1 霹雳怡保培南独立中学 SM POI LAM (SUWA) IPOH FINAL EXAMINATION 2021 ADDITIONAL MATHEMATICS DATE: 18 th November 2021 (Thursday) TIME: 1305 – 1535 (150 minutes) NAME: _________________________ REG. NO: ___________ CLASS: J3 ZI READ THESE INSTRUCTIONS FIRST Write your name, class and student number on all the work you hand in. Write in dark blue or black pen only. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. All answers are to be written on both sides of the writing paper provided. Omission of essential working will result in loss of mark. Calculators should be used where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answers to three significant figures. Give answers in degrees to one decimal place. At the end of the examination, fasten all your work securely together. The number of marks are given in brackets [ ] at the end of each question. The total mark for this paper is 100. Do Not Turn Over This Page Until You Are Told To Do So This paper consists of 4 printed pages (including this cover page) Prepared by: Checked by: _________________ __________________ (Mr. Chen Wai Mun) (Mr. Ong Eik Hooi)
2 1. Solve the following simultaneous equations. 4 2 + 3 + 2 = 8 + 4 = 0 [6] 2. The function is defined, for all real , by () = 13 − 4 − 2 2 . (a) Write () in the form + ( + ) 2 , where , and are constants. [3] (b) Hence write down the range of . [1] 3. Write (27 3 ) 5 3 √81 5 4 in the form 3 × × where , and are constants. [3] 4. DO NOT USE A CALCULATOR IN THIS QUESTION. () = 2 3 − 3 2 − 23 + 12 (a) Find the value of ( 1 2 ) . [1] (b) Write () as the product of three linear factors and hence solve () = 0. [5] 5. Solve the following inequalities. (a) |3 + 2| > 8 + [3] (b) (2 − 3)( + 2) < 4 [3] 6. Find the values of constant for which (2 − 1) 2 + 6 + + 1 = 0 has real roots. [5] 7. Find the exact values of the constant for which the line = 2 + 1 is a tangent to the curve = 4 2 + + − 2. [6] 8. Solve the following equations. (a) |3 − 5| = | + 2| [3] (b) |4 − 1| + | + 2| = 10 [4] 9. A function is defined by : ↦ 2 + 2 − 1 , ≠ 1. Find the value of −1 (3). [3] 10. Given that = 2(4 2 ) − 6(4 +1 ) + 65, find the value of when = 1. [4]
3 11. DO NOT USE A CALCULATOR IN THIS QUESTION. In this question all lengths are in centimetres. In the diagram above = √3 − 1, = √3 + 1, angle = 15° and angle = 90°. (a) Show that tan 15° = 2 − √3. [3] (b) Find the exact length of . [2] 12. The diagram shows the graph of = (), where () is a cubic polynomial. (a) Find (). [3] (b) Write down the values of such that () < 0. [2] 13. DO NOT USE A CALCULATOR IN THIS QUESTION. A curve has equation = (2 − √3) 2 + − 1. The -coordinate of a point on the curve is √3 + 1 2 − √3 . Show that the coordinates of can be written in the form ( + √3, + √3), where , , and are integers. [5]
4 14. The functions and are defined, for > 0, by () = 2 2 − 1 3 () = 1 (a) Find and simplify an expression for (). [2] (b) Given that −1 exists, write down the range of −1 . [1] (c) Show that −1 () = + √ 2 + 4 , where , and are integers. [4] 15. The polynomial () = 6 3 + 2 + + 2, where and are integers, has a factor of − 2. (a) Given that (1) = −2(0), find the value of and . [5] Using your values of and from part (a), (b) find the remainder when () is divided by 2 − 1, [2] (c) factorise () completely. [2] 16. () = 2 + 2 − 3 for ≥ −1 (a) Given that the minimum value of 2 + 2 − 3 occurs when = −1, explain why () has an inverse. [1] (b) Sketch the graph of = () and the graph of = −1 () on the same diagram. Label each graph and state the intercepts on the coordinate axes. [4] 17. (a) On the same axes, sketch the graphs of = |2 − 3| and = + 3. [3] (b) Using your graph to write down the set of values of that satisfy the inequality |2 − 3| ≥ + 3. [2] 18. (a) Sketch the graph of = |(2 − 3) 2 − 4|. [3] (b) Find the set of values of for which |(2 − 3) 2 − 4| = has four solutions. [1] 19. (a) Sketch the graph of = |( 2 − 4)( − 2)|, stating the intercepts with the coordinate axes. [4] (b) Write down number of solutions should be obtained for which |( 2 − 4)( − 2)| = 8. [1] --------------------------------------End of Paper------------------------------------
