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O-LevelAdditional MathematicsDifferentiation and integration4037/12 · Oct/Nov 2024 · Q56 Marks

Show that (1+cot²θ)/cot²θ = sec²θ.

  1. (a)Show that (1+cot²θ)/cot²θ = sec²θ.[1]
  2. (b)Write down the derivative of tanθ with respect to θ.[1]
  3. (c)Using part (a) and part (b), find the exact value of ∫₀^(π/3) ( (1+cot²θ)/cot²θ - sinθ ) dθ.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

  1. [B1] Sufficient detail is needed (Alternative 1: sin²θ+cos²θ / sin²θ / cos²θ / sin²θ = 1/cos²θ = sec²θ. Alternative 2: 1/cot²θ+1 = tan²θ+1 = sec²θ)
  2. [B1] sec²θ
  3. [B1] ∫(sec²θ-sinθ)dθ soi
  4. [B2] B1 for each: tanθ + cosθ
  5. [B1] √3 - 1/2 or exact equivalent

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About this question

Topic
Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 6 marks.
Source
O-Level Additional Mathematics, Oct/Nov 2024 Paper 1 Variant 2, question 5.

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