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

`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`

Find `dy/dx` if, y = `x^(e^x)`

Find `dy/dx if, y =  x^(e^x)`

Evaluate the following.

`intx^3  e^(x^2) dx`

Find the area of the regions bounded by the line y = −2x, the X-axis and the lines x = −1 and x = 2.

Complete the following activity:

`int_0^2 dx/(4 + x - x^2) `

= `int_0^2 dx/(-x^2 + square + square)`

= `int_0^2 dx/(-x^2 + x + 1/4 - square + 4)`

= `int_0^2 dx/ ((x- 1/2)^2 - (square)^2)`

= `1/sqrt17 log((20 + 4sqrt17)/(20 - 4sqrt17))`

Three new machines M1, M2, M3 are to be installed in a machine shop. There are four vacant places A, B, C, D. Due to limited space, machine M2 can not be placed at B. The cost matrix (in hundred rupees) is as follows:

Determine the optimum assignment schedule and find the minimum cost.

Determine the minimum value of the function.

f(x) = 2x³ – 21x² + 36x – 20

Find `dy / dx` if, `y = x^(e^x)`

Evaluate the following:

`intx^3e^(x^2)dx` 

Find `dy/dx` if, y = `x^(e^x)`

Evaluate the following.

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