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HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011 SEMI-FINAL SECTION 1 - STARTERS (INDIVIDUAL) Marks: 2 marks to either or both competitors for the correct answer Time: 30 seconds. Year 7-9 1) 10-11 2) Find the least possible number of coins in my change if I buy three items costing £3.17 each with a £10 note. Find the length of the hypotenuse of this triangle. 7 cm 24 cm 12 3) Find the area under the curve between x = 0, x = 1 and y = 0. 13 4) Find the value of sin15ºcos15º. 7-9 5) Find half of a half of a half of 100. 10-11 6) A cylindrical plastic pipe has outer diameter 4 cm and inner diameter 3 cm. Find the volume of plastic in 4 m of pipe. 12 7) Find exact expressions for the solutions of the equation x2 + 2x – 1 = 0 . 13 8) Find the sum of the infinite geometric series HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011 SEMI-FINAL SECTION 2 - GEOMETRY AND TRIGONOMETRY (PAIRS) Marks: 2 marks to either or both pairs for the correct answer Time: 90 seconds. Year 7-11 1) 12-13 2) 7-11 3) 12-13 4) The figure shows a regular hexagon with centre O. Given that area A : area B = 1 : x , find x. O The figure shows a regular octagon with centre O. Given that area A : area B = 1 : x , find an exact expression for x. A B O The figure shows a circle with centre O and one of its tangents. Given that area A : area B = 1 : x , find an exact expression for x. The figure shows a regular octagon with centre O. Given that area A : area B = 1 : x , find an exact expression for x. [Hint: triangles OIK, JHO are similar.] A B A B 45º O H J A B K O I HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011 SEMI-FINAL SECTION 3 - MENTAL ARITHMETIC AND PROBABILITY (INDIVIDUAL) Marks: 2 or 1 to opponent Time: 60 seconds All questions are to be done mentally Year 7-9 1) A list starts with the year 2011 and then goes back 11 years at a time, so that the list begins 2011, 2000, 1989, … . Find the earliest year of the twelfth century to appear in the list. 2) Find the least integer greater than 100 which is 5 more than a multiple of 6 and 6 more than a multiple of 7. 10-11 3) 12 13 Find how many integers from 12 to 789 inclusive which (like these two numbers) have digits which are consecutive and increase from left to right. 4) 16 and 64 are two 2-digit squares where the units digit of one is the tens digit of the other. Find how many other pairs like this there are. 5) The numbers 1, 2, 3, 4 are arranged in random order. Find the probability that 1 and 2 are next to each other. 6) The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 are arranged in random order. Find the probability that the numbers 1, 2, 3 come in that order, though not necessarily next to each other. 7) Find how many distinct ordered pairs of integers (x, y) satisfy the equation [Reminder: .] 8) Find how many distinct ordered pairs of integers (x, y) satisfy the equation HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011 SEMI-FINAL SECTION 4 - TEAM QUESTION Time: 5 minutes. Towers are made from unit cubes, called bricks. The top layer has 1 brick, the second layer down is a 2 × 2 square of 4 bricks, the third layer is a 3 × 3 square of 9 bricks, and so on. The following table gives the total number of bricks in a tower with a given number of layers, and also the number of bricks in a large cube of the same height. No. of layers 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 No. of bricks in tower 1 5 14 30 55 91 140 204 285 385 506 650 819 1015 1240 No. of bricks in large cube 1 8 27 64 125 216 343 512 729 1000 Your task is start with a tower of bricks and show how to make a set of smaller towers and cubes that uses all of the bricks. There can be at most one tower or cube of any particular size. For example 3 layer tower (14 bricks) = cube of side 2 (8 bricks) + 2 layer tower (5 bricks) + cube of side 1 (1 brick). This solution is written as 8 + 5 + 1. You are to do this for towers of from 5 to 15 layers, entering