The questions given below consists of a question followed by three statements. You have to study the question and the statements and decide which of the statement(s) `"is"//"are"` necessary to answer the question.
How many marks did Tarun secure in English?
I. The average marks obtained by Tarun in four subjects including English is 60.
II. The total marks obtained by him in English and Mathematics together are 170.
III. The total marks obtained by him in Mathematics and Science together are 180.

Options

A. I and II only
B. II and III only
C. I and III only
D. None of these

Answer

To determine how many marks Tarun secured in English, we will analyze the information given in the statements step by step.

### Step 1: Calculate the Total Marks for All Subjects
We know that the average marks obtained by Tarun in four subjects including English is 60.

To find the total marks for all four subjects, we use the formula for average:

\[ \text{Average} = \frac{\text{Total Marks}}{\text{Number of Subjects}} \]

Given that the average is 60 and the number of subjects is 4, we can calculate the total marks:

\[ \text{Total Marks} = \text{Average} \times \text{Number of Subjects} = 60 \times 4 = 240 \]

### Step 2: Use the Information from the Statements
Now we will use the information from the statements to find out how many marks Tarun secured in English.

- **Statement I**: The average marks obtained by Tarun in four subjects including English is 60. (We already calculated the total marks as 240.)

- **Statement II**: The total marks obtained by him in English and Mathematics together are 170.

- **Statement III**: The total marks obtained by him in Mathematics and Science together are 180.

### Step 3: Set Up Equations
Let:
- \( E \) = Marks in English
- \( M \) = Marks in Mathematics
- \( S \) = Marks in Science

From Statement II:
\[ E + M = 170 \quad \text{(1)} \]

From Statement III:
\[ M + S = 180 \quad \text{(2)} \]

### Step 4: Express Total Marks in Terms of Variables
We know the total marks for all subjects:
\[ E + M + S + \text{(other subject)} = 240 \]

Since we only have three subjects mentioned, we can assume the fourth subject's marks are included in the total. However, we need to express \( S \) in terms of \( M \) using equation (2):
\[ S = 180 - M \quad \text{(3)} \]

### Step 5: Substitute and Solve
Substituting equation (3) into the total marks equation:
\[ E + M + (180 - M) = 240 \]
This simplifies to:
\[ E + 180 = 240 \]
Thus, we can find \( E \):
\[ E = 240 - 180 = 60 \]

### Conclusion
Tarun secured **60 marks in English**.

### Summary of Necessary Statements
To answer the question, we needed all three statements:
- Statement I provided the total marks.
- Statement II gave a relationship between English and Mathematics.
- Statement III provided a relationship between Mathematics and Science.