HSC Commerce (English Medium) इयत्ता १२ वी - Maharashtra State Board Question Bank Solutions

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`

`int(xe^x)/((1+x)^2)  dx` = ______

Find `dy/dx,"if"  y=x^x+(logx)^x`

If Mr. Rane order x chairs at the price p = (2x² - 12x - 192) per chair. How many chairs should he order so that the cost of deal is minimum?

Solution: Let Mr. Rane order x chairs.

Then the total price of x chairs = p·x = (2x² - 12x- 192)x

= 2x³ - 12x² - 192x

Let f(x) = 2x³ - 12x² - 192x

∴ f'(x) = `square` and f''(x) = `square`

f'(x ) = 0 gives x = `square` and f''(8) = `square` > 0

∴ f is minimum when x = 8

Hence, Mr. Rane should order 8 chairs for minimum cost of deal.

`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`

Find the area of the region lying in the first quadrant and bounded by y = 4x², x = 0,y = 2 and y = 4.

The rectangle has area of 50 cm². Complete the following activity to find its dimensions for least perimeter.

Solution: Let x cm and y cm be the length and breadth of a rectangle.

Then its area is xy = 50

∴ `y =50/x`

Perimeter of rectangle `=2(x+y)=2(x+50/x)`

Let f(x) `=2(x+50/x)`

Then f'(x) = `square` and f''(x) = `square`

Now,f'(x) = 0, if x = `square`

But x is not negative.

∴ `x = root(5)(2)   "and" f^('')(root(5)(2))=square>0`

∴ by the second derivative test f is minimum at x = `root(5)(2)`

When x = `root(5)(2),y=50/root(5)(2)=root(5)(2)`

∴ `x=root(5)(2)  "cm" , y = root(5)(2)  "cm"`

Hence, rectangle is a square of side `root(5)(2)  "cm"`

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

Evaluate the following.

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

Solve the following

`int_0^1 e^(x^2) x^3 dx`