11th Business Maths Monthly Test - 1 ( Applications of Differentiation )

MABS Institution

11th Business Maths Monthly Test - 1 ( Applications of Differentiation )-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

If the average revenue of a certain firm is Rs 50 and its elasticity of demand is 2, then their marginal revenue is

Rs 50
Rs 25
Rs 100
Rs 75

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

The maximum value of f(x)= sinx is

1
\(\frac { \sqrt { 3 } }{ 2 } \)
\(\frac { 1 }{ \sqrt { 2 } } \)
\(-\frac { 1 }{ \sqrt { 2 } } \)

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

20–6x
20–3x
20+6x
20+3x

If u = \({ e }^{ { x }^{ 2 } }\) then \(\frac { \partial u }{ \partial x } \) is equal to

\({ 2x }e^{ { x }^{ 2 } }\)
\({ e }^{ { x }^{ 2 } }\)
\({2 e }^{ { x }^{ 2 } }\)
0

If the demand function is said to be inelastic, then

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

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

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

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

Part - B

Generic 2 - Answer All Question

If f (x, y) = 3x² + 4y³ + 6xy - x³y³ + 7 then show that f_xy (1,1) = 18.

The demand for a quantity A is q=13 - 2P₁ - \({ 3p }_{ 2 }^{ 2 }\) Find the partial elasticities \(\frac { { E }q }{ { E }p_{ 1 } } \)and \(\frac { { E }q }{ { E }p_{ 2 } } \) when P₁ = P₂ = 2

If f(x,y) = 3x² + 4y³ + 6xy - x²y³ + 6. Find f_xy(2,1)

The demand for a commodity A is q = 80 - \({ p }_{ 1 }^{ 2}\) + 5p₂ - p₁p₂. Find the partial elasticities \(\frac { { E }q }{ { E }p_{ 1 } } \) and \(\frac { { E }q }{ { E }p_{ 2 } } \) when p₁=2, p₂ = 1.
A dealer has to supply his customer with 400 units of a product per every week. The dealer gets the product from the manufacturer at a cost of Rs 50 per unit. The cost of ordering from the manufacturers in Rs 75 per order. The cost of holding inventory is 7.5 % per year of the product cost. Find (i) EOQ (ii) Total optimum cost.

Verify \(\frac { \partial ^{ 2 }u }{ \partial x\partial y } \frac { { \partial }^{ 2 }u}{ \partial y\partial x } \) for u = x³ + 3x² y² + y³

Given C(x)= \(\frac { { x }^{ 3 } }{ 6 } \)5x+200 and p(x) = 40–x are the cost price and selling price when x units of commodity are produced. Find the level of the production that maximize the profit.

The cost function of a firm is \(C={1\over3}x^3-3x^2+9x\)Find the level of output (x>0) when average cost is minimum

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

For the function y=x³ +19 find the values of x when its marginal value is equal to 27.

Part - C

Generic 3 - Answer All Question

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

Verify Euler’s theorem for the function \(u=\frac{1}{\sqrt{x^2+y^2}}\)

The production function for a commodity is P = 10L + 0.1 L² +15K-0.2K² +2KL where L is labour and K is Capital.
(i) Calculate the marginal products of two inputs when 10 units of each of labour and Capital are used
(ii) If 10 units of capital are used, what is the upper limit for use of labour which a rational producer will never exceed?

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.

Find the stationary value and the stationary points f(x)=x²+2x–5.
The total cost of x units of output of a firm is given by c = \(\frac { 2 }{ 3 } x+\frac { 35 }{ 2 } \) find the
(i) cost when output is 4 units
(ii) average cost when output is 10 units
(iii) marginal cost when output is 3 units

\(\bar C=0.05x^2+16+{100\over x}\)is the manufacturer’s average cost function. What is the marginal cost when 50 units are produced and interpret your result

For the production function P= 5(L)^0.7(K)^0.3.Find the marginal productivities of Labour (L) and Capital (K) when L = 10, K = 3 [Use (0.3)^0·3 = 0.6968; (3.33)^0·7 = 2.2322]

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

Find the elasticity of demand in terms of x for the demand law \(p={(a-bx)^{1\over 2}}.\) Also find the values of x when elasticity of demand is unity.

Part - D

Generic 5 - Answer All Question

Find the marginal productivities for Capital (K) and Labour (L) if P = 10K-K² + KL when K = 2 and L = 6.

