1 霹雳怡保培南独立中学 SEKOLAH MENENGAH POI LAM (SUWA) FINAL EXAMINATION 2023 ADDITIONAL MATHEMATICS PAPER ONE ___________________________________________ DATE : 10.11.2023 (FRIDAY) TIME : 1305 – 1505 (2 HOURS) ____________________________________________ NAME : _____________________ REGISTRATION NUMBER: _____________CLASS : S1YI , S1PIN, S1HE, S1AI ________________________________________________________________________________You must answer on the question paper. INSTRUCTIONS Answer all questions. Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs. Write your name, centre number and candidate number at the top of the page. Write your answer to each question in the space provided. Do not use an erasable pen or correction fluid. Do not write on any bar codes. You should use a calculator where appropriate. You must show all necessary working clearly; no marks will be given for unsupportedanswers from a calculator. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal placefor angles in degrees, unless a different level of accuracy is specified in the question. INFORMATION The total number of marks for this paper is 80. The number of marks for each question or part question is shown in brackets [ ]. Do Not Turn Over This Page Until You Are Told To Do So ________________________________________________________________________________This document consists of 15 printed pages (including this page) Prepared by : Ms Koo Kar Kei Checked by : Mr Ong Eik Hooi Signature : _________________ Signature : __________________
2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 + + = 0, = − ± 2 − 4 2 Binomial Theorem + = + 1 −1 + 2 −2 2 + …+ − +. . . +, where n is a positive integer and = !− !! Arithmetic series = + − 1 = 1 2 + = 1 2 2 + − 1 Geometric series = −1 = 1− 1− , ≠ 1 ∞ = 1− , < 1 2. TRIGONOMETRY Identities sin² A + cos² A = 1 sec² A = 1 + tan² A cosec² A = 1 + cot² A Formulae for ABC sin = sin= sin 2 = 2 + 2 − 2 cos ∆ = 1 2 sin
3 1. It is given that = (3 2−2) 2 3−1 , x > 1. (a) Write in the form of = (3 2−2)− 1 3 (−1)2 ( 2 + + ) , where A and B are integers. [5] (b) Find the approximate increase in y as x increases from 2 to 2 + p, where pis small. [2]
4 2. Find the coordinates of the stationary points of the curve 2 y (x 2)(x1) anddetermine the nature of these stationary points. [6]
5 3. The curve = 2 + , where a and b are constants, has a stationary point at (2, 12). (a) Find the value of a and of b. [2] (b) Determine whether (2, 12) is a maximum or a minimum point. [2]
6 4. Figure 1 shows a container is a circular cylinder, open at one end, with a base radiusof r cm and a height of h cm. The volume of the container is 1000 cm3 . Giventhat r and h can vary and that the total outer surface area of the container has a minimumvalue, find this value. [8] Figure 1
7 5. Figure 2 shows a rectangular metal block of length 4x cm, with a cross-sectionareawhich is a square of side x cm. The block is heated and its’ total surface area decreases at a constant rate of 4 cm2s –1 . (a) Express the total surface area, A cm2 , of the block in termof x. [1] (b) Find the rate of which x is decreasing when x = 2. [3] Figure 2
8 (c) Find the rate of decreasing of the volume of the block when x = 2. [4] 6. (a) Sketch, on the same set of axes, the graphs of = cos and =sin 2for 0° ≤ x ≤ 180°. [3] (b) Hence write down the number of solutions of the equation sin 2 − cos = 0 for 0° ≤ x ≤ 180°. [1]
9 7. (a) Sets A and B are such that A = {x : sin x = 0.5 for 0° ≤ x ≤ 360° } , B = {x : cos (x – 30°) = – 0.5 for 0° ≤ x ≤ 360° } . Find the elements of (i) , [2] (ii) ∪ . [2] (b) Set C is such that C = {x : sec 2 3x = 1 for 0° ≤ x ≤ 180° } Find n(C). [3]
10 8. Figure 3 shows three straight lines that form a triangle. AB contains 3 points, AC contains 2 points and BC contains 4 points. Find the number of triangles that can be formed if (a) the vertices of the triangle must be form the points on each straight line. [3] (b) two of the vertices of the triangle must be from the points on the straight lineBC. [3] Figure 3 A B C
11 9. Encik Azlan loves sugary beverages. To support the sugar Reduction Campaign, hewill reduce sugar intake. In the first week, the beverage he drank contained 168gof sugar. For each subsequent week, he reduces his sugar content by one-fourth. (a) How many grams of sugar in his beverages in the 6 th week? [3] (b) What is the total amount of sugar in his beverages in the three months? [3](c) If the sugar reduction practice is continued, estimated the maximumamount of grams of sugar in his beverages. [3]
12 10. (a) (i) Find the first three terms in the expansion of 1 + 7 5 , in ascending powersof x. Simplify the coefficient of each term. [3] (ii) The expansion of 7(1 + ) 1 + 7 5 , where n is a positive integer, is writtenin ascending powers of x. The first two terms in the expansion are 7+89. Find the value of n. [2]
13 (b) In the expansion of ( − 2) 8 , where k is a constant, the coefficient of x 4 divided by the coefficient of x 2 is 5 8 . The coefficient of x is positive. Form an equation and hence find the value of k. [6]
14 11. In this question all lengths are in centimetres and all angles are in radians. (a) The area of a sector of a circle of radius 24cm is 432 cm2 . Find the length of the arc of the sector. [4] (b) Figure 4 shows an isosceles triangle, OAB, with AO = AB = y and height AD. OCD is a sector of the circle with centre O. Angle AOB is α. (i) Find an expression for OB in terms of y and α. [2] Figure 4
15 (ii) Hence show that the area of the shaded region can be written as 2 2 cos α (2 sinα−α cosα ). [4] END OF PAPER
