2015-DSE MATH CP PAPER 1 LONGMAN MATHEMATICS SERIES Candidate Number HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION 2015 Class MOCK PAPER Class Number MATHEMATICS Compulsory Part PAPER 1 Question-Answer book Time allowed : 2 hours 15 minutes 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, Class and Class Number in the spaces provided. 2. This paper consists of THREE sections, A(1), A(2) and B. 3. Do not write in the margins. Answers written in the margins will not be marked. 4. Unless otherwise specified, all working must be clearly shown. 5. Unless otherwise specified, numerical answers should be either exact or correct to 3 significant figures. 6. The diagrams in this paper are not necessarily drawn to scale. 2015-DSE-MATH-CP 1-1 © Pearson Education Asia Limited 2014 SECTION A(1) (35 marks) 2. Simplify ( x −2 y ) 5 and express your answer with positive indices. x8 y 3 (3 marks) Factorize (a) a2 + a − 6 , (b) a 2 + a − 6 + ab 2 − 2b 2 . (3 marks) Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-2 © Pearson Education Asia Limited 2014 Page total Answers written in the margins will not be marked. Answers written in the margins will not be marked. 1. 3. The table below shows the distribution of the number of credit cards owned by some staff of a company. Number of credit cards 0 1 2 3 4 Number of staff 8 18 20 12 2 Find the median, the mean and the standard deviation of the above distribution. 4. The marked price of a watch is 30% higher than its cost. If the watch is sold at a discount of 20% on its marked price, a profit of $130 is made. Find the cost of the watch. (3 marks) Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-3 © Pearson Education Asia Limited 2014 Page total Answers written in the margins will not be marked. Answers written in the margins will not be marked. (3 marks) 5. Consider the formula (a) 1 3 5 − = . h hk k Make k the subject of the above formula. (b) If the value of h is decreased by 4, write down the change in the value of k. 6. Answers written in the margins will not be marked. Answers written in the margins will not be marked. (4 marks) In Figure 1, △ABC is an equilateral triangle. D and E are points on BC and AC respectively such that AE = CD. AD and BE intersect at the point P. (a) Prove that △ABE ≅ △CAD. (b) Find ∠BPD. (4 marks) Figure1 Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-4 © Pearson Education Asia Limited 2014 Page total 7. In a polar coordinate system, O is the pole. The polar coordinates of points A and B are (6, 35°) and (6, 155°) respectively. Let L be the line which bisects ∠AOB. (a) Is L perpendicular to AB? Explain your answer. (b) Find the polar coordinates of the point of intersection of L and AB. Answers written in the margins will not be marked. Answers written in the margins will not be marked. (5 marks) Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-5 © Pearson Education Asia Limited 2014 Page total 8. A pack of salt is termed standard if its weight is measured as 60 g correct to the nearest 10 g, while a pack of sugar is termed standard if its weight is measured as 55 g correct to the nearest 5 g . (a) Find the maximum absolute errors of the weights of a standard pack of salt and a standard pack of sugar. Answers written in the margins will not be marked. Answers written in the margins will not be marked. (b) Is it possible that the total weight of 20 standard packs of salt is greater than that of 25 standard packs of sugar? Explain your answer. (5 marks) Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-6 © Pearson Education Asia Limited 2014 Page total 9. Let g ( x) = x 3 − x 2 − 11x + c , where c is a constant. When g ( x) is divided by x + 1, the remainder is 12. (a) Is x + 3 a factor of g ( x) ? Explain your answer. (b) David claims that all the roots of the equation g ( x) = 0 are real. Do you agree? Explain your answer. Answers written in the margins will not be marked. Answers written in the margins will not be marked. (5 marks) Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-7 © Pearson Education Asia Limited 2014 Page total SECTION A(2) (35 marks) 10. Let $C be the cost of painting a circular tray of radius r m. It is given that C is the sum of two parts, one part varies directly as r and the other part varies directly as r 2 . When r = 2, C = 16 and when r = 4, C = 40. (a) Find the cost of painting a circular tray of radius 3.5 m. (4 marks) Answers written in the margins will not be marked. If the cost of painting a circular tray is $72, find the radius of the tray. (2 marks) Answers written in the margins will not be marked. (b) Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-8 © Pearson Education Asia Limited 2014 Page total There are 30 students in a class. The stem-and-leaf diagram below shows the test scores of the 30 students. Stem (tens) Leaf (units) 1 a 2 3 1 3 4 5 6 9 5 0 1 2 3 7 9 9 6 0 0 3 4 4 6 6 9 7 1 b 3 5 5 5 8 3 6 9 2 (a) If the range and the inter-quartile range of these scores are 81 and 22 respectively, find the values of a and b. (3 marks) (b) Due to a mistake in recording, the score of a student should be 11 instead of 71. Mary claims that for the two statistical measures mentioned in (a), correcting the score from 71 to 11 will ONLY affect the value of the inter-quartile range. Do you agree? Explain your answer. (3 marks) Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-9 © Pearson Education Asia Limited 2014 Page total Answers written in the margins will not be marked. Answers written in the margins will not be marked. 11. Figure 2 shows the graphs of Peter and Susan walking on the same straight road between city A and city B during the period 5:10 to 11:50 on a certain morning. Susan travels at a constant speed during that period. It is given that city A and city B are 20 km apart. Figure 2 (a) Find the duration of Peter’s resting time. (1 mark) (b) Find the distance between city A and the point where Peter and Susan meet. (3 marks) (c) Peter claims that his average speed is greater than that of Susan during the period 5:10 to 11:50 on that morning. Do you agree? Explain your answer. (2 marks) Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-10 © Pearson Education Asia Limited 2014 Page total Answers written in the margins will not be marked. Answers written in the margins will not be marked. 12. Answers written in the margins will not be marked. Answers written in the margins will not be marked. Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-11 © Pearson Education Asia Limited 2014 Page total 13. (a) Figure 3(a) shows a circle centred at O with radius 5 cm. AB is a chord of the circle with length 8 cm. Find the area of the shaded region. Figure 3(a) Answers written in the margins will not be marked. (b) Figure 3(b) shows a rectangular tank with dimensions 15 cm × 30 cm × 15 cm, filled with x cm3 of gear oil. A solid cylindrical metal rod of base radius 5 cm and length 30 cm is fixed horizontally at the bottom of the tank, as shown in Figure 3(c). The depth of the oil level in Figure 3(c) is 8 cm. Find the value of x. Figure 3(b) Figure 3(c) (4 marks) Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-12 © Pearson Education Asia Limited 2014 Page total Answers written in the margins will not be marked. (4 marks) Answers written in the margins will not be marked. Answers written in the margins will not be marked. Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-13 © Pearson Education Asia Limited 2014 Page total 14. In the rectangular coordinate plane, the coordinates of points A, B, C and R are (−1, 0), (7, 6), (7, 0) and (3, 3) respectively. P is a moving point in the rectangular coordinate plane such that AP is perpendicular to BP. Denote the locus of P by Γ. (a) (i) Find the equation of Γ. (ii) Describe the geometric relationship between Γ and R. (4 marks) (i) Find the distance between Q and R. (ii) Winston claims that the ratio of the area of △ARC to the area of △AQC is 1 : 4. Do you agree? Explain your answer. (5 marks) Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-14 © Pearson Education Asia Limited 2014 Page total Answers written in the margins will not be marked. The equation of the straight line L is 4x + 3y + 104 = 0. It is found that Γ and L do not intersect. Let Q be a point lying on L such that Q is nearest to R. Answers written in the margins will not be marked. (b) Answers written in the margins will not be marked. Answers written in the margins will not be marked. Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-15 © Pearson Education Asia Limited 2014 Page total Section B (35 marks) 15. On the sports day of a school, the timekeeping group for running events consists of 1 chief judge, 1 referee and 8 timekeepers. The chief judge and the referee are chosen from 4 teachers while the 8 timekeepers are selected from 12 students. The 12 students include some secondary 4 and secondary 5 students. (a) How many different timekeeping groups can be formed? (2 marks) Answers written in the margins will not be marked. If there are 9 secondary 4 students among the 12 students, find the probability that all the timekeepers in the timekeeping group are secondary 4 students. (2 marks) Answers written in the margins will not be marked. (b) Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-16 © Pearson Education Asia Limited 2014 Page total 16. (a) Let k be a non-zero real constant. 