1:

If x - y = 1, then x3 - y3 - 3xy equals

A. | 0 |

B. | 1 |

C. | 2 |

D. |
x |

x |

2:

If (a, n)! is defined as product of n consecutive numbers starting from a, where a and n are both natural numbers, and if H is the HCF of (a, n)! and n!, then what can be said about H?

A. | h = a! |

B. | h = n! |

C. | h ≥ n! |

D. | h ≥ a * n |

(a. n)! = product of n consecutive natural numbers starting from 'a' which is atleast divisible by n!. (n)! = product of n consecutive natural numbers. For n = 2 : (a. n)! = a(a + 1) and n! = 2 a(a + 1) is divisible by 2!. For n = 3 : (a n)! = a(a + 1)(a + 2) and n! = 6. One of the factors of a(a + 1)(a + 2) is divisible by 3 and other by 2. Thus, proceeding in this manner, (a. n)! and n! have HCF = n! ∴ H = n!. |

3:

If a and b are prime numbers, which of the following is true?

**I.** a2 has three positive integer factors.

**II.** ab has four positive integer factors.

**III.** a3 has four positive integer factors.

Codes

A. | I and II only |

B. | II and III only |

C. | All of these |

D. | None of these |

Factors of a |

4:

If x = b + c, y = c a, z = a b, then

x2 + y2 + z2 - 2xy - 2xz + 2yz is equal to

A. | a + b + c |

B. |
4b |

C. | abc |

D. |
a |

x |

5:

An Egyptian fraction has a numerator equal to 1, and its denominator is a positive integer. What is the maximum number of different Egyptian fraction such that their sum is equal to 1, and their denominators are equal to 10 or less?

A. | 3 |

B. | 5 |

C. | 7 |

D. | 9 |

We ignore 1/7 and 1/9 because no sum of other denominator.numbers is going to give 7ths or 9ths in the denominator |

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