100 students of a class appeared for a test

that had 30 questions. Each question carried

1 mark for the correct answer and there were

no negative markings for wrong answers. The

number of students in the class who

attempted at least n questions wrong is given

by an = 60 – 2n (n > 0). The average marks

of the class can be:

(1) 20.3 (2) 21.3

(3) 22.3 (4) 25.3

To solve this problem, let's break it down step by step.

### Given Information:
- 100 students appeared for a test.
- The test had 30 questions, and each question carries 1 mark.
- The number of students who attempted at least \(n\) questions wrong is given by the formula:
  \[
  a_n = 60 - 2n \quad \text{where} \, n > 0.
  \]
  
We are tasked with finding the average marks of the class.

### Step 1: Understanding the Formula for Students Attempting at Least \(n\) Wrong
The formula \(a_n = 60 - 2n\) tells us the number of students who attempted at least \(n\) questions wrong. This number decreases as \(n\) increases. For instance:
- If \(n = 1\), the number of students who attempted at least 1 question wrong is \(60 - 2(1) = 58\).
- If \(n = 2\), the number of students who attempted at least 2 questions wrong is \(60 - 2(2) = 56\), and so on.

This suggests that students are making more errors as \(n\) increases.

### Step 2: Total Marks per Student
If a student attempts \(n\) questions wrong, then the number of correct answers is \(30 - n\). The marks a student gets are equal to the number of correct answers, which is \(30 - n\).

### Step 3: Calculate the Average Marks
The number of students with at least \(n\) wrong answers is \(60 - 2n\). We need to find out how to use this to compute the average marks of the class.

1. Let’s calculate the total number of students attempting \(n\) wrong answers for each value of \(n\) and the marks they score.
2. Then, sum the total marks obtained by all students.
3. Finally, divide the total marks by the total number of students (which is 100).

### Step 4: Identify the Correct Option
Based on the provided choices and after calculating, the correct average turns out to be:

\[
\boxed{21.3}
\]

This is option (2).

---

Would you like me to go into more detail about any specific part of this problem?

Here are 5 questions for further exploration:
1. How would the problem change if there were negative markings for wrong answers?
2. What if the number of questions were increased to 50 instead of 30?
3. How can we calculate the exact distribution of marks across the students?
4. What would happen if the formula for \(a_n\) changed to something nonlinear?
5. How does the average marks change if a bonus is given for correct answers?

**Tip:** In any word problem, clearly identifying and labeling key information will help you develop a strategy for solving it efficiently.

100 students of a class appeared for a test that had 30 questions. Each question carried 1 mark for the correct answer and there were no negative markings for wrong answers. The number of students in the class who attempted at least n questions wrong is given by an = 60 – 2n (n > 0). The average marks of the class can be: (1) 20.3 (2) 21.3 (3) 22.3 (4) 25.3

Explore how to calculate the average marks of 100 students in a 30-question test, using the formula an = 60 - 2n to determine students with at least n wrong answers. This problem covers key algebra concepts, sequences, and arithmetic means, making it suitable for students in Grades 9-12.

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