A-LevelMathematicsDifferentiation9709/21 · Oct/Nov 2020 · Q810 Marks

A curve has equation y = f(x) where f(x) = (4x³ + 8x - 4) / (2x - 1)

(a)Find an expression for dy/dx and hence find the coordinates of each of the stationary points of the curve y = f(x).[5]
(b)Divide 4x³ + 8x – 4 by (2x – 1), and hence find ∫f(x)dx.[5]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

[*M1] Differentiate using the quotient rule (or product rule)
[A1] Obtain (2x-1)(12x²+8) −2(4x³ +8x-4) / (2x-1)² OE
[DM1] Equate first derivative to zero and attempt solution
[A1] Obtain (0,4) Allow if given separately
[A1] Obtain (3/4, 59/8) Allow A1 if both x-coordinates are given, but y coordinates are omitted.
[M1] Carry out division to obtain quotient of form 2x² + kx + m For non-zero constants k, m
[A1] Obtain correct quotient 2x² + x + 9/2
[A1] Obtain remainder 1/2
[M1] Integrate to obtain at least kₓx³ and k₁ln(2x-1) terms For non-zero constants k₁,k₂
[A1] Obtain 2/3x³ + 1/2x² + 9/4x + 1/4ln(2x-1) as final answer Condone absence of ...+c and modulus signs

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