hans woyda mathematics quiz competition 1999/2000€¦ · web viewhans woyda mathematics quiz...

HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011KNOCK-OUT 1

SECTION 1 - STARTERS (INDIVIDUAL)

Marks: 2 marks to either or both competitors for the correct answerTime: 30 seconds.

Year7-9 1) The sum of the ages of three sisters is now 55 years. Find

how many years it will be before the sum of their ages is 100 years.

10-11 2) The average age of a class of 20 students is 13 years. The average age of the class plus their teacher is 15 years. Find the age of the teacher.

12 3) Find five prime numbers less than 30 which form an arithmetic progression.

13 4) Find the acute angle θ for which cos 1000º = cos θ º.

7-9 5) Find , giving your answer as a decimal.

10-11 6) Find the sum of the marked angles in this squiffy five-pointed star.

12 7) A student who wants to calculate the volume of a cylinder mistakenly uses the formula for the curved surface area instead, but luckily still gets the right answer. Find the radius of the cylinder.

13 8) John’s mother is now fives times as old as John and tentimes as old as the cat. Find how many times as old as John his mother was when the cat was born.


SECTION 2 - GEOMETRY AND TRIGONOMETRY (PAIRS)

Marks: 2 marks to either or both pairs for the correct answerTime: 90 seconds.

Year7-11 1) I am standing 6 m from a vertical wall of height 4.5 m.

My eye level is 1.5 m above the ground. I can just see the top of a tower 31.5 m tall on the other side of the wall. Find how far away from the base of the tower I am standing.

12-13 2) A circle of radius 2 cm just fits inside an equilateral triangle. Find an exact expression for the area of the triangle.

7-11 3) Find the value of x .

12-13 4) An equilateral triangle fits exactly inside a circle of radius 2 cm. Find an exact expression for the area between the triangle and the circle.

40º

xº


SECTION 3 - MENTAL ARITHMETIC AND PROBABILITY (INDIVIDUAL)

Marks: 2 or 1 to opponentTime: 60 seconds

All questions are to be done mentally.

Year

7-9 1) Given that , find x .

2) Given that , find x .

10-11 3) Given that , find one possible value of x .

4) You are reminded that 20112 – 112 = (2011 + 11)(2011 – 11).Evaluate 20112.

12 5) Find the coefficient of x in the expansion of(2x + 5)(x – 2)(3x + 7) .

6) Find the coefficient of x in the expansion of(2x + 5)(2x – 5)(3x + 7)(3x – 7) .

13 7) Given that (cos x + 2sin x)(4cos x + 2sin x) ≡ a + b sin 2x ,

find the values of a and b .

8) Given that

a sin x when cos x = 2 sin x,

find the value of a .


SECTION 4 - TEAM QUESTION

Time: 5 minutes.

For each positive whole number n the number S(n) is the sum of the factors of n, excluding 1 and n.

For example, S(12) = 2 + 3 + 4 + 6 = 15.

Your task is to find S(n) for 21 ≤ n ≤ 89, entering your answers on the sheet provided.

Marks: Give 1 point for each correct answer and -1 point for each wrong answer; no penalty for omissions. Winning team scores 5 marks. Losing team scores 5 – D/5 marks, rounded to the nearest integer, where D is the points difference between the two teams. The minimum score is zero.



Answer sheet

n S(n) n S(n) n S(n)21 44 6722 45 6823 46 6924 47 7025 48 7126 49 7227 50 7328 51 7429 52 7530 53 7631 54 7732 55 7833 56 7934 57 8035 58 8136 59 8237 60 8338 61 8439 62 8540 63 8641 64 8742 65 8843 66 89



Solutions

n S(n) n S(n) n S(n)21 10 44 39 67 022 13 45 32 68 5723 0 46 25 69 2624 35 47 0 70 7325 5 48 75 71 026 15 49 7 72 12227 12 50 42 73 028 27 51 20 74 3929 0 52 45 75 4830 41 53 0 76 6331 0 54 65 77 1832 30 55 16 78 8933 14 56 63 79 034 19 57 22 80 10535 12 58 31 81 3936 54 59 0 82 4337 0 60 107 83 038 21 61 0 84 13939 16 62 33 85 2240 49 63 40 86 4541 0 64 62 87 3242 53 65 18 88 9143 0 66 77 89 0


SECTION 5 - CALCULATORS (INDIVIDUAL)

Marks: 2 to either or both competitors for the correct answerTime: 90 seconds

You are reminded that the written questions are to be given simultaneously to the respective pupils at the beginning of this section.

