SECTION A : 5 marks per question

1.  7 - 0.001 is equal to

(A) 6.99     (B) 6.999     (C) 6.009     (D) 6.9     (E) 6.991

2.  The value of 1-2+3-4+5-6+¼-998+999-1000+1001 is

(A) 500     (B) 501     (C) -501     (D) -1001     (E) 1000

3.  Susan's brother and her grandmother both died young. The sum of their lifespans was 66 years. Susan's brother died 93 years after their grandmother was born. How many years after their grandmother died was Susan's brother born?

(A) 37     (B) 33     (C) 30     (D) 27     (E) 17

4.  Zimbabwe is 2 hours ahead of London time. An aeroplane leaves Harare at 8.00 am local time and arrives in London at 5 pm (local time) the same day. The actual time taken for the journey, in hours, is

(A) 9     (B) 10      (C) 11     (D) 12     (E) 13

5.  Tapiwa's grandfather used to be a maths teacher and he enjoys puzzles. She asked him how old he was and he replied ``See this solid cube? If you multiply the number of edges by 5, add to this four times the number of faces, and subtract twice the number of its vertices, you will find out my age!". How old is he?

(A) 58     (B) 56     (C) 60     (D) 64     (E) 68

6.  One million seconds is about

(A) 3 days     (B) 12 days     (C) 1 year     (D) 2 years     (E) 3 months

7.  The Brilliant Duck in Bertrand Caroll's mathematics fiction book ``Mathland" devised a way of counting her ducklings which was basically a residue evaluation process after division, separately, by 5, 3 and 11. She knew that

· when counted by five's, 2 ducklings remained
· when counted by three's 2 ducklings remained
· when counted by eleven's, 3 ducklings remained

Brilliant Duck had X ducklings, X < 100. What is the sum of the digits of the number X?

(A) 9     (B) 10     (C) 11     (D) 12     (E) 17

8.  The value of [(1001²-999²)/(101²-99²)] is

(A) 10     (B) 11     (C) 20     (D) 40     (E) 100

9.  Zimbabwean millers buy wheat variety Z from Zambia and variety S from South Africa in order to mix these wheat varieties with that variety H grown here in Zimbabwe so as to obtain flour with the recommended protein content. Variety H has a protein content of 13% while that of variety S is 11% and that of variety Z is 9%. The number of tonnes of H, which must be mixed with 20 tonnes of S and 40 tonnes of Z, to give a final mixture protein content of 12% is

(A) 140      (B) 360     (C) 120     (D) 136     (E) 164

10.  The square below is a magic square. This means that the sum of the numbers in any row, column or diagonal must be the same. The value of N is

(A) 13     (B) 10      (C) 17      (D) 9      (E) 14

SECTION B : 6 MARKS PER QUESTION

11. In the diagram on the right above the value of x is

(A) 88     (B) 90     (C) 92     (D) 96     (E) 112

12.  In a test all the questions scored equal marks. If you answered 9 of the first 10 questions correctly, but only 30% of the remaining questions correctly, you would have scored 50% for the whole test. How many questions were in the test?

(A) 60     (B) 40     (C) 20     (D) 50     (E) 30

13.  The figure shown below consists of 9 squares, each of side length 1 unit. It can be divided into two equal areas by a straight line passing through the point X. If PQ=QR=RS=ST=1/4 then the line also passes through

(A) P     (B) Q     (C) R     (D) S     (E) T

14.  A cross-country runner is in open country 2 km west of a long straight fence that runs due north. He is also 5 km south and 8 km east of the finishing post. The rules state that he must touch the fence once before reaching the finishing post. The shortest distance, in kilometres, that he can run to fulfil the conditions of the race is

(A) 5Ö5     (B) 10+Ö{29}     (C) 2+Ö5     (D) 4+Ö{89}     (E) 13

15.  A donkey is tied by a rope to a corner of a rectangular shed as shown above. The shed is 9 metres long and 7 metres wide and the rope is 10 metres long. The shed is surrounded by grass. The area, in square metres, that the donkey can graze is

(A) [ 155/2]p     (B) [ 229/2]p     (C) 75p      (D) 160+[ 5/2]p     (E) [ 309/2]p

16.  The numbers a,  b,  c and d are chosen so that the cube root of abc is 4, and the fourth root of abcd is 2Ö{10}. The value of d is

(A) 25     (B) 100     (C) 2500     (D) 320     (E) 5

17.  Four straight lines intersect as shown in the diagram. The value of x+y+z+w is

(A) 360     (B) 450     (C) 540     (D) 630     (E) 720

18.  If log_b a + log_a b = c then the greatest whole number less than or equal to c for any pair (a,b) where a,  b > 1, is

(A) 1     (B) 2     (C) 3     (D) 4     (E) 5

19.  Suppose P is a polygon with 11 sides. If we connect every vertex of P to every other vertex of P, what is the resulting number of internal crossing points?

