2023-S1 MATH CP Candidate Name: (δΈζε ¨ε) PAPER 1 Candidate Name: (English Full Name) MATHEMATICS REVISION SERIES 2023 Candidate Number: S1 MATHEMATICS Compulsory Part FINAL-EXAM PAPER 1 Question-Answer Book Date: / / 2023 Time Slot: Phone Number: Time allowed: 1 hour 45 minutes This paper must be answered in English INSTRUCTIONS (1) After the announcement made by the invigilator, you should first write your Name, Candidate Number, Examination Date, Time Slot and your Phone Number in the spaces provided on Page 1. Do not start doing the paper until you are told to do so. (2) This paper consists of THREE sections, A, B and C. Question Marker’s Use Only Examinar’s Use Only Marker No. Examiner No. Mark Mark 1 5 (3) Attempt ALL questions in this paper. Write your answers in the spaces provided in this Question-Answer Book. Do not write in the margins. Answers written in the margins will not be marked. 2 5 3 5 (4) Supplementary answer sheets and Graph Paper will be supplied on request. Write your Name, Examination Date and Time Slot, and mark the question number box on each sheet. 4 5 5 6 6 7 7 7 8 5 9 6 10 8 11 8 12 13 13 40 Total 120 (5) Rough Work Sheet will be supplied to all candidates. Write your Candidate Name on the sheet. You are not allowed to take away the Rough Work Sheet after the examination. (6) Unless otherwise specified, all working must be clearly shown and numerical answers should be either exact or correct to 3 significant figures. (7) The diagrams in this paper are not necessarily drawn to scale. (8) No extra time will be given to candidates for writing information after the ‘Stop working’ announcement. (9) There are 3 bonus marks in this paper. However, the maximum mark of this paper is still 120. (10) The following is the suggested time for doing this paper. Section A: 20-25 minutes Section B: 30-35 minutes Section C: 50-55 minutes MATH-CP-S1-FINAL-1 1 SECTION A (40 marks) 1. In a table tennis club, there are 25000 balls and the number of white balls is 50% more than the number of orange balls. Find the difference between the number of white balls and the number of orange balls. (5 marks) _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ 2. The length and breadth of a rectangle are π₯ ππ and π¦ ππ respectively. It is given that the length increases by 10% and the breadth decreases by 10%. (a) Express the new area of the rectangle in terms of π₯ and π¦. (b) Someone claims that the percentage change in the area is 0%. Do you agree? Explain your answer. (5 marks) _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ Answers written in the margins will not be marked. MATH-CP-S1-FINAL-1 2 3. Consider the sequence –8, –19, –30, –41, …. (a) Write down the general term of the sequence. (b) Find the 64th term of the sequence. (c) Determine whether –1989 is a term of the sequence. Explain your answer. (5 marks) _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ 4. It is given that an iron rod is no longer than 50 cm. It is then bent into a rectangle of length π₯ cm and width π¦ cm. (a) Set up an inequality to represent the above situation. (b) Can the iron rod form a rectangle of length 16 cm and width 10 cm? Explain your answer. (c) If the width of the rectangle formed is 11 cm, write down two possible lengths of the rectangle formed.(6 marks) _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ 5. Answer the following question. (5 marks) Part Question Answer (a) Round up 121.008310 to 2 significant figures. (a) (b) Round down 120.038310 to 1 decimal place. (b) (c) Round up 0.079500 to 1 significant figure. (c) (d) Correct 131.28063 to 3 decimal places. (d) (e) Evaluate 0. 3 × 8 ÷ 4 + 3 5 (e) ×2 Answers written in the margins will not be marked. MATH-CP-S1-FINAL-1 3 6. π In the figure, π΄π΅π· is a straight line ∠π΅πΆπ΄ = 60 πππ ∠π΄π΅πΆ = 3∠π΅π΄πΆ. (a) Prove that π΄π· is perpendicular to π΅πΆ. (b) It is given that ∠π΄π·πΆ = 60 − π π π₯ 100 , where π₯ is positive. Someone claims that ∠π΄πΆπ· is an acute angle. Do you agree? Explain your answer. (7 marks) _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ Answers written in the margins will not be marked. MATH-CP-S1-FINAL-1 4 7. 