Joint School Maths Quiz Final Individual (98-99)

1.If[(1+x)/(1-x)]=(x²+2)/(2x²-1), f(6)=?(1 mark)

2.How many distinct positive factors does 1,999,998 have(1 and 1,999,998 inclusive)?(1 mark)

3.In figure (I), BCED is a cyclic quadrilateral. DB and EC produced at A. The ratio of length of AB: AC: BD = 2 : 3 : 4. If the area of triangle ABC=6 sq. units, find the area of BCED.(1 mark)

Figure(I)

4.If x²+xy+y²=18, where x and y are real numbers. Find the maximum value of x²+y²(2 marks)

5.Find the smallest value of n such that f(n)=1999²/n! attains its maximum, where n is any natural number. (recall that n!=n×(n-1)סK×2×1) (2 marks)

6.Consider the system of equations

how many real solutions are there?(2 marks)

7.Find the value of (10.4×10.3×10.2×10.1)-(9.9×9.8×9.7×9.6) (3 marks)

8.If we write 1999¹⁹⁹⁹ into octahedral representation(¤K¶i¦ì¼Æ), what are the last 2 digits?

9.Let [x]denote the integral part of real number x. How many distinct integers are there in the sequence [n²/1993], for n=1,2,¡K,1993?(3 marks)e.g.[3.14]=3, [5.00]=5

10.Given that a function f :R ¡÷R satisfies

(i)f(x) is bounded on (-1,1)

(ii)2000f(2000x)=f(x)+x for all real number x,

Find the value of f(1999).

Remark: R ¡÷R means the function is defined for real numbers and gives real numbers. (3 marks)