Mr. Oberaiappered in CAT for four consecutive years, but coincidently each time his net score was 75. He told me that there was `1/3` rd negative marking for every wrong answer and 1 mark was alloted for every correct answer. He has attempted all the questions every year, but certainly some answer have been wrong due to stress and conceptual problems. Which is not the total number of questions asked for CAT in any year, in that period?

Options

A. 231
B. 163
C. 150
D. 123

Answer

To solve the problem, we need to determine the total number of questions asked in the CAT exam over the four years Mr. Oberai appeared, given that he scored 75 each time with a scoring system that includes negative marking for wrong answers.

### Step-by-Step Solution:

1. **Understanding the Scoring System**:
- For every correct answer, Mr. Oberai earns 1 mark.
- For every wrong answer, he loses 1/3 mark.
- Let \( T \) be the total number of questions attempted each year.
- Let \( C \) be the number of correct answers.
- Therefore, the number of wrong answers will be \( W = T - C \).

2. **Setting Up the Equation**:
- The total score can be represented as:
\[ \text{Score} = C - \frac{1}{3}W \]
- Substituting \( W \) into the equation:
\[ \text{Score} = C - \frac{1}{3}(T - C) = C - \frac{1}{3}T + \frac{1}{3}C \]
- Simplifying this, we get:
\[ \text{Score} = \frac{4}{3}C - \frac{1}{3}T \]
- Setting this equal to 75 (the score Mr. Oberai achieved):
\[ \frac{4}{3}C - \frac{1}{3}T = 75 \]

3. **Clearing the Fraction**:
- Multiply the entire equation by 3 to eliminate the fraction:
\[ 4C - T = 225 \]
- Rearranging gives us:
\[ T = 4C - 225 \]

4. **Finding Possible Values of T**:
- Since \( T \) (total questions) must be a positive integer, \( 4C - 225 > 0 \) implies:
\[ 4C > 225 \implies C > 56.25 \]
- Therefore, \( C \) must be at least 57 (since \( C \) is an integer).

5. **Calculating Possible Values of T**:
- For \( C = 57 \):
\[ T = 4(57) - 225 = 228 - 225 = 3 \]
- For \( C = 58 \):
\[ T = 4(58) - 225 = 232 - 225 = 7 \]
- For \( C = 59 \):
\[ T = 4(59) - 225 = 236 - 225 = 11 \]
- Continuing this way, we can find values for \( C \) and subsequently \( T \).

6. **Identifying Non-possible Values**:
- We need to check which total number of questions (T) does not fit the equation \( T = 4C - 225 \).
- As \( C \) increases, \( T \) will take values that are 4 units apart starting from 3 (i.e., 3, 7, 11, 15, ...).
- We can check the options given in the question to find which one does not fit this pattern.

### Conclusion:
After checking the possible values for \( T \), we can conclude that any number that does not fit the form \( 4C - 225 \) will be the answer.