If [x] denote the greatest integer less than or equal to x then in order that the set of equations `x - 3y = 5, 5x + y = 2, [2pi] x - [e] y = [2a]` 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, we need to ensure that the system of equations is consistent. The equations given are:

1. \( x - 3y = 5 \) (Equation 1)
2. \( 5x + y = 2 \) (Equation 2)
3. \( [2\pi] x - [e] y = [2a] \) (Equation 3)

Where \([x]\) denotes the greatest integer less than or equal to \(x\).

### Step 1: Solve the first two equations for \(x\) and \(y\)

From Equation 1:
\[ x = 3y + 5 \]

Substituting \(x\) into Equation 2:
\[ 5(3y + 5) + y = 2 \]
\[ 15y + 25 + y = 2 \]
\[ 16y + 25 = 2 \]
\[ 16y = 2 - 25 \]
\[ 16y = -23 \]
\[ y = -\frac{23}{16} \]

Now substituting \(y\) back into the expression for \(x\):
\[ x = 3\left(-\frac{23}{16}\right) + 5 \]
\[ x = -\frac{69}{16} + \frac{80}{16} \]
\[ x = \frac{11}{16} \]

### Step 2: Substitute \(x\) and \(y\) into Equation 3

Now we substitute \(x\) and \(y\) into Equation 3:
\[ [2\pi] x - [e] y = [2a] \]

Calculating the greatest integer values:
- \([2\pi] = 6\) (since \(2\pi \approx 6.28\))
- \([e] = 2\) (since \(e \approx 2.71\))

Substituting these values:
\[ 6\left(\frac{11}{16}\right) - 2\left(-\frac{23}{16}\right) = [2a] \]
\[ \frac{66}{16} + \frac{46}{16} = [2a] \]
\[ \frac{112}{16} = [2a] \]
\[ 7 = [2a] \]

### Step 3: Determine the range for \(a\)

Since \([2a] = 7\), this means:
\[ 7 \leq 2a < 8 \]

Dividing the entire inequality by 2 gives:
\[ \frac{7}{2} \leq a < 4 \]

Thus, the value of \(a\) must lie in the interval:
\[ \left[\frac{7}{2}, 4\right) \]

### Final Answer

The value of \(a\) should lie in the interval:
\[ \left[\frac{7}{2}, 4\right) \]