HSC Commerce: Marketing and Salesmanship 12th Standard Board Exam - Maharashtra State Board Important Questions for Mathematics and Statistics

State whether the following statement is true or false.

If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).

Divide 20 into two ports, so that their product is maximum.

Complete the following activity to find MPC, MPS, APC and APS, if the expenditure E_c of a person with income I is given as:

E_c = (0.0003)I² + (0.075)I²

when I = 1000

Slope of the tangent to the curve y = 6 – x² at (2, 2) is ______.

Find the value of x for which the function f(x)= 2x³ – 9x² + 12x + 2 is decreasing.

Given f(x) = 2x³ – 9x² + 12x + 2

∴ f'(x) = `squarex^2 - square + square`

∴ f'(x) = `6(x - 1)(square)`

Now f'(x) < 0

∴ 6(x – 1)(x – 2) < 0

Since ab < 0 ⇔a < 0 and b < 0 or a > 0 and b < 0

Case 1: (x – 1) < 0 and (x – 2) < 0

∴ x < `square` and x > `square`

Which is contradiction

Case 2: x – 1 and x – 2 < 0

∴ x > `square` and x < `square`

1 < `square` < 2

f(x) is decreasing if and only if x ∈ `square`

Find the equation of tangent and normal to the curve y = x² + 5 where the tangent is parallel to the line 4x – y + 1 = 0.

A function f is said to be increasing at a point c if ______.

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

Determine the minimum value of the function.

f(x) = 2x³ – 21x² + 36x – 20

Evaluate the following : `int x^3.logx.dx`

Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`

Evaluate the following.

`int 1/(x(x^6 + 1))` dx 

Evaluate the following.

`int 1/(4x^2 - 20x + 17)` dx

Evaluate the following.

`int x/(4x^4 - 20x^2 - 3) dx`

Evaluate the following.

`int 1/(7 + 6"x" - "x"^2)` dx

Evaluate the following.

`int x^2 *e^(3x)`dx

Evaluate:

`int (2x + 1)/(x(x - 1)(x - 4)) dx`.

Evaluate:

`int x/((x - 1)^2(x + 2)) dx`

`int "dx"/(("x" - 8)("x" + 7))`=

If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______