Exam Tips for andidates on HKSE Mathematics ompulsory Part Question Style in HKSE Paper 1 andidates should pay close attention to the following question types in the examination. Short questions in Section There may be some short questions in Section which carry less marks. Since such questions may not have parts to guide candidates, they are set to differentiate high ability candidates from others. andidates who aim to obtain 5** in the HKSE should strive to master this type of question. Examples in this book: Questions on real-life situations These questions require candidates to apply mathematical knowledge and techniques to real-life situations. Some of them may relate to the topics on Further pplications in the syllabus. Examples in this book: vii Pearson Education sia Limited
Exam Tips for andidates on HKSE Mathematics ompulsory Part Question Style in HKSE Paper ccording to the latest HKSE papers, more questions are set using variables and notations rather than numerical values. andidates should prepare to handle such questions. bout Number and lgebra Examples in this book: bout Measures, Shape and Space Examples in this book: bout ata Handling Examples in this book: x Pearson Education sia Limited
SETIN (35 marks) 15. The graph in Figure 6 shows the linear relation between log 9 x and log 7 y. The slope and the intercept on the horizontal axis of the graph are 4 1 and -4 respectively. Express the relation between x and y in the form y = x k, where and k are constants. (3 marks) log 7 y nswers written in the margins will not be marked. -4 Figure 6 log 9 x nswers written in the margins will not be marked. nswers written in the margins will not be marked. Page total HKSE-MTH-P (Set P1) 16 Pearson Education sia Limited
14. The following stem-and-leaf diagram shows the distribution of the ages of the members of a tennis club. Stem (tens) Leaf (units) 1 7 8 8 9 9 0 0 1 3 3 b 9 3 1 3 5 9 9 4 0 8 8 a It is known that the median and the range of the distribution are 7 and 31 respectively. (a) Find a and b. (3 marks) (b) few days later, both the oldest and the youngest members leave the club, while ennis and his father join the club. It is found that the changes in members have not changed the mode and the median of the distribution. nswers written in the margins will not be marked. (i) (ii) Find the age of ennis s father. Is it possible that ennis is 5 years old? Explain your answer. (iii) The ages of the members of a badminton club are shown as follow: 17,,, 3, 35, 36, 39, 40 ne member from the badminton club and one member from the tennis club are randomly selected to be the club representatives. Find the probability that the sum of the ages of the two representatives exceeds 65. (6 marks) nswers written in the margins will not be marked. nswers written in the margins will not be marked. Page total HKSE-MTH-P (Set 3 P1) 14 Pearson Education sia Limited
8. box contains seven balls marked with the numbers -3, -, -1, 1,, 3 and 4 respectively. If two balls are drawn randomly from the box at the same time, find the probability that the product of the numbers on the balls drawn is positive..... 5 4 1 6 7 3 7 9. The stem-and-leaf diagram below shows the distribution of the numbers of books read by a group of students in the first school term. Stem (tens) Leaf (units) 0 7 1 3 8 9 0 h 4 5 6 8 8 3 0 0 0 0 1 k 7 7 7 8 9 If the inter-quartile range of the above distribution is at least 14, which of the following must be true? I. 0 # h # 4 II. 4 # k # 7 III. 1 # k - h # 3. I only. II only. I and III only. II and III only HKSE-MTH-P (Set 1 P) 11 Pearson Education sia Limited
1. In the figure, is the centre of the circle. If + = 10 and + = 70, find +.. 100. 110. 10. 140. In the figure, is the centre of the circle EF. 3PQR intersects the circle at,,,, E and F. If +PQ = 118 and = = EF, then +PRQ =. 8. P. 56.. 59. F. 6. E R Q 3. If an exterior angle of a regular n-sided polygon is smaller than an interior angle by 90, which of the following is/are true? I. The value of n is 8. II. The number of the diagonals of the polygon is 8. III. The number of axes of reflectional symmetry of the polygon is 8.. I only. II only. I and III only. II and III only HKSE-MTH-P (Set 3 P) 8 Pearson Education sia Limited
Useful Techniques in Tackling MQ Useful Techniques in Tackling MQ In HKSE Mathematics Paper, candidates need to answer 45 multiple choices questions within 1 hour 15 minutes. n average, candidates can only spend about 1.5 minutes on each question. part from direct computation and deductive reasoning, the following six common techniques may help candidates to tackle some of the multiple choice questions quickly. (1) hecking for Questions WITHUT Graphs () hecking for Questions WITH Graphs (3) bserving a Particular ase (4) Substituting a Particular Value (5) Elimination (6) onstruction Technique 1: hecking for Questions WITHUT Graphs Example If the nd term and the 5th term of an arithmetic sequence are 13 and 1 respectively, find the 4th term of the sequence.. 17. 9. 5 Strategy : by hecking. -4 (Reference: HKSE Sample Paper 009 Q36) Solution onsider the options one by one and check which of them satisfies all the given conditions. iv (by hecking) The arithmetic sequence is:, 13,,, 1 ption : The 4th term is 5. Then, the common difference = 1-5 = -4 The following arithmetic sequence obtained satisfies all the given conditions: 17, 13, 9, 5, 1 Ñ The answer is. (by irect omputation) Let a and d be the first term and the common difference of the sequence. Ö nd term = 13 Ñ a + d = 13 (1) Ö 5th term = 1 Ñ a + 4d = 1 () () - (1): 3d = -1 d = -4 From (1), a + d = 13 ( a + d) + d = 13 + d a + 3d = 13 + ( -4) = 5 Ñ The 4th term is 5. Ñ The answer is.
