If [x] denote the greatest integer less than or equal to x then in order that the set of equations `2x - 2y = 4, 7x -3 y = 2, [3pi[ x - [e] y = [4a]` may be consistent then 'a' should lie in

Options

A. `[3, 7//2)`
B. `(3, 7//3)`
C. `(3, 7//3]`
D. none of these

Answer

To solve the problem step by step, we will analyze the given equations and find the conditions for 'a' such that the system of equations is consistent.

### Step-by-Step Solution:

1. **Write down the equations:**
The given equations are:
\[ 2x - 2y = 4 \quad \text{(1)} \]
\[ 7x - 3y = 2 \quad \text{(2)} \]
\[ [3\pi] x - [e] y = [4a] \quad \text{(3)} \]

2. **Solve the first two equations for x and y:**
From equation (1):
\[ 2x - 2y = 4 \implies x - y = 2 \implies y = x - 2 \quad \text{(4)} \]

Substitute (4) into equation (2):
\[ 7x - 3(x - 2) = 2 \]
Simplifying this:
\[ 7x - 3x + 6 = 2 \implies 4x + 6 = 2 \implies 4x = 2 - 6 \implies 4x = -4 \implies x = -1 \]
Now substitute \( x = -1 \) back into (4) to find y:
\[ y = -1 - 2 = -3 \]

3. **Substitute x and y into the third equation:**
Now we substitute \( x = -1 \) and \( y = -3 \) into equation (3):
\[ [3\pi](-1) - [e](-3) = [4a] \]
We need to find the greatest integer values of \( [3\pi] \) and \( [e] \).

- The value of \( \pi \) is approximately \( 3.14 \), so \( 3\pi \approx 9.42 \) and thus \( [3\pi] = 9 \).
- The value of \( e \) is approximately \( 2.71 \), so \( [e] = 2 \).

Now substituting these values:
\[ 9(-1) - 2(-3) = [4a] \]
This simplifies to:
\[ -9 + 6 = [4a] \implies -3 = [4a] \]

4. **Determine the range for 'a':**
Since \( [4a] = -3 \), this means:
\[ -3 \leq 4a < -2 \]
Dividing the entire inequality by 4:
\[ -\frac{3}{4} \leq a < -\frac{1}{2} \]

5. **Final answer:**
Therefore, the value of 'a' should lie in the range:
\[ a \in \left[-\frac{3}{4}, -\frac{1}{2}\right) \]