1 霹雳怡保培南独立中学 SM POI LAM (SUWA) IPOH MID-YEAR EXAMINATION 2022 ADDITIONAL MATHEMATICS DATE: 10 th May 2022 (Tuesday) TIME: 1305 – 1535 (150 minutes) NAME: _________________________ REG. NO: ___________ CLASS: S1 ZI / LI 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 equation e 3 = 6e . [3] 2. Solve the inequality ( + 5)( − 2) > 3 + 6. [3] 3. Without using a calculator, express 6(1 + √3) −2 in the form + √3, where and are integers to be found. [4] 4. Sketch the graph of = − 1 4 (2 + 1)( − 3)( + 4) stating the intercepts with the coordinate axes. [3] 5. By using the substitution = log3 , or otherwise, find the values of for which 3(log3 ) 2 + log3 5 − log3 9 = 0. [6] 6. (a) Given that 3 2 + (1 − 2) = −3, show that, for to be real, 2 − 3 − 9 ≥ 0. [3] (b) Hence find the set of values of for which is real, expressing and simplifying your answer in exact form. [3] 7. The number of bacteria, , present in a culture can be modelled by the equation = 7000 + 2000e −0.05 , where is measured in days. Find (a) the initial number of bacteria, [1] (b) the number of bacteria when = 10, [2] (c) the value of when the number of bacteria reaches 7500, [3] 8. The diagram shows a circle, centre , radius 8 cm. Points and lie on the circle such that the chord = 12 cm and angle = radians. (a) Show that = 1.696, correct to 3 decimal places. [2] (b) Find the perimeter of the shaded region. [3] (c) Find the area of the shaded region. [3]
3 9. The functions f and g are defined by f() = 2 + 1 for > 0, g() = √ + 1 for > −1. (a) Find fg(8). [2] (b) Find an expression for f 2 (), giving your answer in the form + , where , and are integers to be found. [3] (c) Find an expression for g −1 (), stating its domain and range. [4] (d) On the same axes, sketch the graphs of = g() and = g −1 (), including the geometrical relationship between the graphs [3] 10. (a) On the same axes, sketch the graphs of = 5 + |3 − 2| and = 11 − for −6 ≤ ≤ 6. [4] (b) Using the graphs, or otherwise, solve the inequality 11 − < 5 + |3 − 2|. [2] 11. In this question all lengths are in centimetres. In the triangle shown above, = √3 + 1, = √3 − 1 and angle = 60°. (a) Without using a calculator, show that the length of = √6. [3] (b) Show that angle = 15°. [2] (c) Without using a calculator, find the area of triangle . Take sin 60° = √3 2 . [2]
4 12. The expression 2 3 + 2 + + 12 has a factor − 4 and leaves a remainder of −12 when divided by − 1. Find the value of each of the constants and . [5] 13. (a) Given that + 1 is a factor of 3 3 − 14 2 − 7 + , show that = 10. [1] (b) Show that 3 3 − 14 2 − 7 + 10 can be written in the form ( + 1)( 2 + + ), where , and are constants to be found. [2] (c) Hence solve the equation 3 3 − 14 2 − 7 + 10 = 0. [2] 14. The line = − 4, where is a positive constant, passes through the point (0, −4) and is a tangent to the curve 2 + 2 − 2 = 8 at the point . Find (a) the value of , [5] (b) the coordinates of , [3] (c) the length of . [2] 15. Solutions to this question by accurate drawing will not be accepted. The points (, 1), (1,6), (4, ) and (5,4), where and are constants, are the vertices of a kite . The diagonals of the kite, and , intersect at the point . The line is the perpendicular bisector of . Find (a) the coordinates of , [2] (b) the equation of the diagonal , [3] (c) the area of the kite . [3] 16. Answer the whole of this question on a sheet of graph paper. The table shows values of variables and . 10 50 100 200 95.0 8.5 3.0 1.1 (a) By plotting a suitable straight-line graph, show that and are related by the equation = , where and are constants. [4] Use your graph to find (b) the value of , [2] (c) the value of when = 35. [2] --------------------------------------End of Paper------------------------------------
