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 ______.

Evaluate the following definite integral:

`int_1^3 logx.dx`

Choose the correct alternative:

`int_2^3 x/(x^2 - 1)  "d"x` =

`int_"a"^"b" "f"(x)  "d"x` = ______

Evaluate:

`int_1^2 1/(x^2 + 6x + 5)  dx`

Evaluate `int_2^3 x/((x + 2)(x + 3))  "d"x`

Evaluate `int_0^"a" x^2 ("a" - x)^(3/2)  "d"x`

By completing the following activity, Evaluate `int_2^5 (sqrt(x))/(sqrt(x) + sqrt(7 - x))  "d"x`.

Solution: Let I = `int_2^5 (sqrt(x))/(sqrt(x) + sqrt(7 - x))  "d"x`     ......(i)

Using the property, `int_"a"^"b" "f"(x) "d"x = int_"a"^"b" "f"("a" + "b" - x)  "d"x`, we get

I = `int_2^5 ("(  )")/(sqrt(7 - x) + "(  )")  "d"x`   ......(ii)

Adding equations (i) and (ii), we get

2I = `int_2^5 (sqrt(x))/(sqrt(x) - sqrt(7 - x))  "d"x + (   )  "d"x`

2I = `int_2^5 (("(    )" + "(     )")/("(    )" + "(     )"))  "d"x`

2I = `square`

∴ I =  `square`

`int_(-5)^5  x^7/(x^4 + 10)  dx` = ______.

`int_a^b f(x) dx = int_a^b f (t) dt`

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