HSC Commerce (Marathi Medium) इयत्ता १२ वी - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

State whether the following statement is True or False:

If y = 7x + 1, then the rate of change of demand (x) of a commodity with respect to its price (y) is 7

State whether the following statement is True or False:

If y = x², then the rate of change of demand (x) of a commodity with respect to its price (y) is `1/(2x)`

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 5 + x²e^–x + 2x

Find rate of change of demand (x) of a commodity with respect to its price (y) if y = `(3x + 7)/(2x^2 + 5)`

The rate of change of demand (x) of a commodity with respect to its price (y), if y = 20 + 15x + x³.

Solution: Let y = 20 + 15x + x³

Diff. w.r.to x, we get

`("d"y)/("d"x) = square + square  + square`

∴ `("d"y)/("d"x)` = 15 + 3x²

∴ By derivative of the inverse function,

`("d"x)/("d"y)  1/square, ("d"y)/("d"x) ≠ 0`

∴ Rate of change of demand with respect to price = `1/(square + square)`

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 ______.

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 function f(x) = `3/x` + 10, x ≠ 0 is decreasing

The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing

State whether the following statement is True or False: 

If the function f(x) = x² + 2x – 5 is an increasing function, then x < – 1

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

Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing

By completing the following activity, find the values of x such that f(x) = 2x³ – 15x² – 84x – 7 is decreasing function.

Solution: f(x) = 2x³ – 15x² – 84x – 7

∴ f'(x) = `square`

∴ f'(x) = 6`(square) (square)`

Since f(x) is decreasing function.

∴ f'(x) < 0

Case 1: `(square)` > 0 and (x + 2) < 0

∴ x ∈ `square`

Case 2: `(square)` < 0 and (x + 2) > 0

∴ x ∈ `square`

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

`int (7x + 9)^13  "d"x` ______ + c

To find the value of `int ((1 + logx))/x` dx the proper substitution is ______

State whether the following statement is True or False:

If `int x  "f"(x) "d"x = ("f"(x))/2`, then f(x) = `"e"^(x^2)`