A-LevelMathematicsDifferentiation9709/22 · May/June 2025 · Q69 Marks

A curve has equation (x²-3)lny+6x = 14.

(a)Show that there is no point on the curve at which the y-coordinate is e⁻¹.[3]
(b)Find the equation of the tangent to the curve at the point (2, e²). Give your answer in the form y = mx +c, where m and c are exact constants.[6]

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

[*M1] Substitute y = e⁻¹ in equation and simplify to quadratic equation in x
[DM1] Evaluate discriminant OE, such as completion of square or use of formula.
[A1] Obtain -x² +6x−11=0, giving discriminant –8 and confirm result OE, such as (x−3)² + 2 = 0, etc. AG - necessary detail needed.
[*M1] Attempt use of product rule for differentiation of (x² – 3) lny May see in part (a)
[A1] Obtain 2x lny + (x² - 3)/y dy/dx OE
[A1] Obtain complete 2x lny + (x² - 3)/y dy/dx + 6 = 0 FT Following their derivative of (x² – 3) lny.
[DM1] Substitute x = 2 and y = e² to find value of gradient of tangent Need to see attempt at substitution unless correct. -103 implies M1.
[A1] Obtain gradient -14e²
[A1] Obtain equation y = -14e²x+29e² or equivalent of required form

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