A-LevelMathematicsDifferential equations9709/31 · May/June 2025 · Q108 Marks

Find the quotient and remainder when x³ + 5x² – 2x – 15 is divided by x² -3.

(a)Find the quotient and remainder when x³ + 5x² – 2x – 15 is divided by x² -3.[3]
(b)The variables x and y satisfy the differential equation dy/dx = (x³+5x²-2x-15) / (6y(x²-3)). It is given that y = 2 when x = 2. Solve the differential equation to obtain an expression for y² in terms of x.[5]

Mark scheme

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

[M1] Divide by (x² - 3) to obtain x+k (k≠0)
[A1] Obtain quotient x + 5
[A1] Obtain remainder x ISW Allow x / (x²-3)
[B1] Separate variables correctly and obtain ∫ 6y dy = ∫ (x³+5x²-2x-15) / (x²-3) dx
[B1FT] Obtain ∫ x² + 5x dx Follow their linear quotient.
[B1FT] Obtain ∫ x / (x²-3) dx From the x term in their remainder ax + b.
[M1] Use y = 2, x = 2 to evaluate the constant of integration in an integral containing k ln(x²-3) 12 = 2 + 10 + (1/2)ln|1| + C => C = 0
[A1] Obtain y² = x² + 5x + (1/2)ln(x²-3) OE

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Topic: Differential equations, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.
Source: A-Level Mathematics, May/June 2025 Paper 3 Variant 1, question 10.

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