8 Quadratic Equations in ne Unknown and omplex Numbers 8.3 Quadratic Equations Solving quadratic equations by the factor method If (ax + b)(cx - d) = 0, where a, c! 0, then b d x =- or x =. a c Forming quadratic equations with given real roots If a and b are the roots of a quadratic equation in x, then the equation can be written as: (x - a)(x - b) = 0 ommon Mistakes Students may mistakenly write (x + a)(x + b) = 0 when forming a quadratic equation with two given roots a and b. Quick rill 3 If 4(x + 5)(x - 7) = 0, then x = 7. 5 or.. 4 or. 7. -5 or 7.. 4 or 7. Quick rill 4 Form a quadratic equation in x whose roots are -4 and. 5. (x - 4)(5x + ) = 0. (x + 4)(5x - ) = 0. (x - 4)(x + 5) = 0. (x + 4)(x - 5) = 0 Solving quadratic equations by the quadratic formula If ax + bx + c = 0, where a! 0, then b! b 4ac x = - -. a Exam Tips Some M questions require students to find roots of a quadratic equation. If the given options involve surds, that means the equation may not be easily solved by the factor method. In such case, we should solve the equation using the quadratic formula. Quick rill 5 Solve x + 3x - 1 = 0. 3! 5. x = -... 3! 5 x = 3! 13 x = - 3! 13 x = Solving quadratic equations by the graphical method y y = ax + bx + c r 0 s If the x-intercepts of the graph of y = ax + bx + c are r and s, then the roots of the quadratic equation ax + bx + c = 0 are also r and s. ommon Mistakes Some careless students may think a = r and b = s. In fact, the values of a and b cannot be found directly by reading the graph of y = ax + bx + c. x Quick rill 6 Which of the following equations may be represented by the given graph? y. y = (x - 1)(x + 3). y = (x - 1)(x - 3). y = (x + 1)(x + 3). y = (x + 1)(x - 3) 0 x Quick rill nswers: 3. 4. 5. 6. 99
17 asic Properties of ircles HKSE and HKEE Mathematics paper Questions istribution 17.1 hords of a ircle Perpendiculars to chords istances between chords and centre 17. ngles in a ircle ngles at the centre and angles at the circumference ngles in the same segment 17.3 Relationships among rcs, hords and ngles Equal arcs, equal chords and equal angles rcs proportional to angles at the centre 10, Q49 Past Public Exam Questions 05, Q5; 06, Q46; 09, Q48, Q49; Sample Paper, Q1 01, Q3; 0, Q9; 03, Q5; 04, Q50; 05, Q51 17.4 yclic Quadrilaterals 03, Q50, Q51; 07, Q48; 08, Q50; Sample Paper, Q 17.5 Tests for oncyclic Points Exercise Part I Sectional Exercise 17.1 hords of a ircle 1. In the figure, M and N are straight lines. = 10 cm and = 16 cm. Find the area of rectangle MN.. 30 cm. 40 cm. 80 cm. 160 cm. In the figure, M and M are straight lines. If M = M = 8 cm and M = 6 cm, find M.. 10 cm. 1 cm. 15 cm. 16 cm M 10 cm 6 cm 8 cm M N 8 cm 16 cm 3. In the figure, is a diameter of the circle and M is a straight line. If = 1 cm and = 10 cm, then M =. 3 cm.. 11 cm.. 15 cm.. 4 cm. 4. In the figure, M and M are straight lines. If M = M = 1 cm and = 15 cm, find M.. 6 cm. 7.5 cm. 9 cm. 10.5 cm M M 07
17 asic Properties of ircles 40. The figure shows 3. F intersects E at G. If F = and E =, which of the following is/are true? F G E 43. In the figure, is a straight line. If = 3 and = 4, find.. 1.4. 1.6. 1.8. 4 3 I., F, G and E are concyclic. II. E, G, and are concyclic. III., E, and are concyclic.. I only. II only. I and III only. II and III only Part II Miscellaneous Questions 41. In the figure, the radius of the circle is 7. K and K are straight lines. If = 1 and = 10, then K =. 3.. 37.. 41.. 44. 4. In the figure, intersects the chord at M. M = M = 3 and M =. Find M.. 1.5.. 3. 3.5 3 3 M K 44. In the figure, + = + and //. Which of the following is/are true? I. = II. = III. //. I only. II only. I and II only. I, II and III 45. In the figure, M and N are the mid-points of the chords and respectively. NM is a straight line. If : = 1 :, which of the following is/are true? % % I. : = 1 : II. N : M = 1 : III. cos x : cos y = 1 :. I only. III only. I and III only. II and III only M N x y 13
