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

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

`int1/sqrt(x^2 - a^2) dx` = ______

`intsqrt(1+x)  dx` = ______

Solution of the equation `xdy/dx=y log y` is ______

Evaluate the following.

`int x^3 e^(x^2) dx`

Evaluate: 

`int(1+logx)/(x(3+logx)(2+3logx))  dx`

A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.

If the area enclosed by y = f(x), X-axis, x = a, x = b and y = g(x), X-axis, x = a, x = b are equal, then f(x) = g(x).

To solve the problem of maximization objective, all the elements in the matrix are subtracted from the largest element in the matrix.

Calculate the Cost of Living Index Number for the following data.

`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`

`int1/(x+sqrt(x))  dx` = ______

Find `dy/dx, "if"  y=sqrt((2x+3)^5/((3x-1)^3(5x-2)))`

Evaluate `int(3x-2)/((x+1)^2(x+3))  dx`

Complete the following activity to divide 84 into two parts such that the product of one part and square of the other is maximum.

Solution: Let one part be x. Then the other part is 84 - x

Letf (x) = x² (84 - x) = 84x² - x³

∴ f'(x) = `square`

and f''(x) = `square`

For extreme values, f'(x) = 0

∴ x = `square  "or"    square`

f(x) attains maximum at x = `square`

Hence, the two parts of 84 are 56 and 28.

Solve the differential equation (x² + y²) dx - 2xy dy = 0 by completing the following activity.

Solution: (x² + y²) dx - 2xy dy = 0

∴ `dy/dx=(x^2+y^2)/(2xy)`                      ...(1)

Puty = vx

∴ `dy/dx=square`

∴ equation (1) becomes

`x(dv)/dx = square`

∴ `square  dv = dx/x`

On integrating, we get

`int(2v)/(1-v^2) dv =intdx/x`

∴ `-log|1-v^2|=log|x|+c_1`

∴ `log|x| + log|1-v^2|=logc       ...["where" - c_1 = log c]`

∴ x(1 - v²) = c

By putting the value of v, the general solution of the D.E. is `square`= cx

`inte^(xloga).e^x dx` is ______

The area enclosed by the parabola x² = 4y and its latus rectum is `8/(6m)` sq units. Then the value of m is ______.

The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.

`int logx  dx = x(1+logx)+c`