November 15, 2017

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**Answer the questions independently of each other.**

## Mathematics Tests-23

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Question 1 |

**There are 6 boxes numbered 1, 2, …6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is...**

A | 21 |

B | 5 |

C | 60 |

D | 33 |

Question 1 Explanation:

GRRRRR, RGRRRR, RRGRRR, RRRGRR, RRRRGR, RRRRRG GGRRRR, RGGRRR, RRGGRR, RRRGGR, RRRRGG GGGRRR, RGGGRR, RRGGGR, RRRGGG GGGGRR, RGGGGR, RRGGGG GGGGGR, RGGGGG GGGGGG Hence 21 ways.

Question 3 |

**A graph may be defined as a set of points connected by lines called edges. Every edge connects a pair of points. Thus, a triangle is a graph with 3 edges and 3 points. The degree of a point is the number of edges connected to it. For example, a triangle is a graph with three points of degree 2 each. Consider a graph with 12 points. It is possible to reach any point from any other point through a sequence of edges. The number of edges, e, in the graph must satisfy the condition...**

A | 11? e? 66 |

B | 10? e? 66 |

C | 11? e? 65 |

D | 0? e? 11 |

Question 3 Explanation:

The least number of edges will be when one point is connected to each of the other 11 lines, giving a total of 11 lines. One can move from any point to any other point via the common point. The maximum edges will be when a line exists between any two points. Two points can be selected from 12 points in 12C2 i.e. 66 lines.

Question 4 |

**The number of positive integers n in the range 12 ? n? 40 such that the product (n – 1) (n – 2)…3.2.1 is not divisible by n is ...**

A | 7 |

B | 5 |

C | 14 |

D | 13 |

Question 4 Explanation:

From 12 to 40, there are 7 prime number, i.e. 13, 17, 19, 23, 29, 31, 37, which is not divisible by (n– 1)!

Question 5 |

**Let T be the set of integers {3, 11, 19, 27,…451, 459, 467} and S be a subset of T such that the sum of no two elements of S is 470. The maximum possible number of elements in S is...**

A | 29 |

B | 32 |

C | 30 |

D | 28 |

Question 5 Explanation:

Tn = a + (n – 1)d 467 = 3 + (n – 1)8 n = 59 Half of n = 29 terms 29th term is 227 and 30th term is 243 and when these two terms are added the sum is more than 470. Hence the maximum possible values the set S can have are 30.

Question 6 |

A | 9 |

B | 11 |

C | 12 |

D | 10 |

Question 6 Explanation:

Let us say there are only 3 questions. Thus there are 23–1 = 4 students who have done 1 or more questions wrongly, 23–2 = 2 students who have done 2 or more questions wrongly and 23–3 = 1 student who must have done all 3 wrongly. Thus total number of wrong answers = 4 + 2 + 1 = 7 = 23 – 1 = 2n – 1. In our question, the total number of wrong answers = 4095 = 212 – 1. Thus n = 12.

Question 7 |

A | The two curves intersect once |

B | The two curves do not intersect |

C | The two curves intersect twice |

D | The two curves intersect thrice |

Question 7 Explanation:

When we substitute two values of x in the above curves, at x = –2 we get y = –8 + 4 + 5 = 1 y = 4 – 2 + 5 = 7 Hence at x = –2 the curves do not intersect. At x = 2, y = 17 and y = 11 At x = –1, y = 5 and 5 When x = 0, y = 5 and y = 5 And at x = 1, y = 7 and y = 7 Therefore, the two curves meet thrice when x = –1, 0 and 1.

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