Elegantia College (Sponsored by Education Convergence and patronised by H.K.Yee’s Five Tong Assn.) 2019-2020 Mock Examination Form 6 Mathematics Paper 1 Time Limit：08:30 – 10:45 am (2 hours 15 minutes) Name：_____________________ Class：______ Class No.：_____ Marks： Date ： 2020 – 02 – 25 ____________________________________________________________________________________ 2020-DSE MATH CP PAPER 1 Please stick the barcode label here MATHEMATICS Compulsory Part PAPER 1 Question-Answer Book Candidate Number (2¼ hours) This paper must be answered in English INSTRUCTIONS 1. After the announcement of the start of the examination, you should first write your Candidate Number in the space provided on Page 1 and stick barcode labels in the spaces provided on Pages 1, 3, 5, 7 and 9 (if any). 2. This paper consists of THREE sections, A(1), A(2) and B. 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. 4. Graph paper and supplementary answer sheets will be supplied on request. Write your Candidate Number, mark the question number box and stick a barcode label on each sheet, and fasten them with string INSIDE this book. 5. Unless otherwise specified, all working must be clearly shown. 6. Unless otherwise specified, 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 sticking on the barcode labels or filling in the question number boxes after the ‘Time is up’ announcement. Subject: Mathematics Paper I Setter: CCP,FCY -1- SECTION A(1) (35 marks) a7 a 3 1. Simplify 2 ( ) and express your answer with positive indices. b b 2. 3. x Make x the subject of the formula z = x 2y + 4 . (3 marks) (3 marks) Factorize (a) 4a2 – 9b2, (b) 4a2 – 9b2 – 6a + 9b. (3 marks) Subject: Mathematics Paper I Setter: CCP,FCY -2- Please stick the barcode label here 4. (a) Solve the inequality 1 7x 4 > 4x + 19 3 . (b) Find the number of integers satisfying both the inequalities 1 7x 4 > 4x + 19 3 and 5x + 15 0 . (4 marks) 5. (a) A man intends to deposit $50000 in a bank for 7 years at an interest rate of 2% per annum on simple interest. Find the total interest earned after 7 years. (b) The bank now provides a new savings plan with an interest rate of 1.8% per annum compounded yearly. Will he receive a larger interest by using the new plan? Explain your answer. (4 marks) Subject: Mathematics Paper I Setter: CCP,FCY -3- 6. The ratio of the number of apples to that of oranges in a bag is 7 : 2 . If 3 apples are taken out and 7 oranges are put into the bag, the ratio of the numbers becomes 5 : 3 . Find the number of apples in the bag originally. (4 marks) 7. The weights of three goods are 61450 g , 48900 g and 58530 g , correct to the nearest 10 g . (a) Find the upper limit of the sum of the actual weights of the three goods. (b) It is known that the maximum load of a lift is 170 kg , correct to the nearest kg . Determine whether the lift will overload if all the three goods are placed into the lift. Explain your answer. (4 marks) Subject: Mathematics Paper I Setter: CCP,FCY -4- Please stick the barcode label here 8. In a polar coordinate system, O is the pole. The polar coordinates of point Q are (6, 48). P and R lies on the same polar coordinate plane such that OPQR is a rhombus and PR = 6√3 . (a) Describe the geometric relationship between PR and OQ . (b) If the polar angle of P is greater than 90 but less than 180 , find the polar coordinates of P . (5 marks) 9. In the figure, AB // CD . BAE = 28° and CDE = 112° . A B E C D (a) Find AED . (b) If DE is produced to cut AB at F and DAE is twice of BAE , prove that DAF is similar to AEF . (5 marks) Subject: Mathematics Paper I Setter: CCP,FCY -5- SECTION A(2) (35 marks) 10. The time (in minutes) of doing homework of 18 students is represented by the stem-and-leaf diagram below. Stem (10 mins) Leaf (min) 5 x 6 7 6 1 5 5 6 7 9 7 1 3 4 5 y 8 0 1 2 4 It is known that the range and the mean of the time of doing homework are 29 minutes and 70 minutes respectively. (a) Find the values of x and y . (2 marks) (b) Find the inter-quartile range of the time of doing homework. (1 mark) (c) The time of doing homework of 2 more students is added. If both the mean and the median are not changed, is it possible that the interquartile range remains unchanged? Explain your answer. (3 marks) Subject: Mathematics Paper I Setter: CCP,FCY -6- Please stick