Vasu has a rectangular farmland A. The length of this farmland is x²y - 2xy + 3x² and its perimeter is 2x²y + 6x² - 2xy - 4y². Today he purchased the adjacent farmland B and combined the two farmlands into one. The length of his farmland now increased by x² + xy, while the breadth remained the same.

(1) Find the breadth of farmland A :

1. x²y + 3x² - 4y²
2. xy - 2y²
3. 12x² - 4xy
4. 6x² + xy - 2y²

(2) Find the length of the combined farmland owned by Vasu :

1. x² + 2xy - 2y²
2. 4x² - 3xy + x²y
3. 4x² - xy + x²y
4. 2x² - 3xy + x²y

(3) The perimeter of Vasu's combined farmland is :

1. 8x² + 2x²y - 4y²
2. 4y² + 4xy + 2x²y + 8x²
3. 4x² - 2x²y + 4y²
4. 4x² + 4xy - 2x²y - 8x²

(4) The change in the perimeter after combining farmlands A and B is :

1. 4x² + 2xy + 4x²y - y²
2. 2x² - 2xy + y²
3. 4x² + 2xy - y²
4. 2x² + 2xy

Given for Farmland A:

Length (L_A) = x²y - 2xy + 3x²

Perimeter (P_A) = 2x²y + 6x² - 2xy - 4y²

(1)

We know the formula:

Perimeter of a rectangle = 2(Length + Breadth)

$\Rightarrow \text {Breadth} = \dfrac{\text{Perimeter - 2(Length)}}{2}$

First, find 2(Length):

2(Length) = 2(x²y - 2xy + 3x²) = 2x²y - 4xy + 6x²

Now, calculate Perimeter - 2(Length):

$\begin{array}{rcccccc} 2x^2y & + & 6x^2 & - & 2xy & - & 4y^2 \\ +2x^2y & + & 6x^2 & - & 4xy & + & 0 \\ -\phantom{2x^2y} & - & & + & & - \\ \hline 0 & & 0 & & 2xy & - & 4y^2 \\ \hline \end{array}$

Perimeter - 2(Length) = 2xy - 4y²

Now we have:

Breadth = $\dfrac{2xy - 4y^2}{2}$

Breadth = xy - 2y²

Hence, option 2 is the correct option.

(2)

Original Length (L_A) = x²y - 2xy + 3x²

Increase = x² + xy

Combined Length = Original Length + Increase

Substituting the values above, we get:

Combined Length = (x²y - 2xy + 3x²) + (x² + xy)

= (3x² + x²) + (- 2xy + xy) + x²y $\quad$[Arranging like terms together]

= (3 + 1)x² + (-1)xy + x²y

= 4x² - xy + x²y

Hence, option 3 is the correct option.

(3)

Combined Length = 4x² - xy + x²y $\quad$[From step 2]

Breadth = xy - 2y² $\quad$[From step 1]

Perimeter of combined farmland = ?

Let's apply the perimeter of a rectangle formula:

Perimeter of combined farmland = 2(Length + Breadth)

Let's first calculate Length + Breadth.

We have:

Length + Breadth = (4x² - xy + x²y) + (xy - 2y²)

= 4x² + (- xy + xy) + x²y + (- 2y²) $\quad$[Arranging like terms together]

= 4x² + 0xy + x²y - 2y²

= 4x² + x²y - 2y²

Now we have:

Perimeter of combined farmland = 2 x (4x² + x²y - 2y²) = 8x² + 2x²y - 4y²

Hence, option 1 is the correct option.

(4)

Original Perimeter = 2x²y + 6x² - 2xy - 4y²

Combined Perimeter = 8x² + 2x²y - 4y²

Change in perimeter = Combined Perimeter - Original Perimeter

Substituting the values above, we get:

Change in perimeter = (8x² + 2x²y - 4y²) - (2x²y + 6x² - 2xy - 4y²)

= 8x² + 2x²y - 4y² - 2x²y - 6x² + 2xy + 4y² $\quad$[Simplifying brackets]

= (8x² - 6x²) + (2x²y - 2x²y) + 2xy + (- 4y² + 4y²) $\quad$[Arranging like terms together]

= 2x² + 0x²y + 2xy + 0y²

= 2x² + 2xy

Hence, option 4 is the correct option.