bounded by the region y = 4x^{2}, x = 0, y = 0 and y = 4

Find the area bounded by the curve y = x^{2} and the line y = 4

### 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]

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

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`

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

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 – 0.03x^{2}, the daily holding cost per scooter is ₹ 0.3. Determine the total cost for maintaining inventory for 30 days

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. (e^{0.25} = 1.284)

The marginal cost function of a product is given by `"dc"/("d"x)` = 100 – 10x + 0.1x^{2} 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

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

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

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

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)

If the marginal revenue function for a commodity is MR = 9 – 4x^{2}. Find the demand function

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

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

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

If the marginal revenue function is R'(x) = 1500 – 4x – 3x^{2}. Find the revenue function and average revenue function

Find the revenue function and the demand function if the marginal revenue for x units is MR = 10 + 3x – x^{2}

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

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

If MR = 20 – 5x + 3x^{2}, Find total revenue function

If MR = 14 – 6x + 9x^{2}, Find the demand function

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

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

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

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

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

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

If the supply function for a product is p = 3x + 5x^{2}. Find the producer’s surplus when x = 4

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

The demand and supply functions under perfect competition are p_{d} = 1600 – x^{2} and p_{s} = 2x^{2} + 400 respectively. Find the producer’s surplus

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

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

Find the consumer’s surplus and producer’s surplus for the demand function p_{d} = 25 – 3x and supply function p_{s} = 5 + 2x

### 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

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

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

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

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`

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"`

Choose the correct alternative:

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

2

3

4

5

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

9x

^{2}+ 54x9x

^{2}– 54x`54x - (9x^2)/2`

`54x - (9x^2)/2 + "k"`

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`

Choose the correct alternative:

If the marginal revenue MR = 35 + 7x – 3x^{2}, then the average revenue AR is

`35x + (7x^2)/2 - x^3`

`35x + (7x)/2 - x^2`

`35x + (7x)/2 + x^2`

35 + 7x + x

^{2}

Choose the correct alternative:

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

MC – MR = 0

MC = 0

MR = 0

MC + MR = 0

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

Choose the correct alternative:

The demand function for the marginal function MR = 100 – 9x^{2} is

100 – 3x

^{2}100x – 3x

^{2}100x – 9x

^{2}100 + 9x

^{2}

Choose the correct alternative:

When x_{0} = 5 and p_{0} = 3 the consumer’s surplus for the demand function p_{d} = 28 – x^{2} is

250 units

`250/3` units

`251/2` units

`251/3` units

Choose the correct alternative:

When x_{0} = 2 and P_{0} = 12 the producer’s surplus for the supply function P_{s} = 2x^{2} + 4 is

`31/5` units

`31/2` units

`32/3` units

`30/7` units

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

Choose the correct alternative:

The producer’s surplus when the supply function for a commodity is P = 3 + x and x_{0} = 3 is

`5/2`

`9/2`

`3/2`

`7/2`

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)`

Choose the correct alternative:

The demand and supply function of a commodity are P(x) = (x – 5)^{2} and S(x) = x^{2} + x + 3 then the equilibrium quantity x_{0} is

5

2

3

19

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 p_{0} is

5

2

3

10

Choose the correct alternative:

If MR and MC denote the marginal revenue and marginal cost and MR – MC = 36x – 3x^{2} – 81, then the maximum profit at x is equal to

3

6

9

5

Choose the correct alternative:

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

MR

MC

C(x)

AC

Choose the correct alternative:

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

η

_{d}-η

_{d}`(-1)/eta_"d"`

`1/eta_"d"`

Choose the correct alternative:

Area bounded by y = e^{x} 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

Choose the correct alternative:

The area bounded by the parabola y^{2} = 4x bounded by its latus rectum is

`16/3` sq.units

`8/3` sq.units

`73/3` sq.units

`1/3` sq.units

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

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

A manufacture’s marginal revenue function is given by MR = 275 – x – 0.3x^{2}. Find the increase in the manufactures total revenue if the production is increased from 10 to 20 units

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

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`

For the marginal revenue function MR = 6 – 3x^{2} – x^{3}, Find the revenue function and demand function

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

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

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]

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.

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

Find the area of the region bounded by the curve y^{2} = 27x^{3} and the lines x = 0, y = 1 and y = 2

## Chapter 3: Integral Calculus – 2

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

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Concepts covered in Class 12th Business Mathematics and Statistics Answers Guide chapter 3 Integral Calculus – 2 are The Area of the Region Bounded by the Curves, Application of Integration in Economics and Commerce.

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