A monopolist has a demand curve x = 106 – 2p and average cost curve AC = 5+\(\frac { x }{ 50 } \) where p is the price per unit output and x is the number of units of output. If the total revenue is R = px, determine the most profitable output and 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.

For the total revenue function R = - 90 + 6x² - x³ find when R is increasing and when it is decreasing. Also, discuss the behaviour of marginal revenue.
The demand for a quantity A is q = 16- 3P_I - 2P₂². Find the partial elasticities \({Eq\over EP_1}\) and \({Eq\over EP_2}\)

The demand and the cost function of a firm are p = 497-0.2x and C = 25x+10000 respectively. Find the output level and price at which the profit is maximum

The total revenue function for a commodity is R=15x +\(\frac { { x }^{ 2 } }{ 3 } -\frac { 1 }{ 36 } { x }^{ 4 }\) Show that at the highest point average revenue is equal to the marginal revenue

If \(u=tan^{-1} \left(x^2+y^2\over x+y\right)\), then using Euler's theorem, prove that \(x{∂u\over ∂x}+y{∂u\over ∂y}={1\over 2}sin2u\)

A company buys in lots of 500 boxes which is a 3 month supply. The cost per box is Rs 125 and the ordering cost in Rs 150. The inventory carrying cost is estimated at 20% of unit value.
(i) Determine the total amount cost of existing inventory policy
(ii) How much money could be saved by applying the economic order quantity?

For the production function P = C(L)^α(K)^β where C is a positive constant and if α + β = 1, show that \(K{∂P\over ∂ K}+L{∂P\over ∂L}=P\)

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11th Business Maths Monthly Test - 1 ( Applications of Differentiation )

MABS Institution

11th Business Maths Monthly Test - 1 ( Applications of Differentiation )-Aug 2020

Part - A

Multiple Choice Question - Answer All Question

If the average revenue of a certain firm is Rs 50 and its elasticity of demand is 2, then their marginal revenue is

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

The maximum value of f(x)= sinx is

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

If u = \({ e }^{ { x }^{ 2 } }\) then \(\frac { \partial u }{ \partial x } \) is equal to

If the demand function is said to be inelastic, then

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

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

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

Part - B

Generic 2 - Answer All Question

If f (x, y) = 3x2 + 4y3 + 6xy - x3y3 + 7 then show that fxy (1,1) = 18.

The demand for a quantity A is q=13 - 2P1 - \({ 3p }_{ 2 }^{ 2 }\) Find the partial elasticities \(\frac { { E }q }{ { E }p_{ 1 } } \)and \(\frac { { E }q }{ { E }p_{ 2 } } \) when P1 = P2 = 2

If f(x,y) = 3x2 + 4y3 + 6xy - x2y3 + 6. Find fxy(2,1)

The demand for a commodity A is q = 80 - \({ p }_{ 1 }^{ 2}\) + 5p2 - p1p2. Find the partial elasticities \(\frac { { E }q }{ { E }p_{ 1 } } \) and \(\frac { { E }q }{ { E }p_{ 2 } } \) when p1=2, p2 = 1.

Verify \(\frac { \partial ^{ 2 }u }{ \partial x\partial y } \frac { { \partial }^{ 2 }u}{ \partial y\partial x } \) for u = x3 + 3x2 y2 + y3

Given C(x)= \(\frac { { x }^{ 3 } }{ 6 } \)5x+200 and p(x) = 40–x are the cost price and selling price when x units of commodity are produced. Find the level of the production that maximize the profit.

The cost function of a firm is \(C={1\over3}x^3-3x^2+9x\)Find the level of output (x>0) when average cost is minimum

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

For the function y=x3 +19 find the values of x when its marginal value is equal to 27.

Part - C

Generic 3 - Answer All Question

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

Verify Euler’s theorem for the function \(u=\frac{1}{\sqrt{x^2+y^2}}\)

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.

Find the stationary value and the stationary points f(x)=x2+2x–5.