the barcode label here 11. Let f(x) = (x – a)2 1 – a2, where a is a constant. (a) Does the graph of y = f(x) cut the x-axis? Explain your answer. (3 marks) (b) It is given that the graph of y = f(x) passes through (a, –10) and its vertex lies on Quadrant IV. Find the value of a . (3 marks) Subject: Mathematics Paper I Setter: CCP,FCY -7- 12. The straight line L1: 2x + 5y + k = 0 passes through A(6, 20) , where k is a constant. the straight line L2: x 2y + 26 = 0. (a) Find the coordinates of G . G lies on It is given that AG is perpendicular to L1 . (4 marks) (b) P and Q are two different points on the y-axis. PAQ . Find the ratio of the areas of PGQ and (2 marks) Subject: Mathematics Paper I Setter: CCP,FCY -8- Please stick the barcode label here 13. Figure (a) shows an inverted right circular conical vessel made by soft plastic sheet. The height of the vessel is 27 cm. The dotted circle XY is parallel to the base circle AB . A , X and V lie on a straight line and AX : XV = 1 : 2 . Someone pushes up the lower portion VXY of the cone along circle XY to form the new vessel in Figure (b). vessel and the vessel is just filled completely. He then pours 900 cm3 of milk into the new V A B X A Y B X V Figure (a) Y Figure (b) (a) Find the original volume of the vessel VAB in Figure (a) in terms of . (5 marks) (b) Someone claims that the area of the wet curved surface of the new vessel in Figure (b) is at least 1200 cm2 . Do you agree? Subject: Mathematics Paper I Explain your answer. Setter: CCP,FCY -9- (2 marks) 14. Let f(x) = ax3 – 5x2 + b , where a and b are constants. When f(x) is divided by x , the remainder is 2 . It is given that f(x) (x 1)(lx + mx + n), where l , m and n are constants. (a) Find l , m and n . (4 marks) 2 (b) It is given that g(x) is the sum of two parts, where one part varies as x3 and the other part varies as x2 . It is given that g(3) = 72 and g(–1) = 8 . (i) Find g(x) . (ii) Someone claims that the equation f(x) = g(x) has only one real root. Explain your answer. Do you agree? (6 marks) Subject: Mathematics Paper I Setter: CCP,FCY - 10 - Section B (35 marks) 15. Let a and b be constants. Denote the graph of y a logb x by G. G passes through the point (256, 3). Express x in terms of y. Subject: Mathematics Paper I Setter: CCP,FCY - 11 - The x-intercept of G is 4 and (5 marks) 16. The password of a computer consists of 2 upper case letters followed by 4 digits. (a) If repeated use of characters is allowed, how many different passwords are possible if there are no restrictions? (1 mark) (b) If repeated use of characters is NOT allowed, how many different passwords are possible if the first character must be neither ‘I’ nor ‘O’ and the last character must not be less than 7 ? (2 marks) Subject: Mathematics Paper I Setter: CCP,FCY - 12 - 17. For a positive integer n, let an denote the maximum number of regions obtained when a circle is cut by n straight lines. The figures below illustrate the cases for n = 1, 2 and 3. a1 = 2 a2 = 4 a3 = 7 (a) Write down the value of a4. (1 mark) (b) By observing the value of an+1 – an for n = 1, 2, 3, or otherwise, deduce that an Subject: Mathematics Paper I Setter: CCP,FCY - 13 - n2 n 2 . 2 (4 marks) 18. In the figure, ABC is a triangular ground, AD is a vertical pole. E is the foot of perpendicular from A to BC. (a) Prove that DE BC. (2 marks) D (b) It is given that AB = 3 m, AC = 2 m, BAC = 60 and AD = 1 m. (i) Find the angle between the planes BCD and ABC. B (ii) F is a variable point on BC. (1) Show that AFD ≤ AED. A (2) Find the smallest possible value of AFD. E (9 marks) C Subject: Mathematics Paper I Setter: CCP,FCY - 14 - Subject: Mathematics Paper I Setter: CCP,FCY - 15 - 19. A point X moves such that it keeps a constant distance of 1 unit from a fixed point I (1, 1). (a) Describe the locus of the point X. (1 mark) (b) The locus of the point X touches OA, OB and AB at P, Q and R respectively, where the coordinates of O, A and B are (0, 0), (4, 0) and (0, 3) respectively. (i) Prove that OPIQ is a square. (ii) A student claims that the point R is the circumcentre of OAB. Do you agree? Explain your answer. (iii) Find the equation of the circumcircle of OAB. (10 marks) Subject: Mathematics Paper I Setter: CCP,FCY - 16 - - END OF PAPER - Subject: Mathematics Paper I Setter: CCP,FCY - 17 -