O-LevelAdditional MathematicsDifferentiation and integration4037/22 · May/June 2023 · Q87 Marks

A curve has equation y = cos(x/4) where x is in radians. The normal to the curve at the point where x = 4π/3 cuts the x-axis at the point P. Find the exact coordinates of P.

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

[M2] dy/dx = −(1/4)sin(x/4)
[M1] for dy/dx = k sin(x/4), k<0 or k = 1/4
[B1] y = 0.5
[M1] dy/dx at x=4π/3 = −(1/4)sin(π/3) = −(1/4)(√3/2) or −√3/8 FT (their k) × √3/2 providing at least M1 awarded
[M1] m_normal = 8/√3 soi FT their -1/ (dy/dx at x=4π/3)
[A1] y - 0.5 = (8/√3)(x - 4π/3) oe dep on previous M1; must have exact values
[FT] their m_normal and their 0.5 providing both are non-zero, exact values
[A1] (4π/3 + √3/16, 0) or exact equivalent; mark final answer

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Topic: Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 7 marks.
Source: O-Level Additional Mathematics, May/June 2023 Paper 2 Variant 2, question 8.

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