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MHT CET202617 April 2026Evening ShiftMathematicsApplication of DerivativesActual

If the line y = 4x - 5 is tangent to the curve y^2 = ax^3 + b at the point (2,3) , then the value of 7a - 2b is...

Options

  1. A0
  2. B7
  3. C14
  4. D28

Correct answer

D. 28

Step-by-step solution

The point (2,3) lies on the curve y^2 = ax^3 + b . Substituting x = 2 and y = 3 into the curve equation: 3^2 = a(2^3) + b 8a + b = 9 Differentiating the curve equation with respect to x : 2y dy dx = 3ax^2 dy dx = 3ax^2 2y At the point (2,3) , the slope of the tangent is: dy dx = 3a(2^2) 2(3) = 2a The given tangent line is y = 4x - 5 , which has a slope of 4 . Equating the slopes: 2a = 4 a = 2 Substituting a = 2 into 8a + b = 9 : 8(2) + b = 9 b = -7 The value of 7a - 2b is: 7(2) - 2(-7) = 14 + 14 = 28 Answer: 28

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