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MHT CET202615 April 2026Morning ShiftMathematicsDifferential EquationsActual

If the differential equation vmatrix f(x) & f'(x) f'(x) & f''(x) vmatrix = 0 for all x and f(0) = 1, f'(0) = 2 , then...

Options

  1. Af'(x) = -f(x)
  2. Bf'(x) = f(x)
  3. Cf'(x) = 2f(x)
  4. Df'(x) = 0

Correct answer

C. f'(x) = 2f(x)

Step-by-step solution

The given differential equation is vmatrix f(x) & f'(x) f'(x) & f''(x) vmatrix = 0 Expanding the determinant, we get f(x)f''(x) - (f'(x))^2 = 0 Dividing both sides by (f(x))^2 , we get f(x)f''(x) - (f'(x))^2 (f(x))^2 = 0 This can be written as d dx ( f'(x) f(x) ) = 0 Integrating both sides with respect to x , we get f'(x) f(x) = c Given f(0) = 1 and f'(0) = 2 Substituting x = 0 , we get c = f'(0) f(0) = 2 1 = 2 Therefore, f'(x) f(x) = 2 f'(x) = 2f(x) Answer: f'(x) = 2f(x)

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