Measures, Shape and Space 60. In the figure, the two circles intersect at G and. E is a triangle. Which of the following is/are true? G F Quiz Time allowed: 30 min 1. In the figure, = 0 and = 6. Find the shortest distance from to chord.. 6. 69. 4. 194 6 0 E I. EF is a cyclic quadrilateral. II. GF is a cyclic quadrilateral. III. 3F ~ 3E. I only. II only. I and III only. I, II and III. The figure shows two concentric circles with common centre. M is a straight line with M = M = 6. = 1 and = 10. Find.. 8 5. 10 5. 5 10. 6 10 M 3. In the figure, //. Find +.. 15. 0. 35. 75 35 4. In the figure, is a diameter of the circle. E and E are straight lines. Find x.. 5. 6. 7. 8 7 x 46 E 16
Mock Test 1 There are 30 questions in Section and 15 questions in Section. The diagrams in this paper are not necessarily drawn to scale. hoose the best answer for each question. Section 356 J1 666 3 1. (- ) K N = L P. -1. 0. 1.. If 3x - 4y = 5xy, then y =.. 1 441 3x 5x + 4.. 3x 5x - 4. 5x + 4 5x - 4... 3x 3x 3. ab - bc - ad + cd =. (a - c)(d - b). (a - c)(b - d). (a + c)(d - b). (a + c)(b - d) 4. Which of the following is an identity/are identities? I. 4x - 9 = 0 II. 4x - 9 = (x - 3) III. 4x - 9 = (x - 3)(x + 3). II only. III only. I and II only. II and III only 5. Let f (x) = -x + x - 1. If f (a) = f (-a), find the value(s) of a.. 0. 1 1 1. 0 or. 0 or - 6. Let f (x) = -x 3 + x - x + k. If x + 1 is a factor of f (x), find the remainder when f (x) is divided by x -.. -18. -1. 18. 4 7. Which of the following about the graph of quadratic function y = -1 - (x - 3) are true? I. The graph has no x-intercepts. II. The coordinates of the vertex are (-3, -1). III. The axis of symmetry of the graph is x - 3 = 0.. I and II only. I and III only. II and III only. I, II and III 8. Find the range of values of k such that the quadratic equation x - 4x + k = has real roots.. k # 4. k $ 4. k # 6. k $ 6 9. If a and b are real numbers such that ab 1 0, which of the following must be true? I. a 1 0 b II. a + b 1 0 III. a - b 1 0. I only. II only. I and III only. II and III only 10. If x is a positive integer satisfying the inequality 3x - 10x - 8 1 0, then the smallest possible value of x is. 0.. 1.. 3.. 4.
Mock Test 1 Section 31. x + x - 3 - = x - 5x + 6 x - x -.... 3x + 11 ( x - 3)( x - )( x + 1) 3x - 7 ( x - 3)( x - )( x + 1) - 3x + 11 ( x - 3)( x - )( x + 1) 9x - 7 ( x - 3)( x - )( x + 1) 3. The graph in the figure shows the linear relation between x and log 4 y. If y = ab x, find the values of a and b. 34. i 01 =. i. -i. 1. -1 35. Which region in the figure represents the Z ] x $ y solutions of [ x + y # 1? ] 3x + y $ 1 \ 1 0 y I (6, 6) III II IV x 4 1 log 4 y 1. Region I. Region II. Region III. Region IV 0. a = 1, b =. a = 1, b = -. a = 4, b = - 1. a = 4, b = 33. Which of the following have the same value as 01 (10)? I. # 10 4 + 0 # 10 3 + 1 # 10 + # 10 1 x 36. Let a, b and c be positive integers. If a, b, c is an arithmetic sequence, which of the following must be true? I. a, ab, ac is an arithmetic sequence. II. a, b, c is a geometric sequence. III. a, b, c is not an arithmetic sequence.. I and II only. I and III only. II and III only. I, II and III II. 7 (16) III. 11111011100 (). I and II only. I and III only. II and III only. I, II and III 37. The sum of all the negative terms in the 1 3 geometric sequence -,, -,... is 3 8 8 3. -.. -. 1 1. 8 -.. 3 3 -. 75 360