Answer:

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Let's break down each part of the problem:

(a) To find the area of Durga's square land, you can use the formula for the area of a square, which is side length squared. In this case, the side length is 64 meters.

[tex]{ \red{ \boxed{ \tt \: Area \: of \: square = Side \: length^2}}}[/tex]

Area = 64²

[tex]{ \boxed{ \bold{= 4096 \: square \: meters}}}[/tex]

So, the area of her square land is 4096 square meters.

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(b) If Durga wants to exchange her square land for a rectangular land of equal area, and the length of the rectangular land is 128 meters, you can find the breadth of the rectangular land by dividing the area by the length.

[tex]{ \red{ \boxed{ \tt \: Area \: of \: rectangle = Length × Breadth}}}[/tex]

4096 square meters = 128 meters × Breadth

Now, solve for Breadth:

[tex] \tt \: Breadth = \dfrac{4096 \: square \: meters}{128 \: meters} [/tex]

[tex]{ \boxed{ \bold{= 32 \: meters}}}[/tex]

So, the breadth of the rectangular land is 32 meters.

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(c) To find the required length of wire to fence the rectangular land three times, you need to calculate the perimeter of the rectangular land and then multiply it by 3 (since it needs to be fenced three times).

[tex]{ \red{ \boxed{ \tt \: Perimeter \: of \: rectangle = 2 × (Length + Breadth)}}}[/tex]

Perimeter = 2 × (128 meters + 32 meters)

Perimeter = 2 × 160 meters

[tex]{ \underline{ \tt \: Perimeter = 320 \: meters}}[/tex]

Now, to fence it three times:

Required length of wire = 3 × Perimeter

Required length of wire = 3 × 320 meters

[tex]{ \boxed{ \bold{= 960 \: meters}}}[/tex]

So, the required length of wire to fence the rectangular land tree times is 960 meters.

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(d) If Durga wanted to fence the rectangular field one time for Rs 31,000 but Rs 1,000 was not enough, we need to find the cost of fencing per meter.

Let C be the cost per meter. So, the total cost for fencing is given by:

Total cost = Length of fencing × Cost per meter

320 meters × C = Rs 31,000 - Rs 1,000

320C = Rs 30,000

Now, divide both sides by 320 to find the cost per meter:

[tex] \tt \: C = \dfrac{(Rs \: 30,000) }{320} [/tex]

[tex]{ \boxed{ \bold{C = Rs \: 93.75 \: per \: meter}}}[/tex]

So, the cost of fencing the land per meter is Rs 93.75.

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[tex]{ \orange{ \sf \: hope \: it \: helps \: you}}[/tex]