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

MABS Institution

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

Time : 1.30 Hours

Part - A

Multiple Choice Question - Answer All Question

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 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

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

20–6x
20–3x
20+6x
20+3x

The maximum value of f(x)= sinx is

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

Relationship among MR, AR and ηd is

\({ n }_{ d }=\frac { AR }{ AR-MR } \)
n₄ = AR - MR
MR = AR = n₄
\(AR=\frac { MR }{ { n }_{ 4 } } \)

Profit P(x) is maximum when

MR = MC
MR = 0
MC = AC
TR = AC

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

If the demand function is said to be inelastic, then

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

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

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

Part - B

Generic 2 - Answer All Question

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

For the cost function C = 2000 + 1800 x - 75x² + x³ find when the total cost (C) is increasing and when it is decreasing

The demand and cost functions of a firm are x = 6000–30p and C = 72000+60x respectively. Find the level of output and price at which the profit is maximum.
The cost function of a firm is C =x³ - 12x²+ 48x Find the level of output (x>0) at which average cost is minimum

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

Part - C

Generic 3 - Answer All Question

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

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

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.
If the demand law is given by p = 10e^{\(-\frac { x }{ 2 } \)} then find the elasticity of demand.

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

Part - D

Generic 5 - Answer All Question

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

For the cost function \(C=2x\left(x+5\over x+2\right)+7\), prove that marginal cost (MC) falls continuously as the output x increases

Verify the relationship of elasticity of demand, average revenue and marginal revenue for the demand law p = 50-3x.
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.

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

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

MABS Institution

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

Part - A

Multiple Choice Question - Answer All Question

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

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

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

The maximum value of f(x)= sinx is

Relationship among MR, AR and ηd is

Profit P(x) is maximum when

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

If the demand function is said to be inelastic, then

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

Part - B

Generic 2 - Answer All Question

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

For the cost function C = 2000 + 1800 x - 75x2 + x3 find when the total cost (C) is increasing and when it is decreasing

The demand and cost functions of a firm are x = 6000–30p and C = 72000+60x respectively. Find the level of output and price at which the profit is maximum.

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

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

Part - C

Generic 3 - Answer All Question

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

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

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.

If the demand law is given by p = 10e\(-\frac { x }{ 2 } \) then find the elasticity of demand.

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

Part - D

Generic 5 - Answer All Question

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

For the cost function \(C=2x\left(x+5\over x+2\right)+7\), prove that marginal cost (MC) falls continuously as the output x increases

Verify the relationship of elasticity of demand, average revenue and marginal revenue for the demand law p = 50-3x.

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

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Tamilnadu Stateboardszs - Class

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

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

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

For the cost function C = 2000 + 1800 x - 75x² + x³ find when the total cost (C) is increasing and when it is decreasing

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

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

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

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

If the demand law is given by p = 10e^{\(-\frac { x }{ 2 } \)} then find the elasticity of demand.