In a class, there are 80 students. The following table gives the details regarding the distribution of the marks scored (in percentage terms) by the students in each of the six subjects — Mathematics, Physics, Chemistry, History, English and Geography. Every student wrote all the six subjects.

Subject | The number of students scoring marks (m) in the range | ||||

90% < m | 75% < m ≤ 90% | 60% < m ≤ 75% | 40% < m ≤ 60% | 20% < m ≤ 40% | |

Mathematics | 14 | 15 | 13 | 17 | 13 |

Physics | 7 | 23 | 13 | 8 | 17 |

Chemistry | 9 | 10 | 16 | 12 | 18 |

History | 11 | 17 | 15 | 11 | 20 |

English | 13 | 19 | 10 | 7 | 27 |

Geography | 8 | 14 | 14 | 9 | 25 |

**Q1. **Which of the following is a possible number of students who scored more than 20% but not more than 90% of marks in each of the six subjects?

- 53
- 57
- 61
- 63

**Q2. **What is the minimum possible number of students who could have scored more than 40% but not more than 75% of marks in at least two of the six subjects?

- 11
- 9
- 15
- 13

**Q3. **What is the maximum possible number of students who could have scored more than 40% but not more than 60% of marks in at least four of the six subjects?

- 16
- 15
- 14
- None of these

**Q4. **What is the minimum possible number of students who scored more than 60% but not more than 90% in at most three of the six subjects?

- 40
- 38
- 36
- None of these

**Q5. **What is the maximum possible number of students for each of who the marks scored in all the six subjects are the same?

- 47
- 60
- 51
- None of these

**Q1. **If we consider the number of students who scored more than 20% but not more than 90% of marks for each of the six subjects. it is the lowest (56) for Chemistry.

Therefore, at most 56 students would have scored more than 20% but not more than 90% of marks in each of the six subjects. Choice (1)

**Q2.**The number of students who scored more than 40% but not more than 75% of marks in Mathematics, Physics, Chemistry, History, English and Geography are 30, 21, 28, 26, 17 and 23 respectively, which add up to 145.

To minimize the number of students who scored in the given range in at least two subjects, we have to maximize the number of students who scored in the given range in exactly one subject. To maximize the number of students who scored in the given range in exactly one subject, we have to maximize the number of students who scored in the given range in each of the six subjects and minimize the number of students who scored in the given range in exactly two, three, four and five subjects.

Let a and b be the number of students who scored in the given range in exactly one subject and all the six subjects respectively.

Therefore, a + b = 80

a+6b=145=>5b=65=>b=13 Choice(4)

**Q3. **The number of students who scored more than 40% but not more than 60% of marks in at least four of the six subject is at most $\frac { 17 + 8 + 12 + 11 + 7 + 9 } { 4 }$ i.e., $\frac { 64 } { 4 } = 16$, which occurs if the number of students who scored in the given range in each of the subjects is 16 or less.

But for Mathematics, it is 17.

Therefore, At most 15 students scored more than 40% but not more than 60% of marks in at least four subjects. Choice (2)

**Q4**. To minimize the number of students, who scored in the given range in at most three subjects, we have to maximize the number of students who scored in the given range in exactly four subjects. The maximum possible number of students who scored more than 60% but not more than 90% in exactly four subjects is $\frac { ( 15 + 13 + 23 + 13 + 10 + 16 + 17 + 15 + 19 + 10 + 14 + 14 ) } { 4 }$ i.e., $\frac { 179 } { 4 }$ i.e., 44

As 44 is not less than the number of students who scored more than 60% but not more than 90% of marks in any of the six subjects, it is possible.

Therefore, the minimum possible value of the required number is 80- 44 i.e., 36. Choice (3)

**Q5. **To get the required number, we have to add the least number in each column (i.e., ranges of marks).

The number of students who scored less than or equal to 20% of marks in Mathematics, Physics, Chemistry, History, English and Geography are 8, 12, 15, 6, 4 and 10 respectively. Hence, in this range at most 4 students could have got the same marks in all the six subjects.

Therefore, the sum of the least numbers in each column is 7+10+10+7+13+4i.e.,51. Choice(3)

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