HSC Commerce (Marathi Medium) १२ वीं कक्षा - Maharashtra State Board Important Questions for Mathematics and Statistics

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 (2x + 1)/((x + 1)(x - 2))  "d"x`

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

`int 1/sqrt(x^2 - 9) dx` = ______.

State whether the following statement is true or false.

If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2e^x – 5| + c, where c is the constant of integration, then A = 5.

`int x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`

The value of `int ("d"x)/(sqrt(1 - x))` is ______.

Choose the correct alternative:

Area of the region bounded by the parabola y² = 25x and the lines x = 5 is ______

The area of the shaded region bounded by two curves y = f(x), and y = g(x) and X-axis is `int_"a"^"b" "f"(x) "d"x + int_"a"^"b" "g"(x)  "d"x`

Find the area of the region bounded by the curve y = `sqrt(9 - x^2)`, X-axis and lines x = 0 and x = 3

Find the area of the region bounded by the curve y = (x² + 2)², the X-axis and the lines x = 1 and x = 3

Find the area of the region bounded by the curve x = `sqrt(25 - y^2)`, the Y-axis lying in the first quadrant and the lines y = 0 and y = 5

The slope of a tangent to the curve y = 3x² – x + 1 at (1, 3) is ______.

The area of the region bounded by the curve y = x², x = 0, x = 3, and the X-axis is ______.

Find the area between the two curves (parabolas)

y² = 7x and x² = 7y.

The differential equation of `y = k_1e^x+ k_2 e^-x` is ______.

Solve the differential equation `("d"y)/("d"x) + y` = e^−x 

Solve the differential equation (x² – yx²)dy + (y² + xy²)dx = 0

Solve: `("d"y)/("d"x) + 2/xy` = x² 

Choose the correct alternative:

The integrating factor of `("d"^2y)/("d"x^2) - y` = e^x, is e^–x, then its solution is