HSC Commerce (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions

If f'(x) = 4x³ − 3x²  + 2x + k, f(0) = 1 and f(1) = 4, find f(x).

If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).

Evaluate the following.

`int "x" sqrt(1 + "x"^2)` dx

Evaluate the following.

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

Evaluate the following.

`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx

Evaluate the following.

`int (1 + "x")/("x" + "e"^"-x")` dx

Fill in the blank:

The order of highest derivative occurring in the differential equation is called ___________ of the differential equation.

Evaluate the following.

∫ (x + 1)(x + 2)⁷ (x + 3)dx

Fill in the blank:

The power of the highest ordered derivative when all the derivatives are made free from negative and / or fractional indices if any is called __________ of the differential equation.

Evaluate the following.

`int 1/("x" log "x")`dx

Fill in the blank:

Order and degree of a differential equation are always __________ integers.

Evaluate the following.

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

Evaluate the following.

`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx

Evaluate the following.

`int 1/(sqrt"x" + "x")` dx

Evaluate the following.

`int 1/(x(x^6 + 1))` dx 

State whether the following statement is true or false:

Order and degree of a differential equation are always positive integers.

State whether the following is True or False:

The order of highest derivative occurring in the differential equation is called degree of the differential equation.

Evaluate the following.

`int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt

State whether the following is True or False:

The power of the highest ordered derivative when all the derivatives are made free from negative and / or fractional indices if any is called order of the differential equation.

State whether the following is True or False:

The degree of the differential equation `e^((dy)/(dx)) = dy/dx +c` is not defined.