11th Business Maths Monthly Test - 2 ( Applications of Differentiation )

MABS Institution

11th Business Maths Monthly Test - 2 ( Applications of Differentiation )-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

If demand and the cost function of a firm are p= 2–x and c = 2x² +2x +7 then its profit function is

x² + 7
x² - 7
- x² + 7
- x² - 7

Marginal revenue of the demand function p= 20–3x is

20–6x
20–3x
20+6x
20+3x

If f(x,y) is a homogeneous function of degree n, then \(x\frac { \partial f }{ \partial x } +y\frac { \partial f }{ \partial y } \) is equal to

(n–1)f
n(n–1)f
nf
f

Instantaneous rate of change of y = 2x² + 5x with respect to x at x = 2 is

The demand function is always

Increasing function
Decreasing function
Non-decreasing function
Undefined function

if u = 4x² + 4xy + y² + 32 + 16 , then \(\frac { \partial ^{ 2 }u }{ \partial y\partial x } \) is equal to

8x + 4y + 4
4
2y + 32
0

Average fixed cost of the cost function C(x) = 2x³ +5x² - 14x +21 is

\(\frac { 2 }{ 3 } \)
\(\frac { 5}{ x } \)
\(\frac { -14 }{ x } \)
\(\frac { 21 }{ x } \)

if q = 1000 + 8p₁ - p₂ then, \(\frac { \partial q }{ \partial { p }_{ 1 } } \) is

-1
8
1000
1000 - p₂

A company begins to earn profit at

Maximum point
Breakeven point
Stationary point
Even point

If the demand function is said to be inelastic, then

|n_d|>1
|n_d|=1
|n_d|<1
|n_d| = 0

Part - B

Generic 2 - Answer All Question

If f(x,y) = 3x² + 4y³ + 6xy - x²y³ + 6. Find f_yy(1,1)

The cost function of a firm is C =x³ - 12x²+ 48x Find the level of output (x>0) at which average cost is minimum

Find out the elasticity of demand for the following functions
(i) p = xe^x
(ii) p = xe^-x
(iii) p = 10\({ e }^{ -\frac { x }{ 3 } }\)

Show that the function f(x) = x³ −3x² + 4x,x\(\varepsilon \) R is strictly increasing function on R.

If u = x²(y–x) + y2 (x–y), then show that

A tour operator charges Rupees 136 per passenger with a discount of 40 paisa for each passenger in excess of 100. The operator requires at least 100 passengers to operate he tour. Determine the number of passenger that will maximize the amount of money the tour perator receives.

let u= x²y³ cos \(\left( \frac { x }{ y } \right) \) by using Euler’s theorem show that \(x.\frac { \partial u }{ \partial x } +y.\frac { \partial u }{ \partial y } =5u\)

Part - C

Generic 3 - Answer All Question

If u = log(x²+y²) show that \(\frac { { \partial }^{ 2 }u }{ { \partial x }^{ 2 } } +\frac { { \partial }^{ 2 }u }{ \partial { y }^{ 2 } } =0\)

The demand curve of a commodity is given by p = \(\frac { 50-x }{ 5 } \) , find the marginal revenue for any output x and also find marginal revenue at x = 0 and x = 25?

Find the interval in which the function f(x)=x2–4x+6 is strictly increasing and strictly decreasing.

Find the intervals in which the function f given by f(x)=4x³–6x²–72x+30 is increasing or decreasing.

If \(y={2x+1\over 3x+2}\) then, obtain the value of elasticity at x = 1.

For the demand function \(x={20\over p+1}\), find the elasticity of demand with respect to price at a point p = 3. Examine whether the demand is elastic at p = 3.

For the given demand function p = 40–x, find the value of the output when η_d=1

Part - D

Generic 5 - Answer All Question

The demand function of a commodity is p = 200-\(\frac { x }{ 100 } \) and its cost is C=40x+12000 where p is a unit price in rupees and x is the number of units produced and sold. Determine (i) profit function (ii) average profit at an output of 10 units (iii) marginal profit at an output of 10 units and (iv) marginal average profit at an output of 10 units.

If \(z=e^{x^3+y^3}\) then prove that \(x{{\partial} z\over \partial x}+y{\partial z\over \partial y}=3z\ logz\) by using Euler's theorem

The Cost function, when the output of x is given by C = x(2e^x + e^-x). Show that the minimum average cost is 2√2

Find the stationary values and stationary points for the function f(x) = 2x³ +9x² +12x+1
A company has to supply 1000 item per month at a uniform rate and for each time, a production run is started with the cost of Rs 200. Cost of holding is Rs 20 per item per month. The number of items to be produced per run has to be ascertained. Determine the total of setup cost and average inventory cost if the run size is 500, 600, 700, 800. Find the optimal production run size using EOQ formula.

A firm has revenue function R = 8x and production cost function \(C = 150000 + 60\left(x^2\over 900\right)\) Find the total profit function and the number of units to be sold to get the maximum profit.

The total cost function of a firm C(x) = \(\frac { { x }^{ 3 } }{ 3 } -{ 5x }^{ 2 }+28x+10\), where x is the output. A tax at the rate of Rs. 2 per unit of output is imposed and the producer adds it to his cost. If the market demand function is given by p = 2530 – 5x, where p is the price per unit of output, find the profit-maximizing the output and price.

