UPSC CDS 2 2022 Math Paper

The perimeter of square =4a
Perimeter of rectangle =2(1+b)
Area of square =a2
Area of rectangle =1×b

Let the side of the square be a and the length and breadth of the rectangle be 1 and b respectively.
The perimeter of a square = perimeter of a rectangle
⇒4a=2(1+b)
⇒2a=(1+b)
⇒a=(1+b)∕2‌‌‌⋅⋅⋅⋅⋅⋅⋅(i)
They differ in area = 1 square cm
⇒a2−lb=1
⇒[(1+b)∕2]2−lb=1
⇒(l2+b2+2‌lb)∕4−lb=1
⇒12+b2+2‌lb−4‌lb=4
⇒12+b2−2‌lb=4
⇒(1−b)2=4‌‌‌‌[∵(a−b)2=a2+b2−2‌ab]
⇒1−b=2