HSC Commerce (English Medium) इयत्ता १२ वी - Maharashtra State Board Important Questions for Mathematics and Statistics

Choose the correct alternative:

The function f(x) = x³ – 3x² + 3x – 100, x ∈ R is

The price P for the demand D is given as P = 183 + 120D − 3D², then the value of D for which price is increasing, is ______.

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

The slope of tangent at any point (a, b) is also called as ______.

If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______

The total cost function for production of articles is given as C = 100 + 600x – 3x², then the values of x for which the total cost is decreasing is  ______

State whether the following statement is True or False:

The equation of tangent to the curve y = x² + 4x + 1 at (– 1, – 2) is 2x – y = 0 

Find the values of x such that f(x) = 2x³ – 15x² + 36x + 1 is increasing function

Find the values of x such that f(x) = 2x³ – 15x² – 144x – 7 is decreasing function

A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.

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 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`

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 ______.