(A) 121     (B) 22     (C) 440      (D) 330      (E) 220

20.  In the diagram, angle PQR is 12^o and a sequence of isosceles triangles is drawn as shown. What is the largest number of such triangles that can be drawn?

(A) 4     (B) 5     (C) 6     (D) 7     (E) 8

SECTION C : 8 marks per question

21.  How many three element subsets drawn from the set S = {1,2,3,¼,29,30} are such that the sum of the three elements is a multiple of 3?

(A) 1 000     (B) 120     (C) 1 120     (D) 30     (E) 1 360

22.  If n ¹ 0 then the expression ⁿÖ{[ 20/(2²ⁿ⁺⁴+2²ⁿ⁺²)]}    equals

(A) [ 1/(2n)]     (B) [ 4/n]     (C) nÖ{[ 5/2]}      (D) [ 1/4]     (E) [ 1/4]nÖ{[ 5/2]}

23.  Find the value of the product (1-[ 1/(2²)])(1-[ 1/(3²)])(1-[ 1/(4²)])¼(1-[ 1/(2001²)]).

(A) [ 2003/4004]     (B) [ 2001/4004]     (C) [ 2000/2001]     (D) [ 2002/2003]     (E) [ 2002/4002]

24.  A sequence is defined by the rule u_n+1=Ö{3u_n} and u₁ = 1, where u₁ is the first term in the sequence, u₂ is the second term, and so on. Which of the following numbers is the smallest whole number which is greater than u_n, for all values of n?

(A) 1     (B) 2     (C) 3     (D) 4     (E) 5

25.  Let P denote the set of all prime numbers. How many ordered pairs (p,q) with p > q and p, q Î P, satisfy p-q = 3?

(A) 0     (B) 1     (C) 2     (D) 3      (E) 4

26.  If ||x-2|-1|=a, where a is a constant integer, has exactly 3 distinct roots, then a equals

(A) 0     (B) 1     (C) 2     (D) 3     (E) 4

27.  While trying to solve a quadratic equation, Jubelah inadvertently interchanged the coefficient of x² with the constant term, causing the equation to change. He solved this changed equation correctly - one of the roots he got was 2 and the other was a root of the original equation. The sum of the squares of the roots of the original equation is

(A) 5     (B) 1     (C) [ 1/4]     (D) 1[ 1/4]      (E) 1[ 1/2]

28.  A square PQRS of side length 1m sits inside a circle of radius 1 m as shown in the diagram. The square marches around the inside of the circle as follows: rotate the square clockwise about the point R until S touches the circle; then rotate clockwise about S until P touches the circle, and so on until the square arrives back at a position where two of its vertices occupy the original position of Q and R. The length of the locus traced out by the point P, in metres, is

(A) (1+[(Ö2)/3])p     (B) ([ 2/3]+[(Ö2)/2])p     (C) Ö2p      (D) ([ 1/2]+[(Ö2)/3])p     (E) (1+[(Ö2)/2])p

29.  If Ö{12+Ö{12+Ö{12+Ö{12+¼} }}} = aÖ{a+Ö{a+Ö{a+¼}}}, then one possible value of a is

(A) Ö5     (B) 3     (C) 2     (D) [(1+Ö3)/4]     (E) [(1+Ö2)/3]

30.  The equation lim_x®¥([ 1/x]) is read as the limit of [ 1/x] as x approaches infinity is equal to zero. The expression is simply a shorthand way of representing the statement that, as x takes on progressively larger values, the value of [ 1/x] gets closer and closer to zero. Now suppose that lim_x®¥(1+[ a/x])=e^a  where e is a real number. Find, in terms of e, the value of lim_x®¥([(x+1)/(x+2)])^3x.