MATHEMATICAL INDUCTION-(E)-04-Assignment

1. is divisible by (a) 5 (b) 7 (c) 9 (d) 11 2. For every natural number n, is divisible by (a) 6 (b) 12 (c) 24 (d) 5 3. For every natural number n, is divisible by (a) 16 (b) 128 (c) 256 (d) None of these 4. is divisible by (a) 3 (b) 19 (c) 64 (d) 29 5. For all positive integral values of n, is divisible by (a) 8 (b) 16 (c) 24 (d) None of these 6. For all positive integral values of n, is divisible by (a) 2 (b) 4 (c) 8 (d) 12 7. If , then is divisible by (a) (b) (c) (d) 8. For each , is divisible by (a) 23 (b) 41 (c) 49 (d) 98 9. For each , is divisible by (a) (b) (c) (d) 10. If , then the greatest integer which divides n(n – 1)(n – 2) is (a) 2 (b) 3 (c) 6 (d) 8 11. If , then . is always divisible by (a) 25 (b) 35 (c) 45 (d) None of these 12. If , then is divisible by (a) 113 (b) 123 (c) 133 (d) None of these 13. For every natural number n, is divisible by (a) 4 (b) 6 (c) 10 (d) None of these 14. The difference between an integer and its cube is divisible by (a) 4 (b) 6 (c) 9 (d) None of these 15. For every natural number n (a) (b) (c) (d) 16. For each , the correct statement is (a) (b) (c) (d) 17. For natural number n, , if (a) n < 2 (b) n > 2 (c) n  2 (d) Never 18. If n is a natural number then is true when (a) n > 1 (b) n  1 (c) n > 2 (d) n  2 19. For positive integer n, , if (a) n > 5 (b) n  5 (c) n < 5 (d) n > 6 20. For every positive integer n, when (a) n < 4 (b) n  4 (c) n < 3 (d) None of these 21. For every positive integral value of n, when (a) n > 2 (b) n  3 (c) n  4 (d) n < 4 22. For natural number n, , if (a) n > 3 (b) n > 4 (c) n  4 (d) n  3 23. The value of the n natural numbers n such that the inequality is valid is (a) For n  3 (b) For n < 3 (c) For mn (d) For any n 24. Let P(n) denote the statement that is odd. It is seen that , is true for all (a) n > 1 (b) n (c) n > 2 (d) None of these 25. If {x} denotes the fractional part of x then , is (a) 3/8 (b) 7/8 (c) 1/8 (d) None of these 26. If p is a prime number, then is divisible by p when n is a (a) Natural number greater than 1 (b) Irrational number (c) Complex number (d) Odd number 27. is divisible by for (a) n > 1 (b) n > 2 (c) All n  N (d) None of these 28. Let P(n) be a statement and let P(n)  p(n + 1) for all natural numbers n, then P(n) is true (a) For all n (b) For all n > 1 (c) For all n > m , m being a fixed positive integer (d) Nothing can be said 29. If P(n) = 2 + 4 + 6 +….+ 2n, n  N, then P(k) = k(k + 1) + 2  P(k + 1) = (k + 1)(k + 2) + 2 for all k  N. So we can conclude that P(n) = n(n + 1) + 2 for (a) All n  N (b) n > 1 (c) n > 2 (d) Nothing can be said 30. For every natural number n, n(n + 1) is always (a) Even (b) Odd (c) Multiple of 3 (d) Multiple of 4 31. The statement P(n) “ ” is (a) True for all n > 1 (b) Not true for any n (c) True for all n  N (d) None of these 32. If , then for any n  N, equals (a) (b) (c) (d) None of these 33. The least remainder when is divided by 5 is (a) 1 (b) 2 (c) 3 (d) 4 34. The remainder when is divided by 13 is (a) 6 (b) 8 (c) 9 (d) 10 35. when divided by 7 leaves the remainder (a) 1 (b) 6 (c) 5 (d) 2