HSC Commerce (English Medium) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

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

`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)`

State whether the following statement is True or False:

`int3^(2x + 3)  "d"x = (3^(2x + 3))/2 + "c"`

State whether the following statement is True or False:

`int"e"^(4x - 7)  "d"x = ("e"^(4x - 7))/(-7) + "c"`

State whether the following statement is True or False:

`int sqrt(1 + x^2) *x  "d"x = 1/3(1 + x^2)^(3/2) + "c"`

Evaluate `int(3x^2 - 5)^2  "d"x`

Evaluate  `int"e"^x (1/x - 1/x^2)  "d"x`

`int x^3"e"^(x^2) "d"x`