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MHT CET202526 Apr 2025Evening ShiftMathematicsApplication of DerivativesActual

If x and y are sides of two squares such that y =x-x^2 , then the rate of change of area of the second square with respect to that of the first square is

Options

  1. A2 x^2+3 x+1
  2. B3 x^2+2 x-1
  3. C2 x^2-3 x+1
  4. D3 x^2+2 x+1

Correct answer

C. 2 x^2-3 x+1

Step-by-step solution

Let x represent the side length of the first square and y the side length of the second. The area of the first square is A₁ = x^2 , and the area of the second is A₂ = y^2 . Given that y = x - x^2 , we express A₂ in terms of x as A₂ = (x - x^2)^2 = x^2(1 - x)^2 = x^2(1 - 2x + x^2) = x^4 - 2x^3 + x^2 . The rate of change dA₂ dA₁ is found using the chain rule: dA₂ dA₁ = dA₂/dx dA₁/dx . Differentiating A₁ = x^2 gives dA₁ dx = 2x . Differentiating A₂ = x^4 - 2x^3 + x^2 yields dA₂ dx = 4x^3 - 6x^2 + 2x . Substituting the

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