G8 Math Middle Exam Paper 2023-2024 PDF

# Diocesan Boys' School ## 2023-24 Mid-year Examination ### Grade 8 Mathematics #### Part I (Conventional Questions) **PP, u.** Dec 2023 Name: Cheung Yung Class: 85 Class Number: 7 ### Instructions: 1. Write your name, class and class number in the spaces provided. 2. Time allowed: 1 hour 40 minu...

# Diocesan Boys' School ## 2023-24 Mid-year Examination ### Grade 8 Mathematics #### Part I (Conventional Questions) **PP, u.** Dec 2023 Name: Cheung Yung Class: 85 Class Number: 7 ### Instructions: 1. Write your name, class and class number in the spaces provided. 2. Time allowed: 1 hour 40 minutes (Part I and Part II) 3. This part carries a total of 100 marks. 4. Answer the questions in the spaces provided in this Question-Answer Booklet. Supplementary answer sheets will only be provided upon request. 5. Unless otherwise specified, all working must be clearly shown. 6. The diagrams in this paper are not necessarily drawn to scale. Measurement does not constitute as answer or solution unless otherwise stated in the question. 7. Unless otherwise specified, numerical answers should be exact. 8. The use of HKEAA approved calculators or non-CAS version of TI-inspire calculator is allowed. ## Section A: Answer-only Questions (20 marks) Write down the answers clearly. No working step is required. | Answer | | |---------------------------|---| | 6.41×10⁻² | 1 | | | 2 | | 4 | 3 | | 343 cm³ | 4 | | (5x-3)/(x+1)(x-1) | 5 | | 3 | 6 | | 48 | 7 | | (2a+3b)(a-3b) | 8 | | | 9 | | 40x(x+2)² | 10 | ## Section B: Short Questions (30 marks) All working steps must be shown clearly. 1. Simplify $x^2y/-7x^3y^{-2}$ and express your answer with positive indices. (4 marks) 2. Solve $1+y/4 = y/3$. (3 marks) 3. (a) Prove that $(a^n+1)(a^n-1)=a^{2n}-1$ where $a≠0$. (2 marks) (b) Hence, simplify $2a^n/(a^{2n}-1)(a^n-1)$. (3 marks) 4. (a) Factorise $4a^3-12a^2b$. (1 mark) (b) Hence, factorize $4a^3-12a^2b-ab^2+3b^3$. (4 marks) 5. It is given that $(x−2)(x^2+5x−1)= Bx(x + 1)^2 − 7x(x + A) + A$, where A and B are constants. Find A and B. (4 marks) 6. Ian has two pieces of wire of the same length. He bents one piece of wire into an equilateral triangle and then bents the other piece of wire into a square. The length of a side of the triangle is 6 cm longer than that of the square. Find the area of the square. (4 marks) 7. The marked price of a bag is higher than its cost by $400. The bag is sold at a discount of 25% on its market price. After selling the bag, the percentage loss is 10%. Find the marked price of the bag. (5 marks) ## Section C: Long Questions (max. 50 marks) All working steps must be shown clearly. 1. (a) Factorize: (i) $b-ab^2$ (1 mark) (ii) $b^2-1/2$ (1 mark) (iii) $1 -a^2/ab^2$ (2 marks) (b) Using the results of (a), simplify $(b^2 −1/2)÷(1 −a^2/ab^2 )÷(b − ab^2)$. (3 marks) 2. (a) Solve $3^{3x+1} = 81$ (3 marks) (b) Solve $9(3^{2x}) − 80(3^{3x+1})− 81 = 0$. (4 marks) (c) Hence, or otherwise, solve $3^{3x+1}/3^{2x} = 80$. (3 marks) 3. A house is with 2 bedrooms. The size of the large bedroom is 120 m³. (a) Given that there are about 3.4×10⁹ dust particles per m³, find the number of dust particles present in the large bedroom. Express your answer in scientific notation. (2 marks) (b) If the number of dust particles in the small bedroom is about 9.82×10¹⁰. Find the estimated total number of dust particles present in two bedrooms. Express your answer in scientific notation. (3 marks) (c) If the mass of a dust particle is about 7.53×10⁻¹⁰ g, find the total weight of the dust particles