QBM031

Q1. Let S_n = 1 + 1/2 + 1/3 + _ _ _+ 1/n (n = 1; 2; _ _ _). then S_n − S_m is
(i)Always a prime number for m(ii)Always an odd number for m (iii) Always an even number for m(iv)never an integer for m2 - 10x - 22 ? Q4. You may recall that a real number is called rational if it can be written as a fraction with integers in both the numerator and denominator, otherwise it's called irrational . There are many irrational real numbers; for example, if n is any positive integer which is not a perfect square, then is irrational. Find two irrational numbers, a and b, so that ab is an integer. Q5. Find the smallest integer N such that N and N2 together contain each digit its own number of times, that is, 0 zeros, 1 one, 2 twos, ..., 9 nines. Q6. Two players take turns choosing one number at a time (without replacement) from the set {-4, -3, -2, -1, 0, 1, 2, 3, 4}. The first player to obtain three numbers (out of three, four, or five) which sum to 0 wins. Does either player have a forced win? Q7. Determine all the square integers whose decimal representations end in 2001. What is the smallest of these numbers? Q8. In an apartment building there are seven elevators, each stopping at no more than six floors. If it is possible to go from any one floor to any other floor without changing elevators, what is the maximum number of floors in the building? Q9. How many numbers between 1 to 1000 (both excluded) are both squares and cubes?
(A)none
(B)1
(C)2
(D)3 Q10. 5⁶ - 1 is divisible by
(1)13
(2)31
(3)5
(4) None of these Q11. Find the smallest whole numbers, M and N such that you can rearrange the digits of M to get N, and you can rearrange the digits of M3 to get N3. Q12. Is there a set S of positive integers such that a number is in S if and only if it is a sum of two distinct members of S or a sum of two distinct positive integers not in S? Q13. At a movie theater, the manager announces that a free ticket will be given to the first person in line whose birthday is the same as someone in line who has already bought a ticket. You have the option of getting in line at any time. Assuming that you don't know anyone else's birthday, and that birthdays are uniformly distributed throughout a 365 day year, what position in line gives you the best chance of being the first duplicate birthday? Q14. 1/m + 1/n = 1/94, where m and n are positive integers. Find m + n, given that m is half of n.
(1)49
(2)282
(3)423
(4)927 Q15. The units digit of 2006 ²⁹ + 97²⁰⁰⁶ is:
a.2
b.4
c.6
d.8
e.0 END