Mock A Math Paper

Mock Preliminary Examination 2024 NAME: __________________________________________________________________________ CLASS: ___________ REGISTER NUMBER: ( ) ADDITIONAL MATHEMATICS 4047/01 Paper 1 20 August 2024 Secondary 4 Express 2 hours Additional Materials: Nil READ THESE INSTRUCTIONS FIRST Write your name, registration number and class on all the work you hand in. Write in dark blue or black pen. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Write your answers in the space provided. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an approved scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 80. This document consists of 18 printed pages

1 Find the range of values of the constant p for which the line y = px – 1 intersects the curve 𝑦 = 2𝑥2 − 5𝑥 + 1 at two points. [4]

2 A trapezoidal prism has a base area of (5 + √5) cm2. The volume of the prism is (20 + 9√5) cm3. Find the height of the prism in the form (𝑎 + 𝑏√5) cm, where a and b are integers. [3]

3 Express 𝑥3+4𝑥2+7 𝑥2(𝑥−1) in partial fractions. [5]

4 The polynomial f(x) = 2𝑥3 + 𝑎𝑥2 + 𝑏𝑥 − 6, where a and b are constants, has two factors (x + 1) and (x – 2). (i) Find the values of a and b. [4]

(ii) Using the values of a and b found in part (i), explain why the equation f(x) has two distinct roots. Find the two roots. [4] (iii) Hence, solve 2𝑦3 + 𝑎𝑦2 + 𝑏𝑦 − 6 = 0. [2]

5 Solve the equation 2ln (x – 3) = ln (x + 5) + ln 4. [5]

6 In the diagram, AB and CB are tangents to the circle at point E and C respectively. The two tangents both meet at point B. Points C, D, F and E lie on the circle. FGC and DGE are straight lines that intercept at point G. ED = EC. (i) Prove that triangle FDG and triangle CEG are similar. [4]

(ii) What type of quadrilateral DCBE? Give reasons to support your answer. [5]