Each of the following question is followed by two Statements.' Give answer
Total marks obtained by P, Q, R and Sin Mathematics is 360 How many marks did P secure in Mathematics?
I P secured one-third marks of the total of Q, Rand N
II Average marks obtained by Q and Rare 20 more than that secured by S.

Options

A. A)if Statement I alone is sufficient
to answer the question
B. B)if Statement II alone is sufficient
to answer the question
C. C)if both Statements I and II
together are necessary to
answer the question
D. D)if either the Statement I
alone or Statement II alone
is sufficient to answer the question

Answer

To solve the problem step by step, let's analyze the information provided in the question and the statements.

### Given:
- Total marks obtained by P, Q, R, and S in Mathematics = 360

### Question:
How many marks did P secure in Mathematics?

### Statements:
1. **Statement I**: P secured one-third of the total marks of Q, R, and S.
2. **Statement II**: The average marks obtained by Q and R are 20 more than that secured by S.

### Step 1: Analyze Statement I
From Statement I, we can express the relationship mathematically. Let the marks obtained by P, Q, R, and S be represented as follows:
- Let P's marks = P
- Let Q's marks = Q
- Let R's marks = R
- Let S's marks = S

According to Statement I:
\[ P = \frac{1}{3}(Q + R + S) \]

### Step 2: Express Total Marks
We know the total marks:
\[ P + Q + R + S = 360 \]

### Step 3: Substitute P in Total Marks Equation
Substituting the expression for P from Statement I into the total marks equation:
\[ \frac{1}{3}(Q + R + S) + Q + R + S = 360 \]

### Step 4: Simplify the Equation
To eliminate the fraction, multiply the entire equation by 3:
\[ (Q + R + S) + 3Q + 3R + 3S = 1080 \]
\[ 4Q + 4R + 4S = 1080 \]

### Step 5: Divide by 4
Dividing the entire equation by 4:
\[ Q + R + S = 270 \]

### Step 6: Find P's Marks
Now, substitute back to find P's marks:
\[ P = \frac{1}{3}(Q + R + S) = \frac{1}{3}(270) = 90 \]

Thus, P secured **90 marks** in Mathematics.

### Conclusion from Statement I
Statement I alone is sufficient to answer the question.

### Step 7: Analyze Statement II
Now let's analyze Statement II:
- The average marks obtained by Q and R are 20 more than that secured by S.

Mathematically, this can be expressed as:
\[ \frac{Q + R}{2} = S + 20 \]

However, this statement does not provide any direct information about P's marks. We cannot derive P's marks from this statement alone.

### Conclusion from Statement II
Statement II alone is not sufficient to answer the question.

### Final Answer
Since Statement I alone is sufficient to answer the question, while Statement II is not, the answer is:
**Option A**: Statement I alone is sufficient, but Statement II alone is not sufficient.

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