Edexcel IALMathematicsGeneral Applications9 Marks

A curve C has equation y = x³ - 6x² + 5x - 2.

(a)Find ∣±y/∣±xⁱ.[3]
(b)Find the equation of the tangent to the curve C at the point P(1, -2), giving your answer in the form y = mx + c.[4]
(c)Verify that the tangent to the curve at the point Q(5, -2) has equation y = 20x - 102.[2]

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

M1 for attempting to differentiate y = x³ - 6x² + 5x - 2 to the form Ax² + Bx + C.
A1 for ∣±y/∣±xⁱ = 3x² - 12x + 5 (any two terms correct, unsimplified).
A1 for ∣±y/∣±xⁱ = 3x² - 12x + 5 (fully correct).
M1 for substituting x=1 into their ∣±y/∣±xⁱ to find the gradient of the tangent at P.
A1 for gradient = 3(1)² - 12(1) + 5 = -4.
M1 for using y - y₁ = m(x - x₁) with their gradient and point P(1, -2).
A1 for y - (-2) = -4(x - 1) leading to y = -4x + 2.
M1 for substituting x=5 into their ∣±y/∣±xⁱ to find the gradient at Q.
A1 for showing that gradient at Q is 3(5)² - 12(5) + 5 = 75 - 60 + 5 = 20, and showing that the equation y - (-2) = 20(x - 5) simplifies to y = 20x - 102.

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