Question

Five students appeared for an examination. The average mark obtained by these five students is 40. The maximum mark of the examination is 100, and each of the five students scored more than 10 marks. However, none of them scored exactly 40 marks. Based on the information given, which of the following MUST BE true?

• A. At least, three of them scored a maximum of 40 marks
• B. At least, three of them scored more than 40 marks
• C. At least, one of them scored exactly 41 marks
• D. At most, two of them scored more than 40 marks
• E. At least, one of them scored less than 40 marks

Answer 1

Answer: (E)

To solve this problem, let's analyze the given details and evaluate each option to determine which must be true.

**Given Information:**

• 5 students appeared for the examination.
• The average mark obtained by these 5 students is 40.
• Each scored more than 10 marks.
• No one scored exactly 40 marks.

**Calculating Total Marks:**

The average of the five students is 40. This means the total marks obtained by these students is:

\(Total\ Marks = 5 \times 40 = 200\)

Since none of them scored exactly 40, we need to investigate the distribution of their marks under this total. Since each student must score more than 10, assume marks of students are \(a, b, c, d, e\) such that \(a + b + c + d + e = 200\) and none of these are 40.

**Evaluating the Options:**

1. At least, three of them scored a maximum of 40 marks.
2. At least, three of them scored more than 40 marks.
3. At least, one of them scored exactly 41 marks.
4. At most, two of them scored more than 40 marks.
5. At least, one of them scored less than 40 marks.

Let's verify each of these options:

• Option 1: This is incorrect. Scores can be distributed in ways where fewer or even none scored maximum marks.
• Option 2: This is also incorrect. It's possible all scores are around or below 40 but not exactly 40 due to the given average, making it feasible that less than 3 students scored more than 40.
• Option 3: False. There's no requirement in the problem indicating that someone must score exactly 41 marks.
• Option 4: This is theoretically reasonable (all around and one below), but not necessarily true from the information given. More could have scored less than 40.
• Option 5: At least one must score less than 40 marks to balance the average at 40 when considering none can score exactly 40.

For the average to be 40 without anyone scoring it exactly, combining scores greater and less than 40 is necessary. To balance higher scores, at least one lower score is required.

Thus, the correct answer is: At least, one of them scored less than 40 marks.

Answer 2

The average mark obtained by the five students is 40. This implies that the total sum of their marks is 40 * 5 = 200. We need to determine which option must be true based on the given information.

We have the following conditions:

• None of the students scored exactly 40 marks.
• Each scored more than 10 marks.

Let's analyze the options one by one:

Option	Analysis
At least, three of them scored a maximum of 40 marks	None of them scored exactly 40 marks, so this option isn't valid.
At least, three of them scored more than 40 marks	Let's assume three students scored more than 40. If so, then two would score less than 40. It does not consider average, so this isn't a must.
At least, one of them scored exactly 41 marks	This doesn't align with the given average unless adjusted by other exact values, not necessarily true.
At most, two of them scored more than 40 marks	This could still meet the requirements but isn't necessarily true given the flexibility.
At least, one of them scored less than 40 marks	If no student scored exactly 40, some would have to score below for the average to remain exactly 40.

Conclusion: Given an average of 40 but none scored exactly 40, it's essential that at least one student scored less than 40 to balance scores otherwise above 40 so the correct option is that at least, one of them scored less than 40 marks.