If a , b, c, d are non-zero integers such that `a lt b lt c lt d` and mean and median of a, b, c and d are both equal to zero, then which one of the following is correct?

Options

A. b = -c
B. a = -d
C. both (a) and (b)
D. none of these

Explanation

To solve the problem, we need to analyze the conditions given for the integers \( a, b, c, d \).

### Step-by-step Solution:

1. **Understanding the Mean Condition**:
The mean of \( a, b, c, d \) is given to be zero. The formula for the mean is:
\[ \text{Mean} = \frac{a + b + c + d}{4} = 0 \]
This implies:
\[ a + b + c + d = 0 \quad \text{(1)} \]

2. **Understanding the Median Condition**:
Since \( a, b, c, d \) are in increasing order (\( a < b < c < d \)), the median for four numbers is calculated as the average of the two middle numbers:
\[ \text{Median} = \frac{b + c}{2} = 0 \]
This implies:
\[ b + c = 0 \quad \text{(2)} \]

3. **Finding Relationships Between Variables**:
From equation (2), we can express \( b \) in terms of \( c \):
\[ b = -c \quad \text{(3)} \]

4. **Substituting Back into the Mean Equation**:
Substitute \( b \) from equation (3) into equation (1):
\[ a + (-c) + c + d = 0 \]
Simplifying this gives:
\[ a + d = 0 \quad \text{(4)} \]
Therefore, we can express \( a \) in terms of \( d \):
\[ a = -d \quad \text{(5)} \]

5. **Summarizing Relationships**:
From equations (3) and (5), we have:
- \( b = -c \)
- \( a = -d \)

6. **Verifying the Order**:
Since \( a < b < c < d \):
- From \( a = -d \), we know \( a \) is negative if \( d \) is positive.
- From \( b = -c \), if \( c \) is positive, then \( b \) is negative.
- This means \( a < b < c < d \) holds true as long as \( d > 0 \) and \( c > 0 \).

### Conclusion:
The relationships derived from the conditions are:
- \( b = -c \)
- \( a = -d \)

Thus, the correct option from the given choices is:
- **Option 3: Both \( a = -d \) and \( b = -c \)**.