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HARRISON COLLEGE INTERNAL EXAMINATION, APRIL 2013 CARIBBEAN ADVANCED PROFICIENCY EXAMINATION SCHOOL BASED ASSESSMENT PURE MATHEMATICS UNIT 2 - TEST 1 TIME: 1 Hour & 30 minutes This examination paper consists of 2 printed pages. The paper consists of 3 questions. The maximum mark for this examination is 60. INSTRUCTIONS TO CANDIDATES 1. Write your name clearly on each sheet of paper used. 2. Answer ALL questions. 3. Number your questions carefully and do NOT write your solutions to different questions beside one another. 4. Unless otherwise stated in the question, any numerical answer that is not exact, MUST be written correct to three (3) significant figures. EXAMINATION MATERIALS ALLOWED 1. Mathematical formulae 2. Electronic calculator (non-programmable, non-graphical) 1. (a) (b) One root of the quadratic equation π§ 2 + ππ§ + π = 0, where p and q are real, is the complex number (3 − π). (i) Write down the other root. [1] (ii) Find the values of p and q. [4] Use De Moivre’s theorem to express (√3 + π)5 in the form (−π√3 + ππ) where a is a real number. (c) (d) [5] Sketch and describe the following loci in separate Argand diagrams: (i) |π§ − 3 − 4π|2 = 9 [2] (ii) |π§ − 1| = |π§ − π| [2] (iii) π§ + π§ ∗ = 4 where π§ ∗ is the conjugate of z. [2] By expressing πππ π in terms of π ππ and π −ππ , show that 1 πππ 5 π = 16 (πππ 5π + 5 cos 3π + 10πππ π) [5] TOTAL 21 marks 1 2. (a) (b) ππ¦ Find ππ₯ π₯3 (i) π¦= π (ii) π¦= when: + π‘ππ−1 (π₯) [3] πππ₯ [3] π ππ−1 π₯ The equation of a curve is given by 4π¦ − π₯ = π₯π¦ Find the equation of the tangent to the given curve at the point (3, 3). (c) [4] A curve is defined by the parametric equations π¦ = π‘ 3 − 3π‘ and π₯ = 2π‘ Find in terms of t (i) (ii) (d) ππ¦ [3] ππ₯ π2 π¦ [2] ππ₯ 2 Let π(π₯, π¦) = 4π₯ − 5π₯ 2 π¦ 3 + 2π¦ 2 , find π2 π [2] ππ₯ππ¦ TOTAL 17 marks 3. (a) (i) Given that (π₯ − 2) is a factor of π₯ 3 − 3π₯ 2 + 4 completely factorise π₯ 3 − 3π₯ 2 + 4 (b) 3 (ii) Express f(x) = (iii) Hence find ∫ π(π₯)ππ₯ π₯ 3 −3π₯ 2 +4 [2] in partial fractions. [4] [3] It is given that for non-negative integers n, π πΌπ = ∫1 (πππ₯)π ππ₯ (i) Show that for π ≥ 1 πΌπ = π − ππΌ(π−1) (ii) (c) [4] Find πΌ3 in terms of e. [4] Use the trapezium rule with 3 intervals of equal width to estimate the value of 3 ∫0 √π₯ 2 + 3 ππ₯ . Give your answer to 2 decimal places. [5] TOTAL 22 marks 2