12π¦+14 9π¦−3 = 16 3 . (a) Solve for the equation (2 marks) (b) Hence, solve for the following set of equations. 7π₯ + 6π¦ = 14............................................... (1) 12π¦ + 14 = 16(3π¦ − 1)............................ (2) 54π¦ − 36π¦π§ + 42 81 (5 marks) π₯π§ = –43..................... (3) _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ Answers written in the margins will not be marked. MATH-CP-S1-FINAL-1 5 SECTION B (40 marks) 8. Consider a regular n-sided convex polygon. (a) Express each interior angle of a regular polygon in terms of π. (2 marks) (b) Find the sum of all exterior angles in a regular nonagon. (3 marks) _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ 9. In the figure, a metal solid rectangular block is melted and recast to form a triangular prism. (a) Find the value of π. (2 marks) (b) Find the total surface area of rectangular block and hypotenuse of the triangular prism. (4 marks) _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ Answers written in the margins will not be marked. MATH-CP-S1-FINAL-1 6 10. The following figure shows the stem-and-leaf diagram of the exam scores of 30 students in a school. (a) If 14 students scored less than 66, find the values of π₯ and π¦ . (2 marks) _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ (b)(i) Complete the following table. Data Answer Lowest exam score (1) Difference between the highest exam score and the lowest exam score (2) Your expected score in this paper π (3) /120 (ii) The above table is used to construct a pie chart of the exam scores of 30 students in the school. Find π. _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ (iii) If 52 is the passing score of this exam. Find the percentage of students who failed the exam. (6 marks) _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ Answers written in the margins will not be marked. MATH-CP-S1-FINAL-1 7 11. For question 11, choose EITHER part (a) or part (b) to answer. NO extra marks will be given for doing BOTH parts. 2 2 15 6 25−3π +3(π −10) −5 −4×13÷2 (a) Consider the expression (i) (π π π ) π −6 2 0 17 3π π π π . Simplify the following expression and express your answer with positive indices, where π, π, π ≠ 0. (5 marks) (ii) (b) If π = 3π and π = π, simplify the expression and express your answers in terms of π. (3 marks) 2 Consider the polynomial (π₯ − 2π¦)(3π₯ + 4π¦) − (3π₯ − 4π¦) . (i) Expand and simplify the above polynomial. Arrange the terms in ascending powers of π₯. (ii) (1) (2) (3 marks) If π₯ = π¦, express the polynomial in terms of π₯. 2 Let π = (π₯ − 2π¦)(3π₯ + 4π¦) − (3π₯ − 4π¦) . Is it possible that π has a value larger than 0 for all 2 2 real values of π₯? Explain your answer. (Hint: Is π₯ ≥ 0 ππ π₯ ≤ 0 ?) (5 marks) The part you have chosen (Circle it): Part (a) / Part (b) . _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ Answers written in the margins will not be marked. MATH-CP-S1-FINAL-1 8 _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ Answers written in the margins will not be marked. MATH-CP-S1-FINAL-1 9 12. A retailer sells Apple and Samsung phones. Both the marked price of each iPhone 14 Pro Max and Samsung Galaxy S23 Ultra are $9600. iPhone 14 Pro Max is sold at a profit of 20%, while Samsung Galaxy S23 Ultra is sold at a loss of 20%. (a) Find the cost of an iPhone 14 Pro Max and a Samsung Galaxy S23 Ultra. (4 marks) (b) Originally, the retailer bought 45 iPhone 14 Pro Max and 60 Samsung Galaxy S23 Ultra at their cost price. However, the wholesalers wrongly gave 45 iPhone 14 Pro Max, 35 Samsung Galaxy S23+ and 25 Samsung Galaxy S23 Ultra. (i) Find the total costs of buying 45 iPhone 14 Pro Max and 60 Samsung Galaxy S23 Ultra. (ii) To ensure the retailer has an overall 5% profit in this transaction. The retailer, therefore, decided to increase the marked price of an iPhone 14 Pro Max by 45% and sell them at a 5% discount. Given that the selling price of each Samsung Galaxy S23+ is $8275. Let the new selling price of a Samsung Galaxy S23 Ultra be π. (1) Express the new total selling price of all 60 phones in terms of π . (2) If the new selling price of the Samsung Galaxy S23 Ultra is higher