your solutions on the sheet provided. Marks: Give 1 point for each correct answer and -1 point for each wrong answer; no penalty for omissions. The winning team scores 5, the other team scores 5 – D, where D is the difference in points, with minimum score 0. HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011 SEMI-FINAL SECTION 4 - TEAM QUESTION Answer sheet No. of layers 5 6 7 8 9 10 11 12 13 14 15 No. of bricks in tower 55 91 140 204 285 385 506 650 819 1015 1240 Solution HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011 SEMI-FINAL SECTION 4 - TEAM QUESTION Solutions (not unique) No. of layers 5 6 7 8 9 10 11 12 13 14 15 No. of bricks in tower 55 91 140 204 285 385 506 650 819 1015 1240 Solution 27 + 14 + 8 + 5 + 1 64 + 27 91 + 30 + 14 + 5 140 + 64 216 + 64 + 5 343 + 27 + 14 + 1 343 + 140 + 14 + 8 + 1 385 + 91 + 64 + 55 + 27 + 14 + 8 + 5 + 1 512 + 285 + 14 + 8 650 + 343 + 14 + 8 819 + 385 + 30 + 5 + 1 HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011 SEMI-FINAL SECTION 5 - CALCULATORS (INDIVIDUAL) Marks: 2 to either or both competitors for the correct answer Time: 90 seconds You are reminded that the written questions are to be given simultaneously to the respective pupils at the beginning of this section. Year 7-9 1) 10-11 2) Find the last four digits of the product 314 572 × 2 135 213 . Find the exact value of the product 314 572 × 2 135 213 12 3) Find the largest prime factor of 87 703 . 13 4) A sequence x1, x2, x3, … is defined by x1 = 12 for n ≥ 1 . Find the value of x4 correct to the nearest integer. HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011 SEMI-FINAL SECTION 6 - ALGEBRA AND CALCULUS (INDIVIDUAL) Marks: 2 or 1 for opponent Time: 60 seconds. For Questions 1 – 4 If X is the point (a, b) and Y is the point (c, d) then the gradient of the line XY is given by G(X, Y) = d b c a Give all answers in the simplest form. Year 7-9 1) Find G(X, Y) when X is (3, 4) and Y is (7, 12) . 2) Find G(X, Y) when X is ( 5 , 1) and Y is (7, 13) . 10-11 3) 4) Find G(X, Y) when X is ( m , m(m 2) ) and Y is (3, 3) . X is ( a 2 , ab ) and Y is (1, a 2b) . Find G(X, Y) when 12 5) Find the gradient of the curve 1 2 y 6x x at the point where x 12 . 6) Find the gradient of the curve 2 6 x y 2 2 x at the point where x 2 . In Questions 7 and 8 your answer should be an exact numerical expression. 13 7) Find the gradient of the curve y sin 2 4x at the point where . 8) Find the gradient of the curve at the point where x 0 HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011 SEMI-FINAL SECTION 7 - RACE (INDIVIDUAL) Marks: 2 or 0 Time: 60 seconds. Year 7-9 1) 10-11 2) Find three different positive whole numbers whose product is three times their sum. Write 7 4 4 4 as the product of prime factors. 12 3) A piece of wire 120 cm long is cut into two parts. Each part is then bent to form a square. Find the maximum possible total area of these squares. 13 4) A piece of wire 120 cm long is cut into two parts. Each part is then bent to form an equilateral triangle. Find the maximum possible total area of these triangles. 7-9 5) Each internal angle of a regular polygon is 171º. Find how many sides the polygon has. 10-11 6) 12 7) A cone with base radius r has the same volume as a sphere of radius r . Find the height of the cone in terms of r . Given that y 10 0 13 8) 1 1 x10 dx , find dy . dx The first ten terms of the arithmetic progression 199, 409, 619, … are all prime numbers. Find a prime factor of the eleventh term. HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011 SEMI-FINAL ANSWERS (allow equivalent answers) SECTION 1 SECTION 5 1. 2. 3. 4. 5. 6. 7. 8. 5 25 0.5 0.25 12½ 700π (cm3) -1 ± √2 2 1. 2. 3. 4. SECTION 6 SECTION 2 1. 2. 3. 2 2 + 2√2 1. 2. 3. 2 1 4. 5. 4. SECTION 3 1. 2. 3. 4. 1109 125 15 3 5. 1/2 6. 7. 8. 1/6 8 12 SECTION 4 Please see solutions sheet 3 836 671 678 223 836 67 531 m 1 ab a 1 10 2.5 6. 7. 4 8. 3 3 2 SECTION 7 1. 2. 3. 4. 5. 6. 7. 8. E.g. 2, 3, 5 3 5 11 13 450 (cm2) 200 3 (cm2) 40 4r 0 11 or 19