HSC Commerce: Marketing and Salesmanship 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

State whether the following statement is True or False:

If f'(x) = 3x² + 2x, then by definition of integration, we get f(x) = x³ + x² + c 

If f(x) = k, where k is constant, then `int "k"  "d"x` = 0

State whether the following statement is True or False:

y² = 4ax is the standard form of parabola when curve lies on X-axis

State whether the following statement is True or False:

Standard form of parabola is x² = – 4by, when curve lies in the positive Y-axis

Find the area between the parabolas y² = 5x and x² = 5y

Choose the correct alternative:

Solution of the equation `x("d"y)/("d"x)` = y log y is

Choose the correct alternative:

Differential equation of the function c + 4yx = 0 is

Choose the correct alternative:

General solution of `y - x ("d"y)/("d"x)` = 0 is

A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constant is called ______ solution

The function y = e^x is solution  ______ of differential equation

The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______

State whether the following statement is True or False:

The integrating factor of the differential equation `("d"y)/("d"x) - y` = x is e^–x 

The function y = cx is the solution of differential equation `("d"y)/("d"x) = y/x`

Solve the following differential equation `("d"y)/("d"x)` = x²y + y

Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0

Solve the following differential equation

`y log y ("d"x)/("d"y) + x` = log y

Verify y = `a + b/x` is solution of `x(d^2y)/(dx^2) + 2 (dy)/(dx)` = 0

y = `a + b/x`

`(dy)/(dx) = square`

`(d^2y)/(dx^2) = square`

Consider `x(d^2y)/(dx^2) + 2(dy)/(dx)`

= `x square + 2 square`

= `square`

Hence y = `a + b/x` is solution of `square`

Solve the following differential equation 

sec² x tan y dx + sec² y tan x dy = 0

Solution: sec² x tan y dx + sec² y tan x dy = 0

∴ `(sec^2x)/tanx  "d"x + square` = 0

Integrating, we get

`square + int (sec^2y)/tany  "d"y` = log c

Each of these integral is of the type

`int ("f'"(x))/("f"(x))  "d"x` = log |f(x)| + log c

∴ the general solution is

`square + log |tan y|` = log c

∴ log |tan x . tan y| = log c

`square`

This is the general solution.

Solve the following differential equation `("d"y)/("d"x)` = cos(x + y)

Solution: `("d"y)/("d"x)` = cos(x + y)    ......(1)

Put `square`

∴ `1 + ("d"y)/("d"x) = "dv"/("d"x)`

∴ `("d"y)/("d"x) = "dv"/("d"x) - 1`

∴ (1) becomes `"dv"/("d"x) - 1` = cos v

∴ `"dv"/("d"x)` = 1 + cos v

∴ `square` dv = dx

Integrating, we get

`int 1/(1 + cos "v")  "d"v = int  "d"x`

∴ `int 1/(2cos^2 ("v"/2))  "dv" = int  "d"x`

∴ `1/2 int square  "dv" = int  "d"x`

∴ `1/2* (tan("v"/2))/(1/2)` = x + c

∴ `square` = x + c

Find the particular solution of the following differential equation

`("d"y)/("d"x)` = e^2y cos x, when x = `pi/6`, y = 0.

Solution: The given D.E. is `("d"y)/("d"x)` = e^2y cos x

∴ `1/"e"^(2y)  "d"y` = cos x dx

Integrating, we get

`int square  "d"y` = cos x dx

∴ `("e"^(-2y))/(-2)` = sin x + c₁

∴ e^–2y = – 2sin x – 2c₁

∴ `square` = c, where c = – 2c₁ 

This is general solution.

When x = `pi/6`, y = 0, we have

`"e"^0 + 2sin  pi/6` = c

∴ c = `square`

∴ particular solution is `square`