Square and rectangle with the same perimeter whose areas differ: difference of length and breadth

This type of question, asked 2 times: CDS 2 2022, CDS 1 2026. All 2 questions below, oldest first.

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Question 1
CDS 2 2022
89 $A square and a rectangle have same perimeter. They differ in areas by 1 square cm. The length of the rectangle exceeds its breadth by$
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Question 2
CDS 1 2026
62 $A square and a rectangle have the same perimeter p and their areas differ by q units. What is the square of the difference between the length and breadth of the rectangle ?$
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