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

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).

y² = (x + c)³ is the general solution of the differential equation ______.

Write the converse, inverse, and contrapositive of the statement. "If 2 + 5 = 10, then 4 + 10 = 20."

Solve the following LP.P.

Maximize z = 13x + 9y,

Subject to 3x + 2y ≤ 12,

x + y ≥ 4,

x ≥ 0,

y ≥ 0.

Conditional of p → q is equivalent to p → ∼ q.

If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.

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

The optimal value of the objective function is attained at the ______ of feasible region.

`int (f^'(x))/(f(x))dx` = ______ + c.

`int(7x - 2)^2dx = (7x -2)^3/21 + c`

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`

The set of feasible solutions of LPP is a ______.

If b_xy < 0 and b_yx < 0 then 'r ' is ______.

Solution which satisfy all constraints is called ______ solution.

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

The degree of the differential equation `((d^2y)/dx^2)^2 + (dy/dx)^3` = a^x is 3.

Converse of the statement q `rightarrow` p is ______.

`int(1 - x)^(-2)` dx = `(1 - x)^(-1) + c`

Evaluate `int(1 + x + x^2/(2!) )dx`

Evaluated the following

`int x^3/ sqrt (1 + x^4 )dx`