The total cost of x units of output of a firm is given by c = \(\frac { 2 }{ 3 } x+\frac { 35 }{ 2 } \) find the (i) cost when output is 4 units (ii) average cost when output is 10 units (iii) marginal cost when output is 3 units

\(\bar C=0.05x^2+16+{100\over x}\)is the manufacturer’s average cost function. What is the marginal cost when 50 units are produced and interpret your result

For the production function P= 5(L)0.7(K)0.3.Find the marginal productivities of Labour (L) and Capital (K) when L = 10, K = 3 [Use (0.3)0·3 = 0.6968; (3.33)0·7 = 2.2322]

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

Find the elasticity of demand in terms of x for the demand law \(p={(a-bx)^{1\over 2}}.\) Also find the values of x when elasticity of demand is unity.

Part - D

Generic 5 - Answer All Question

Find the marginal productivities for Capital (K) and Labour (L) if P = 10K-K2 + KL when K = 2 and L = 6.

A monopolist has a demand curve x = 106 – 2p and average cost curve AC = 5+\(\frac { x }{ 50 } \) where p is the price per unit output and x is the number of units of output. If the total revenue is R = px, determine the most profitable output and the maximum profit.

For the total revenue function R = - 90 + 6x2 - x3 find when R is increasing and when it is decreasing. Also, discuss the behaviour of marginal revenue.

The demand for a quantity A is q = 16- 3PI - 2P22. Find the partial elasticities \({Eq\over EP_1}\) and \({Eq\over EP_2}\)

The demand and the cost function of a firm are p = 497-0.2x and C = 25x+10000 respectively. Find the output level and price at which the profit is maximum

The total revenue function for a commodity is R=15x +\(\frac { { x }^{ 2 } }{ 3 } -\frac { 1 }{ 36 } { x }^{ 4 }\) Show that at the highest point average revenue is equal to the marginal revenue

If \(u=tan^{-1} \left(x^2+y^2\over x+y\right)\), then using Euler's theorem, prove that \(x{∂u\over ∂x}+y{∂u\over ∂y}={1\over 2}sin2u\)

For the production function P = C(L)α(K)β where C is a positive constant and if α + β = 1, show that \(K{∂P\over ∂ K}+L{∂P\over ∂L}=P\)

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

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

If f (x, y) = 3x² + 4y³ + 6xy - x³y³ + 7 then show that f_xy (1,1) = 18.

The demand for a quantity A is q=13 - 2P₁ - \({ 3p }_{ 2 }^{ 2 }\) Find the partial elasticities \(\frac { { E }q }{ { E }p_{ 1 } } \)and \(\frac { { E }q }{ { E }p_{ 2 } } \) when P₁ = P₂ = 2

If f(x,y) = 3x² + 4y³ + 6xy - x²y³ + 6. Find f_xy(2,1)

The demand for a commodity A is q = 80 - \({ p }_{ 1 }^{ 2}\) + 5p₂ - p₁p₂. Find the partial elasticities \(\frac { { E }q }{ { E }p_{ 1 } } \) and \(\frac { { E }q }{ { E }p_{ 2 } } \) when p₁=2, p₂ = 1.

Verify \(\frac { \partial ^{ 2 }u }{ \partial x\partial y } \frac { { \partial }^{ 2 }u}{ \partial y\partial x } \) for u = x³ + 3x² y² + y³

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

For the function y=x³ +19 find the values of x when its marginal value is equal to 27.

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

Find the stationary value and the stationary points f(x)=x²+2x–5.

The total cost of x units of output of a firm is given by c = \(\frac { 2 }{ 3 } x+\frac { 35 }{ 2 } \) find the
(i) cost when output is 4 units
(ii) average cost when output is 10 units
(iii) marginal cost when output is 3 units

For the production function P= 5(L)^0.7(K)^0.3.Find the marginal productivities of Labour (L) and Capital (K) when L = 10, K = 3 [Use (0.3)^0·3 = 0.6968; (3.33)^0·7 = 2.2322]

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

Find the marginal productivities for Capital (K) and Labour (L) if P = 10K-K² + KL when K = 2 and L = 6.

For the total revenue function R = - 90 + 6x² - x³ find when R is increasing and when it is decreasing. Also, discuss the behaviour of marginal revenue.

The demand for a quantity A is q = 16- 3P_I - 2P₂². Find the partial elasticities \({Eq\over EP_1}\) and \({Eq\over EP_2}\)

For the production function P = C(L)^α(K)^β where C is a positive constant and if α + β = 1, show that \(K{∂P\over ∂ K}+L{∂P\over ∂L}=P\)