1 in the form a + bi , where a and b are real numbers. Express 1 − ki (2 marks) (b) The roots of the quadratic equation x 2 + px + q = 0 are 5 5 and . 1 − 3i 1 + 3i Find p and q. Answers written in the margins will not be marked. Answers written in the margins will not be marked. (3 marks) Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-17 © Pearson Education Asia Limited 2014 Page total 17. (a) In Figure 4, the equations of the four straight lines are x + y = 17 , x + y = 24 , x = 18 and y = 15 . The shaded region (including the boundary) represents the solution of a system of inequalities. Find the system of inequalities. Figure 4 (b) A merchant has two warehouses P and Q. Warehouse P stores 24 tonnes of flour and warehouse Q stores 16 tonnes of flour. Two cake shops A and B place orders for 18 tonnes and 15 tonnes of flour respectively. The costs of transporting each tonne of flour from the warehouses to the cake shops are shown in the following table: To From Warehouse P Warehouse Q Cake shop A Cake shop B $50 $20 $30 $10 Suppose the merchant transports x tonnes and y tonnes of flour from warehouse P to cake shops A and B respectively. The merchant claims that the transportation cost can be less than $ 900. Do you agree? Explain your answer. (4 marks) Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-18 © Pearson Education Asia Limited 2014 Page total Answers written in the margins will not be marked. Answers written in the margins will not be marked. (2 marks) Answers written in the margins will not be marked. Answers written in the margins will not be marked. Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-19 © Pearson Education Asia Limited 2014 Page total 18. Figure 5 shows a vertical tower AB of height 35 m. A police officer at B is observing the traffic condition along the straight road PQR. A, P, Q and R are points on the horizontal ground. (a) The police officer at B finds that the angle of depression of P from B is 8° while the angle of depression of Q from B is 6°. It is known that ∠PAQ = 145° and Q is due east of P. Find the distance between P and Q. (4 marks) (b) A vehicle travels along the road PR at a uniform speed. The police officer observes that the vehicle only takes 15 seconds to travel from P to Q and suspects that it has exceeded the speed limit of the road. At the very moment when the vehicle passes point Q, a patrol car at A starts moving to R along road AR to stop the vehicle. After two minutes, the patrol car meets the vehicle at R. Find the average speed of the patrol car. (5 marks) Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-20 © Pearson Education Asia Limited 2014 Page total Answers written in the margins will not be marked. Answers written in the margins will not be marked. Figure 5 Answers written in the margins will not be marked. Answers written in the margins will not be marked. Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-21 © Pearson Education Asia Limited 2014 Page total 19. Stanley joins a savings scheme offered by a bank. At the beginning of 2015, Stanley has to put an initial deposit of $P into his account. Every year afterwards, he has to deposit 10% more than he did in the previous year. The interest rate is 5% per annum, compounded yearly. (a) (i) Find, in terms of P, the amounts in Stanley’s account at the end of 2015, 2016 and 2017. (Note: You need not simplify your answers.) (ii) Show that the amount in Stanley’s account at the end of the nth year is $21P (1.1n − 1.05 n ) . Answers written in the margins will not be marked. (b) Stanley plans to buy a parking space which is worth $500 000 at the end of 2014. Suppose the value of the parking space increases by 5% every year. Stanley joins the savings scheme with $P = $50 000 at the beginning of 2015. At the end of which year will Stanley receive an amount from the savings scheme that is just enough to buy the parking space? Explain your answer. (4 marks) Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-22 © Pearson Education Asia Limited 2014 Page total Answers written in the margins will not be marked. (7 marks) Answers written in the margins will not be marked. Answers written in the margins will not be marked. Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-23 © Pearson Education Asia Limited 2014 Page total Answers written in the margins will not be marked. Answers written in the margins will not be marked. END OF PAPER Answers written in the margins will not be marked. 2015-DSE-MATH-CP 1-24 © Pearson Education Asia Limited 2014 Page total