HSC Commerce: Marketing and Salesmanship १२ वीं कक्षा - Maharashtra State Board Important Questions

Find `("d"y)/("d"x)`, if x = e^m, y = `"e"^(sqrt("m"))`

Solution: Given, x = e^m and y = `"e"^(sqrt("m"))`

Now, y = `"e"^(sqrt("m"))`

Diff.w.r.to m,

`("d"y)/"dm" = "e"^(sqrt("m"))("d"square)/"dm"`

∴ `("d"y)/"dm" = "e"^(sqrt("m"))*1/(2sqrt("m"))`    .....(i)

Now, x = e^m

Diff.w.r.to m,

`("d"x)/"dm" = square`    .....(ii)

Now, `("d"y)/("d"x) = (("d"y)/("d"m))/square`

∴ `("d"y)/("d"x) = (("e"sqrt("m"))/square)/("e"^"m")`

∴  `("d"y)/("d"x) = ("e"^(sqrt("m")))/(2sqrt("m")*"e"^("m")`

If x = `sqrt(1 + u^2)`, y = `log(1 + u^2)`, then find `(dy)/(dx).`

If ax² + 2hxy + by² = 0, then prove that `(d^2y)/(dx^2)` = 0.

Solve the following differential equations:

x²ydx – (x³ – y³)dy = 0

Find `(d^2y)/(dy^2)`, if y = e^4x

`int 1/(4x^2 - 1) dx` = ______.

If y = x . log x then `dy/dx` = ______.

If y = 2x² + a² + 2² then `dy/dx` = ______.

If y = (log x)² the `dy/dx` = ______.

Price P for demand D is given as P = 183 +120D - 3D² Find D for which the price is increasing

If x = cos2 θ and y = cot θ then find `dy/dx  at  θ=pi/4` 

Find `dy/dx,if e^x+e^y=e^(x-y)`

Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q 

Find the marginal revenue if the average revenue is 45 and elasticity of demand is 5.

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

Find the elasticity of demand, if the marginal revenue is 50 and price is Rs 75.

The total cost of manufacturing x articles is C = 47x + 300x² − x⁴.  Find x, for which average cost is increasing.

Find MPC ( Marginal propensity to Consume ) and APC ( Average Propensity to Consume ) if the expenditure E_c of a person with income I is given as E_c = ( 0.0003 ) I² + ( 0.075 ) I when I = 1000.

If the demand function is D = 50 - 3p - p², find the elasticity of demand at (a) p = 5 (b) p = 2 ,  Interpret your result. 

Determine the maximum and minimum value of the following function.

f(x) = `x^2 + 16/x`