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11th Business Maths Monthly Test - 2 ( Applications of Differentiation )

MABS Institution

11th Business Maths Monthly Test - 2 ( Applications of Differentiation )-Aug 2020

Part - A

Multiple Choice Question - Answer All Question

If demand and the cost function of a firm are p= 2–x and c = 2x2 +2x +7 then its profit function is

Marginal revenue of the demand function p= 20–3x is

If f(x,y) is a homogeneous function of degree n, then \(x\frac { \partial f }{ \partial x } +y\frac { \partial f }{ \partial y } \) is equal to

Instantaneous rate of change of y = 2x2 + 5x with respect to x at x = 2 is

The demand function is always

if u = 4x2 + 4xy + y2 + 32 + 16 , then \(\frac { \partial ^{ 2 }u }{ \partial y\partial x } \) is equal to

Average fixed cost of the cost function C(x) = 2x3 +5x2 - 14x +21 is

if q = 1000 + 8p1 - p2 then, \(\frac { \partial q }{ \partial { p }_{ 1 } } \) is

A company begins to earn profit at

If the demand function is said to be inelastic, then

Part - B

Generic 2 - Answer All Question

If f(x,y) = 3x2 + 4y3 + 6xy - x2y3 + 6. Find fyy(1,1)

The cost function of a firm is C =x3 - 12x2 + 48x Find the level of output (x>0) at which average cost is minimum

Find out the elasticity of demand for the following functions (i) p = xex (ii) p = xe-x (iii) p = 10\({ e }^{ -\frac { x }{ 3 } }\)

Show that the function f(x) = x3 −3x2 + 4x,x\(\varepsilon \) R is strictly increasing function on R.

If u = x2(y–x) + y2 (x–y), then show that

A tour operator charges Rupees 136 per passenger with a discount of 40 paisa for each passenger in excess of 100. The operator requires at least 100 passengers to operate he tour. Determine the number of passenger that will maximize the amount of money the tour perator receives.

let u= x2y3 cos \(\left( \frac { x }{ y } \right) \) by using Euler’s theorem show that \(x.\frac { \partial u }{ \partial x } +y.\frac { \partial u }{ \partial y } =5u\)

Part - C

Generic 3 - Answer All Question

If u = log(x2+y2) show that \(\frac { { \partial }^{ 2 }u }{ { \partial x }^{ 2 } } +\frac { { \partial }^{ 2 }u }{ \partial { y }^{ 2 } } =0\)

The demand curve of a commodity is given by p = \(\frac { 50-x }{ 5 } \) , find the marginal revenue for any output x and also find marginal revenue at x = 0 and x = 25?

Find the interval in which the function f(x)=x2–4x+6 is strictly increasing and strictly decreasing.

Find the intervals in which the function f given by f(x)=4x3–6x2–72x+30 is increasing or decreasing.

If \(y={2x+1\over 3x+2}\) then, obtain the value of elasticity at x = 1.

For the demand function \(x={20\over p+1}\), find the elasticity of demand with respect to price at a point p = 3. Examine whether the demand is elastic at p = 3.

For the given demand function p = 40–x, find the value of the output when ηd=1

Part - D

Generic 5 - Answer All Question

If \(z=e^{x^3+y^3}\) then prove that \(x{{\partial} z\over \partial x}+y{\partial z\over \partial y}=3z\ logz\) by using Euler's theorem

The Cost function, when the output of x is given by C = x(2ex + e-x). Show that the minimum average cost is 2√2

Find the stationary values and stationary points for the function f(x) = 2x3 +9x2 +12x+1

A firm has revenue function R = 8x and production cost function \(C = 150000 + 60\left(x^2\over 900\right)\) Find the total profit function and the number of units to be sold to get the maximum profit.

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If demand and the cost function of a firm are p= 2–x and c = 2x² +2x +7 then its profit function is

Instantaneous rate of change of y = 2x² + 5x with respect to x at x = 2 is

if u = 4x² + 4xy + y² + 32 + 16 , then \(\frac { \partial ^{ 2 }u }{ \partial y\partial x } \) is equal to

Average fixed cost of the cost function C(x) = 2x³ +5x² - 14x +21 is

if q = 1000 + 8p₁ - p₂ then, \(\frac { \partial q }{ \partial { p }_{ 1 } } \) is

If f(x,y) = 3x² + 4y³ + 6xy - x²y³ + 6. Find f_yy(1,1)

The cost function of a firm is C =x³ - 12x²+ 48x Find the level of output (x>0) at which average cost is minimum

Find out the elasticity of demand for the following functions
(i) p = xe^x
(ii) p = xe^-x
(iii) p = 10\({ e }^{ -\frac { x }{ 3 } }\)

Show that the function f(x) = x³ −3x² + 4x,x\(\varepsilon \) R is strictly increasing function on R.

If u = x²(y–x) + y2 (x–y), then show that

let u= x²y³ cos \(\left( \frac { x }{ y } \right) \) by using Euler’s theorem show that \(x.\frac { \partial u }{ \partial x } +y.\frac { \partial u }{ \partial y } =5u\)

If u = log(x²+y²) show that \(\frac { { \partial }^{ 2 }u }{ { \partial x }^{ 2 } } +\frac { { \partial }^{ 2 }u }{ \partial { y }^{ 2 } } =0\)

Find the intervals in which the function f given by f(x)=4x³–6x²–72x+30 is increasing or decreasing.

For the given demand function p = 40–x, find the value of the output when η_d=1

The Cost function, when the output of x is given by C = x(2e^x + e^-x). Show that the minimum average cost is 2√2

Find the stationary values and stationary points for the function f(x) = 2x³ +9x² +12x+1