O-LevelAdditional MathematicsDifferentiation and integration4037/21 · May/June 2022 · Q1010 Marks

Differentiate xlnx-2x with respect to x. Simplify your answer.

(a)Differentiate xlnx-2x with respect to x. Simplify your answer.[2]
(b)A curve is such that d²y/dx² = (x+1)²/√x. It is given that dy/dx = e²/2 + 2e at the point (e, e³/6 + e²). Using your answer to part (a), find the exact equation of the curve.[8]

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

[B2] ln x - 1
[B1] for [lx]ln x + xx⁻¹ - [-2] oe
[M2] dy/dx = x²/2 + lnx + 2x (+c) (M1 for dy/dx = ... + lnx + ... or for dy/dx = x²/2 + ... + 2x)
[M1] Substitution to find c: e²/2 + 2e = e²/2 + ln e + 2e + c dependent on at least M1
[A1] [c=-1]
[M1] y = ∫(x²/2 + 2x + lnx - 1)dx
[M1] Integrates and uses (a) y = x³/6 + x² + xlnx - 2x + C Dependent on M2M1 FT error in c only
[M1] Substitution to find C: e³/6 + e² = e³/6 + e² + e ln e - 2e + C Dependent on previous M1
[A1] y = x³/6 + x² + xlnx - 2x + e

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Topic: Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 10 marks.
Source: O-Level Additional Mathematics, May/June 2022 Paper 2 Variant 1, question 10.

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