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TANJONG KATONG GIRLS’ SCHOOL MID-YEAR EXAMINATION 2010 SECONDARY THREE 4038 ADDITIONAL MATHEMATICS Friday 30 April 2010 Additional Materials: Answer Paper READ THESE INSTRUCTIONS FIRST s h e f a 2 h 15 min C Write your name, class and register number on all the work you hand in. t a M Write in dark blue or black pen on both sides of the paper, and use a pencil for drawing graphs and diagrams. Do not use staples, highlighters or correction fluid. Answer all the questions. Write your answers on the separate writing paper 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. e h The use of a scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. T 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 marks for this paper is 90. Setter : Markers : Miss Yeo LS Miss Lee SN, Miss Sharifah, Mr Seah CS, Miss Yeo LS This Question Paper consists of 4 printed pages, including this page. 2 Answer all questions. Section A [40 marks] 1 Determine the range of values of m for which 4 x m 1x 8 3 x 7 2 has (i) (ii) 2 Given that ln 2 = a and ln 3 = b, 3 (i) (ii) 3 4 e f a 2 real distinct roots, no real roots. express ln 24e numbers, s h C in the form p(qa + rb + s), where p, q, r and s are real t a M find x such that ln x = 1 2a 3b . 3 e h T 3 2 3 2 (a) (b) [3] [6] cm2. Given that the base BC = (6 2 3 ) cm, find the height of the triangle in the form a b 3 cm, where a and b are real numbers. 5 [3] 2 Find the range of values of p for which the curve y x 3x p 2 intersects the line y 9 x 30 at two distinct points. State the value of p for which the line is a tangent to the curve. Triangle ABC has area [5] [1] [6] Solve the inequality x2 x 3 9 , where x > 0. [4] Calculate the range of values of c for which x3 x 4 6 x c for all real values of x. [4] TKGS Sec 3 Mid-year Exam 2010 4038 Additional Mathematics 3 6 At the beginning of 1990, the number of a certain species of birds was estimated to to be 70000. The population decreases so that after a period of n years, the population 0.08 n p was 70000 e . Estimate (i) the population at the beginning of year 2000, [3] (ii) the year in which the population is one-fifth of those presented at the beginning of 1990. s h Section B [50 marks] t a M 2 7 The roots of the equation 3x 1 8 x are and . (i) State the values of + and . (ii) Find the quadratic equation in x whose roots are e f a [5] C [2] 1 1 and , 2 2 2 leaving your answer in the form ax bx c 0 , where a, b and c e h are integers. 8 T 3 [5] 2 It is given that f(x) = 2 x 11x 5 x 18 . (i) (ii) Show that x 2 is a factor of f(x). 3 2 [1] 2 Given that 2 x 11x 5 x 18 = (x 2) (ax bx c) where a, b and c are constants, find the values of a, b and c. Hence solve the equation f(x) = 0. [5] 2 (iii) 3 Use the solutions of f(x) = 0 to solve the equation 2 5 x 18 x 11x . TKGS Sec 3 Mid-year Exam 2010 [4] 4038 Additional Mathematics 4 9 (a) The cubic polynomial f(x) is such that the coefficient of x3 is 2 and the roots of f(x) = 0 are 1, k and k, where k > 0. It is given that f(x) has a remainder of 42 when divided by x 4. (b) (i) Find the value of k. [4] (ii) Show that f(x) has a remainder of 18 when divided by x . [1] x2 + 2x 3 . x 1 10 11 s h x 8 2 21 . (a) Solve the equation 4 (b) 3 2 2 x 3x 22 x 11 Express in partial fractions. 2 x 2 x 3 e h Solve the equation 4x 2 (a) (b) e f a The polynomial f(x) leaves remainders of 2 and 10 when divided by x 1 and x + 3 respectively. Find the remainder when f(x) is divided by (3)7 2x 3 t a M C [5] [5] [6] , [6] 2 log 9 9 x log 3 4 x 8 log 3 x 5 . [6] 8 T THE END TKGS Sec 3 Mid-year Exam 2010 4038 Additional Mathematics