HSC Commerce (Marathi Medium) 12th Standard Board Exam [इयत्ता १२ वी] - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Evaluate the following:

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

Evaluate `int1/(x(x-1))dx` 

Evaluate:

`int(5x^2-6x+3)/(2x-3)dx`

Evaluate the following

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

Evaluate the following.

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

Evaluate the following.

`int 1/ (x^2 + 4x - 5) dx`

Find the differential equation by eliminating arbitrary constants from the relation x² + y² = 2ax

If x + y = 3 show that the maximum value of x²y is 4.

Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx

An annuity immediate is to be paid for some years at 12% p.a. The present value of the annuity is ₹ 10,000 and the accumulated value is ₹ 20,000. Find the amount of each annuity payment

A person sets up a sinking fund in order to have ₹ 1,00,000 after 10 years. What amount should be deposited bi-annually in the account that pays him 5% p.a. compounded semi-annually? [Given (1.025)²⁰ = 1.675]

Four new machines M₁, M₂, M₃ and M₄ are to be installed in a machine shop. There are five vacant places A, B, C, D and E available. Because of limited space, machine M₂ cannot be placed at C and M₃ cannot be placed at A. The cost matrix is given below:

Find the optimal assignment schedule.

Solve the following problem :

Solve the following assignment problem to maximize sales:

Use quantifiers to convert the given open sentence defined on N into a true statement

n² ≥ 1

Use quantifiers to convert the given open sentence defined on N into a true statement.

3x – 4 < 9

Use quantifiers to convert the given open sentence defined on N into a true statement.

Y + 4 > 6

Divide the number 20 into two parts such that their product is maximum

A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.

A metal wire of 36 cm long is bent to form a rectangle. By completing the following activity, find it’s dimensions when it’s area is maximum.

Solution: Let the dimensions of the rectangle be x cm and y cm.

∴ 2x + 2y = 36

Let f(x) be the area of rectangle in terms of x, then

f(x) = `square`

∴ f'(x) = `square`

∴ f''(x) = `square`

For extreme value, f'(x) = 0, we get

x = `square`

∴ f''`(square)` = – 2 < 0

∴ Area is maximum when x = `square`, y = `square`

∴ Dimensions of rectangle are `square`

By completing the following activity, examine the function f(x) = x³ – 9x² + 24x for maxima and minima

Solution: f(x) = x³ – 9x² + 24x

∴ f'(x) = `square`

∴ f''(x) = `square`

For extreme values, f'(x) = 0, we get

x = `square` or `square`

∴ f''`(square)` = – 6 < 0

∴ f(x) is maximum at x = 2.

∴ Maximum value = `square`

∴ f''`(square)` = 6 > 0

∴ f(x) is maximum at x = 4.

∴ Minimum value = `square`