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KVPY2019MathematicsArea Under Curves

Let A 1 ,   A 2 and A 3 be the regions on R 2 defined by A 1 = x , y : x ≥ 0 , y ≥ 0 , 2 x + 2 y - x 2 - y 2 > 1 > x + y A 2 = x , y : x ≥ 0 , y ≥ 0 , x + y > 1 > x 2 + y 2 A 3 = x , y : x ≥ 0 , y ≥ 0 , x + y > 1 > x 3 + y 3 Denote by A 1 ,   A 2 and A 3 the areas of the regions A 1 ,   A 2 and A 3 respectively. Then

Options

  1. AA 1 > A 2 > A 3
  2. BA 1 > A 3 > A 2
  3. CA 1 = A 2 < A 3
  4. DA 1 = A 3 > A 2

Correct answer

C. A 1 = A 2 < A 3

Step-by-step solution

Area of region defined by A 1 = ( x , y ) : x ≥ 0 , y ≥ 0 , 2 x + 2 y - x 2 - y 2 > 1 > x + y is A 1 . Similarly, A 2 is the area of region defined by A 2 = x , y : x ≥ 0 , y ≥ 0 , x + y > 1 > x 2 + y 2 Here, A 1 = A 2 and area A 3 is the area of region defined by A 3 = x , y : x ≥ 0 , y ≥ 0 , x + y > 1 > x 3 + y 3 then A 3 > A 2 = A 1

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