HSC Commerce: Marketing and Salesmanship 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Fill in the blank :

Multiplicative models of time series _______ independence of its components.

State whether the following is True or False :

Additive model of time series does not require the assumption of independence of its components.

State whether the following is True or False :

Multiplicative model of time series does not require the assumption of independence of its components.

The time interval between starting the first job and completing the last job including the idle time (if any) in a particular order by the given set of machines is called _______.

Fill in the blank :

In a sequencing problem, all machines are of _______ types.

If `sqrt(x) + sqrt(y) = sqrt("a")`, then `("d"y)/("d"x)` is ______

State whether the following statement is True or False:

If `sqrt(x) + sqrt(y) = sqrt("a")`, then `("d"y)/("d"x) = 1/(2sqrt(x)) + 1/(2sqrt(y)) = 1/(2sqrt("a"))`

If the marginal revenue is 28 and elasticity of demand is 3, then the price is ______.

If the elasticity of demand η = 1, then demand is ______.

If 0 < η < 1, then the demand is ______.

If the average revenue is 45 and elasticity of demand is 5, then marginal revenue is ______.

State whether the following statement is True or False:  

If the marginal revenue is 50 and the price is ₹ 75, then elasticity of demand is 4

The manufacturing company produces x items at the total cost of ₹ 180 + 4x. The demand function for this product is P = (240 − 𝑥). Find x for which profit is increasing

A manufacturing company produces x items at a total cost of ₹ 40 + 2x. Their price per item is given as p = 120 – x. Find the value of x for which revenue is increasing

Solution: Total cost C = 40 + 2x and Price p = 120 – x

Revenue R = `square`

Differentiating w.r.t. x,

∴ `("dR")/("d"x) = square`

Since Revenue is increasing,

∴ `("dR")/("d"x)` > 0

∴ Revenue is increasing for `square`

A manufacturing company produces x items at a total cost of ₹ 40 + 2x. Their price per item is given as p = 120 – x. Find the value of x for which profit is increasing

Solution: Total cost C = 40 + 2x and Price p = 120 − x

Profit π = R – C

∴ π = `square`

Differentiating w.r.t. x,

`("d"pi)/("d"x)` = `square`

Since Profit is increasing,

`("d"pi)/("d"x)` > 0

∴ Profit is increasing for `square`

A manufacturing company produces x items at a total cost of ₹ 40 + 2x. Their price per item is given as p = 120 – x. Find the value of x for which elasticity of demand for price ₹ 80.

Solution: Total cost C = 40 + 2x and Price p = 120 – x

p = 120 – x

∴ x = 120 – p

Differentiating w.r.t. p,

`("d"x)/("dp")` = `square`

∴ Elasticity of demand is given by η = `- "P"/x*("d"x)/("dp")`

∴ η = `square`

When p = 80, then elasticity of demand η = `square`

Choose the correct alternative:

`int sqrt(1 + x)  "d"x` =

Choose the correct alternative:

`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?

Choose the correct alternative:

`int ((x^3 + 3x^2 + 3x + 1))/(x + 1)^5 "d"x` =

`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x⁵ – ______ x³ + 5x + c