than the original marked price, the customers would not choose to buy it. Can the retailer sell the Samsung Galaxy S23 Ultra at the new selling price to the customer? Explain your answer. (9 marks) _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ Answers written in the margins will not be marked. MATH-CP-S1-FINAL-1 10 _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ Answers written in the margins will not be marked. MATH-CP-S1-FINAL-1 11 SECTION C (40 marks + 3 bonus marks) Answer ALL parts in this question in the Question 13 Answer Book. Try your best! 13. Consider the following triangular number sequence ππ : 1 , 3 , 6 , 10 … (a) Write down the 5th term of the sequence and deduce the general term ππ of the sequence. (3 marks) (b) In a rectangular coordinate plane, a point π ( π , ππ ) is moving in the plane, where n is a positive integer such that the π represents the π th term of the sequence and y-coordinate of π is indeed the general term ππ found in part (a). A vertical line Γ passes through π' ( 5 , π5 ) cuts x-axis at πΎ. The horizontal line πΏ passes through π΅( 1 , 3 ) intersects with Γ at π. πΏ cuts the y-axis at π. Denote the origin by π. Given that ππ΅π' is a straight line. (i) Prove that βπππ΅ ~ βπ'ππ΅. (ii) If the length of ππ΅ is π‘ units. Express π'π΅ in terms of π‘. (iii) Hence, or otherwise, find the ratio of the area of βπππ΅ to βπ'ππ΅. (8 marks) (c) A student, Matthew, wants to find the shortest distance from π to π'π, he lets ππΈ be the shortest distance, where πΈ is a point on straight line ππ'. (i) By finding the area of βπ'ππ and using the result of (b), express ππΈ in terms of π‘. (ii) Matthew claims that βππΈπ΅~βπππ΅. Using the reason “ratio of 2 sides, included angle”, prove that he is correct. (7 marks) (d) Given that the vertical distance from πΈ to πΏ is 0.3 units. (i) Using the result of (b) and (c), find: 2 (1) the value of π‘ , and (2) prove that π‘ = 3.16. (ii) [This part consists of 3 bonus marks] Find the perimeter of βπππ΅. (iii) Find the total surface area of a prism with βπππ΅ as the base. The height of the prism is 30 cm. (9 marks) and [3 bonus marks] (e) Write down 3 triangles which are also similar to βππΈπ΅ other than βπππ΅. (3 marks) (f) Matthew then cuts out βπππ΅ and βπ'ππ΅ and uses the two triangles to build two triangular prisms with the two triangles as their bases respectively. Using the Euler formula ‘ ππππ‘ππππ + πΉππππ − πΈππππ = 2’ , find the number of faces in a triangular prism. (g) (2 marks) It is known that if any two figures have the same side ratios, then the two figures are also similar to each other. (i) Explain why it is impossible when π moves to a point π'' such that π'' intersects with πΏ at π (14 , 3) and trapezium ππ΅ππΎ is similar to trapezium ππ'π''π . (ii) In order to make trapezium ππ΅ππΎ similar to trapezium ππ'π''π ' , π'' undergoes certain transformation(s). Assume π'' can translate to any points in the rectangular coordinate plane. Describe how π'' can be transformed such that the two trapeziums are similar to each other. (8 marks) Answers written in the margins will not be marked. MATH-CP-S1-FINAL-1 12 Question 13 Marking Grid Marker Total Mark (out of 40) Part Mark Full Mark (a) 3 (b) 8 (c) 7 (d) 9 (e) 3 (f) 2 (g) 8 bonus 3 [Note: You are advised to skip ONE line for EACH PART of the question and present the answer CLEARLY, for easy marking!] [Note: For question 13, NO need to write the units for EVERY answer.] Mark Grid _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ Answers written in the margins will not be marked. MATH-CP-S1-FINAL-1 13 Mark Grid _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ Answers written in the margins will not be marked. 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MATH-CP-S1-FINAL-1 18 Mark Grid _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ Choose the difficulty of this paper (Circle the suitable answer) Very difficult / Difficult / Standard / Easy / Very Easy -END OF PAPERAnswers written in the margins will not be marked. MATH-CP-S1-FINAL-1 19 (0 marks) Do not write on this page. Answers written on this page will not be marked. MATH-CP-S1-FINAL-1 20