in two bedrooms. Express your answer in scientific notation and correct to 3 significant figures. (3 marks) (d) The total number of dust particles in these two bedrooms is reduced to half of the original after every vacuum cleaning. Assume that a vacuum cleaning bag can collect 1 kg of dust particles, at most how many times of vacuum cleaning can be done before the vacuum cleaning bag is full? (2 marks) 4. In this question, answer EITHER part (A) or part (B). DO NOT answer both parts. The full marks of part (A) and part (B) are 12 and 7 respectively. If both parts are attempted, only part (A) will be marked. A. (a) Prove that $ax^2 + bx = a(x + b/2a)^2 − b^2/4a$ where a, b ≠ 0. (2 marks) (b) Using the result of part (a), if a < 0, show that $ax^2 + bx ≤ b^2/4a$. (2 marks) (c) The figure below shows a right-angled triangle XYZ where XY = 12 cm and XZ = 16 cm. A rectangle XABC with XC = h cm is cut from ∆XYZ. (i) Given that XZ : CZ = YX : BC, find BC in terms of h. (2 marks) (ii) Find the area of the rectangle XABC in terms of h. (2 marks) (iii) Ivan claims that it is possible to cut a rectangle with the area of 49 cm². Do you agree? Explain your answer. (4 marks) B. The figure below shows a right-angled triangle PQR where PR = 12 cm and PQ = 8 cm. A rectangle PABC with CR = x cm is cut from ∆PQR, where 0 < x < 6. (a) Given that $CR/PR = BC/QP$, find BC in terms of x. (2 marks) (b) Find the area of the rectangle PABC in terms of x. (2 marks) (c) If the area of the rectangle PABC is 18 cm², find CR. (3 marks) 5. In this question, answer EITHER part (A) or part (B). DO NOT answer both parts. The full marks of part (A) and part (B) are 11 and 6 respectively. If both parts are attempted, only part (A) will be marked. A. (a) Solve $(x − p)(x − q)(x − r) = 0$. (3 marks) (b) If $(1/x) + (1/y) + (1/z) = (1/a)$, show that $xyz = a(yz + xz + xy)$. (2 marks) (c) Given that $x + y + z = 3$ and $(1/x) + (1/y) + (1/z) = (1/3)$, where x, y, z ≠ 0. Using the results of (a) and (b), show that at least one in x, y, z is equal to 3. (6 marks) B. (a) Simplify $a/(a + b) + b/(a – b)$. (2 marks) (b) If $a/(a + b) + b/( a − b) = 5/4$, find the value of $(a − b)/(a + b)$. (4 marks) # Diocesan Boys' School ## 2023-24 Mid-year Examination ### Grade 8 Mathematics #### Part II (Multiple Choice Questions) Name: Cheung Yung Class: 85 Class Number: 7 ### Instructions: 1. Time allowed: 1 hour 40 minutes (Part I and Part II) 2. There are 36 questions in this part and all questions carry equal marks. 3. Use a PENCIL to mark your answers clearly on the MC answer sheet provided. No ball pen should be used. You will lose marks if the answers cannot be captured. 4. Write your name, class and class number on both the cover of the question booklet and the MC answer sheet. 5. All figures are not necessarily drawn to scale. Measurement does not constitute as answer or solution unless otherwise stated in the question. 6. Students should always select the best option as the answer. If you mark more than one answer, you will receive no marks for that question. 7. HKEAA approved calculators or non-CAS version of TI-inspire calculators can be used. | Answer | | |---------|---| | D | 1 | | B | 2 | | D | 3 | | A | 4 | | C | 5 | | C | 6 | | A | 7 | | B | 8 | | D | 9 | | A | 10 | | C | 11 | | C | 12 | | B | 13 | | B | 14 | | A | 15 | | C | 16 | | B | 17 | | C | 18 | | A | 19 | | C | 20 | | C | 21 | | B | 22 | | D | 23 | | D | 24 | | A | 25 | | A | 26 | | B | 27 | | C | 28 | | A | 29 | | C | 30 | | B | 31 | | D | 32 | | B | 33 | | A | 34 | | D | 35 | | B | 36 |