Joint School Maths Quiz Final Group (98-99)

Instructions:

This section consists 15 questions, answer ALL questions. Put your answers on the answer sheet. The use of calculator is STRICTLY prohibited.Time allowed: 1 Hour

1.In a sequence {a₁,a₂¡K}, a_na_n-1=[½(a_n+a_n-1)]²-1 and a_n+1>a_n for all positive integer n>1. If a₁=1 and 3a_k=a_k=5, find k.(1 mark)

2.A belt fits around three circles x²+y²=1, (x-14)²+y²=1 and (x-9)²+(y-12)²=1.Find the length of the belt.(1 mark)

3.Given that the "*" operation is associative, i.e.a*(b*c)=(a*b)*c. Fill in the blanks:

4.The perimeter of a triangle is 36 units. Given that one of the sides is 10 units. Find the greatest possible area of such a triangle. (1 mark)

5.How many integers between 1 and 1999 (1 and 1999 inclusive), give a perfect square when multiplied by 338? (1 mark)

6.There are x sweets in a bag. When they are divided into 8 equal groups, 4 sweets are left. When they sre divided into 7 equal groups, 5 sweets are left and when they are divided into 9 equal groups, 7 sweets left. Find the smallest number of x. (1 mark)

7.If a₁=2, a₂=6, a_n =2a_n-1-a_n-2+2 where nis aan integer greater than 2. Find (1/a₁)+(1/a₂)+¡K+(1/a₁₉₉₉). (2 marks)

8.A function f(x) is defined as:

(i)f(1999)+0

(ii)f(1999+x)=f(1999-x),x¡Ú0

9.k is a 4001-digit number. The firs 2000 and the last 2000 digits are all 8. If k is divisible by 91, find the 2001^th digit. (2marks)

10.2 persons, A and B,are playing a game which has 80 possible outcomes. Among the 80 possible outcomes,there are only 60 favourable outcomes. A will play the game first, the one who gets the favourable outcome wins. If one cannot get the favourable outcomes, he should pass the chance to the other pereson. The game is repeated again unitil someone wins. Find the probaility that A can win the game. (2marks)

11.A fooball team has many team modes, e.g.3-5-2, 4-4-2, where the first number means defender, second number means midfielder and the last number means forwars (check this word). If the sum of all 3 integersare 10 (All numbers must be non-negative),how many team modes are there? (e.g.0-010 is regard as 1 mode). (2Marks)

12.If f(x)+f[(x-1)/x]=1+x, findf(2). (2masrks)

13.The curve y=(n²+n)x²-(2n+1)x+1 cuts x-axis at points A_nand B_n. Find .(sum of distances between A_i and B_i)(2 marks)

14.Find the number which has the following properties:

i)It is a 6-digit number and

ii)After the first digit is shifted to the unit digit, the new number is 9/10 of the original number.

iii)The second digit of the original number is greater than the fourth digit by 4.

Find the original number.(3 marks)

15.Refer to the figure, given that KF=8 and FL=3. FInd the length of LG.(3 marks)