The district-Level test of excellent student school year: 2017 - 2018 subject: Mathematicsgrade 8

PEOPLE’S COMMITTEE OF LE CHAN DISTRICT THE OFFICIAL TEST EDUCATION AND TRAINING DEVISION THE DISTRICT-LEVEL TEST OF EXCELLENT STUDENT SCHOOL YEAR: 2017 - 2018 SUBJECT: MATHEMATICSGRADE 8 Time: 120minutes (not count the time of the examination) Note: Examination consists of 09 pages. Students write directly in the examination Mark Full name, signature Code Examiner 1: .................................................................................... ................... ................... ................... ................... ................... .................... Examiner 2: .................................................................................... CODE 1 PART 1: MULTIPLE-CHOICE (150 marks) (Choose the bold word part is the answer to each question) Question 1. The remainder when the polynomial is divided by is A. 135 B. –135 C. D. Question 2.Suppose that 108n is asquare number. Find the least positive integer n. A. 1 B. 3 C. 12 D. 27 Question 3.Which of the following is a factor of the polynomial A. B. C. D. Question 4.Multiply: A. B. C. D. Question 5.The product of polynomial and monomial is A. B. C. D. Question 6. The smallest real number x satisfying the inequality is A. B. C. D. Question 7. 60% of the pupils in the school are girls. 40% of the girls and 80% of the boys in this school can swim. What is the percentage of the pupils in the school who can swim? A. 80% B. 76% C. 56% D. 64% Question 8. Nguyen is 25 years younger than his mother. In 21 years time, Nguyen will be half of his mother’s age. What is the product of their ages now? A. 112 B. 116 C. 136 D. 166 Question 9.Given the expression The coefficient of x in the expansion of P(x) is: A. 1007 B. 1008 C. 1009 D. 1010 Question 10.The polynomial isdivisible by x and x + 1 if and only if A. B. C. D. Question 11. The remainder when 2x3 + kx2 + 7 is divided by x – 2 is half of the remainder when the same expression is divided by 2x – 1. Find the value of k. A. –5 B. –10 C. –15 D. –20 Question 12. Consider the sequence: 1 2 2 3 3 3 4 4 4 4 5 5 5 5 5 6.............. What is the 2018th term? A. 62 B. 63 C. 64 D. 65 Question 13. Out of 80 children, 60 can swim, 54 can play chess and only 10 can do neither. How many children can swim and play chess? A. 34 B. 44 C. 54 D. 24 Question 14. Nancy and Ellen each start reading a copy of Gone with the Wind on the first day of their summer vacation. Nancy decides to read 7 pages each day, but Ellen only wants to read 5 pages each day. When Nancy is on page 84, what page is Ellen reading? A. 35 B. 40 C. 50 D. 60 Question 15. A girl group baked a batch of cookies to sell at the annual of bake sale. They made between 100 and 150 cookies. One fourth of the cookies were lemon crunch and one fifth of the cookies were chocolate macadamia nut. What is the largest number of cookies the group could have baked? A. 110 B. 120 C. 130 D. 140 Question 16. Two positive integers a and b differ by 5. Suppose that the sum of their square roots is also 5. What is the value of 100(a + b)? A. 1300 B. 3600 C. 7200 D.10800 Question 17.Given that , find the value of . A. 25 B. 36 C. 49 D. 64 Question 18. Given that . Find range of the value of . A. B. C. D. CODE Question 19. The smallest value of is: A. B. C. D. Question 20. If xy = 3 and x + y = 5 then A. B. C. D. Question 21. The area of the isosceles trapezoid ABCD is 144cm2. If AB = 12cm and CD = 24cm then AD = A. 7cm B. 8cm C. 9cm D. 10cm Question 22.ABC is a triangle with AB = 5cm, BC = 3cm, and AC = 4cm. Let G be the centroid of triangle ABC. What is the length of AG? A. B. C. D. Question 23. Given the equilateral triangle ABC, AB = 16cm. The area of the triangle ABC is: A. B. C. D. Question 24. Given a right trapezoid MNPQ (MN // PQ, ), PQ = MN + MQ. The measurement of the angle N is: A. 900 B. 600 C. 450 D. 1350 Question 25.The sum of the exterior angles of a polygon is: A. 1800 B. 3600 C. 5400 D. 7200 Question 26. Given a parallelogram ABCD. Let H be orthogonal projection of A on the side DC. If AH = 4cm, AB = 100cm, BC = 5cm then HC = A. 97cm B. 96cm C. 95cm D. 103cm Question 27. The quadrilateral ABCD has two diagonals that are perpendicular lines. If AB = 16cm, BC = 14cm and AD = 8cm then CD = A. 4cm B. 3cm C. 2cm D. 1cm Question 28. Two squares, each with side 8m, are placed such that a vertex of one lies at the centre of the other. Find the area of the overlapping region. A. 48m222232321 B. 16m2 C. 32m2 D. 8m2 Question 29. ABCD is a rectangle. Find the perimeter of ABCD if its area is 66cm2 and AB – BC = 5cm. A. 34cm B. 17cm C. 48cm D. 66cm Question 30. Given a right angle ABC . Let E, D be the midpoint of AB and AC respectively. Let G be the intersection of BD and EC. Find the area of the quadrilateral AEGD. A. 4,5cm222232321 B. 9cm2 C. 18cm2 D. 6cm2 PART 2: ESSAY (150 marks) Question 31. Given a fraction . Prove that there is a polynomial with integer coefficients such that for every which is a root of the polynomial. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. CODE .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . .. .. .. .. .. .. . . Question 32. Given an isosceles at A triangle ABC. Let H be the midpoint of BC. Let E be the point on the opposite ray of the ray CB such that CE = CA. Let I be the point on the ray AB such that AI = HE. Prove that the line IH passes through the midpoint of AE. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. CODE .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . .. .. .. .. .. .. . . Question 33. Let a, b, c be positive real numberssuch that. Prove that .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . .. .. CODE .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . .. .. .. .. .. .. .. ----- TheEnd ----- LE CHAN DIVISION OF EDUCATION AND TRAINING LE CHAN DISTRICT MATHEMATICS COMPETITION School Year 2017 – 2018 ANSWERS AND MARKS Question Answers Marks 1 to 30 CODE1 1 2 3 4 5 6 7 8 9 10 A B D B C A C B D A 50 11 12 13 14 15 16 17 18 19 20 A C B D D A B A B B 50 21 22 23 24 25 26 27 28 29 30 D C C D B A C B A B 50 1 to 30 CODE2 1 2 3 4 5 6 7 8 9 10 50 11 12 13 14 15 16 17 18 19 20 50 21 22 23 24 25 26 27 28 29 30 50 31 20 Therefore, 20 If is a root of then . Thus,. So there is a polynomial that 10 32 Suppose that IH intersects AE at M We have AI = AB + BI = HE = CE + HC = AC + HB so BI = BH Therefore, 25 Then, we also have (2) From (1) and (2), we have , so is an isosceles triangle at M, Because the triangle AHE is right at H, we have MA = ME. 25 33 We have 10 10 Let (). We have 10 15 The equality holds when . 5