Year7-9 1) A student correctly calculates the area of a circle of radius

38 cm to be 4536 cm2 (to the nearest whole number). If this radius is in fact given to the nearest centimetre, find how many significant figures of this answer can be relied on.

10-11 2) The surface area of a sphere of radius r is 4r2.70 % of the Earth’s surface is covered by oceans. Taking the Earth to be a sphere of radius 6.37 106 m, find the

surface area of the Earth’s oceans in m2, giving your answer in standard form correct to 2 significant figures.

12 3) The radius r of a hemispherical dome on top of a building of

height h is given by the formula ,

where and are the angles of elevation of the highest visible parts of the dome from points on the ground in line with the axis of the dome and a distance a apart. Calculate r, given that h = 56 m, a = 25 m, = 42º and = 48º, giving your answer correct to the nearest metre.

13 4) The first and tenth terms of a geometric sequence are 10 and 100 respectively. Find the twentieth term, giving your answer correct to the nearest integer.

HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011 KNOCK-OUT 1

SECTION 6 - ALGEBRA AND CALCULUS (INDIVIDUAL)

Marks: 2 or 1 for opponentTime: 60 seconds.

Throughout this section a b = .

For example 5 9 = .

Year7-9 1) Evaluate (3 4) 8 .

2) Solve the equation (3 4) x = 8

10-11 3) Solve the equation x (2x – 2) = x + 1.

4) Given that x > 4, simplify the expression ((x – 2) (x + 2)) (x – 4).

12 5) Find the integer a such that a a is an integer.

6) Find a non-zero integer k such that k (3k) is an integer.

13 7) Find the positive integer k such that n (n + k) is an integer for every non-negative integer n .

8) Evaluate .


SECTION 7 - RACE (INDIVIDUAL)

Marks: 2 or 0Time: 60 seconds.In Questions 1 to 4, each letter of the word TRIGONOMETRIC is printed (in the typeface shown) in the centre of a square tile. Some or all of these thirteen tiles can be used to form ‘words’ (which do not need to make sense), with the tiles used put in a horizontal line on the table.

Year7-9 1) Find the largest number of tiles that can be used to form a

‘word’ which has a vertical line of symmetry.

10-11 2) Find the largest number of tiles that can be used to form a ‘word’ which has a horizontal line of symmetry.

12 3) Find the largest number of tiles that can be used to form a ‘word’ which has rotational symmetry of order two and no line of symmetry.

13 4) Find the largest number of tiles that can be used to form a ‘word’ which has rotational symmetry of order two and two lines of symmetry.

In Questions 5 to 8, ribbons of infinite length are decorated with regular shapes (polygons or circles) which touch each other as shown.

For example, this ribbon has hexagonal polygons:

Find the proportion of the area of the ribbon NOT covered by the shapes when the shape is:

7-9 5) Equilateral triangle:

10-11 6) Circle:

12 7) Hexagon: 13 8) Octagon:


ANSWERS (allow equivalent answers)

SECTION 1

1. 15 (yr) 2. 55 (yr)3. 5, 11, 17, 23, 294. 80 (º)5. 0.0004 (must be decimal)6. 180 (º)7. 28. 9

SECTION 2

1. 60 (cm)2. 12√3 (cm2)3. 70 (º)4. 4π - 3√3 (cm2)

SECTION 3

1. 102. 13. 5 or -54. 40441215. -236. 07. a = 4, b = 58. 9

SECTION 4

Please see answer sheet

SECTION 5

1. 0 or none2. 3.6 1014 (m2)

3. 89 (m)4. 1292

SECTION 6

1. 62. 153. 1 or 34. x − 25. 06. e.g. 2 or 87. 48. 7⅓

SECTION 7

1. 7 (e.g. TOIMIOT)2. 6 (e.g. CEIIOO)3. 5 (e.g. OINIO)4. 4 (e.g. OIIO)5. 1/2

6.

7. 1/4

8. (= )

hans woyda mathematics quiz competition 1999/2000€¦ · web viewhans woyda mathematics quiz...

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