If `x,y in (0,30)` such that `[x/3]+[(3pi)/2]+[y/2]+[(3pi)/4]=11/6x+5/4 y` (where {x} denotes greatest integer `<=x )` number of ordered pairs (x, y) is (A) `0` (B) `2` (C) `4` (D) none of these

Options

A. A. 0
B. B. 2
C. C. 4
D. D. none of these

Answer

To solve the given problem step by step, we start with the equation provided:

\[ \left[\frac{x}{3}\right] + \left(\frac{3\pi}{2}\right) + \left[\frac{y}{2}\right] + \left(\frac{3\pi}{4}\right) = \frac{11x}{6} + \frac{5y}{4} \]

### Step 1: Understanding the Greatest Integer Function
The notation \([x]\) denotes the greatest integer less than or equal to \(x\). This means that \([x/3]\) is an integer, and similarly for \([y/2]\).

### Step 2: Simplifying the Equation
We can rearrange the equation to isolate the integer parts on one side:

\[ \left[\frac{x}{3}\right] + \left[\frac{y}{2}\right] = \frac{11x}{6} + \frac{5y}{4} - \left(\frac{3\pi}{2} + \frac{3\pi}{4}\right) \]

### Step 3: Calculate the Constants
Calculating the constants on the right side:

\[ \frac{3\pi}{2} + \frac{3\pi}{4} = \frac{6\pi}{4} + \frac{3\pi}{4} = \frac{9\pi}{4} \]

### Step 4: Setting Up the Equation
Now we have:

\[ \left[\frac{x}{3}\right] + \left[\frac{y}{2}\right] = \frac{11x}{6} + \frac{5y}{4} - \frac{9\pi}{4} \]

### Step 5: Analyzing the Range of \(x\) and \(y\)
Given \(x, y \in (0, 30)\), we can express \(x\) and \(y\) in terms of their integer parts:

Let \(x = 3k + r\) where \(k = \left[\frac{x}{3}\right]\) and \(0 \leq r < 3\).

Let \(y = 2m + s\) where \(m = \left[\frac{y}{2}\right]\) and \(0 \leq s < 2\).

### Step 6: Finding Integer Values
From the equation, we need to ensure that both sides remain integers. Therefore, we can analyze the values of \(x\) and \(y\) that satisfy the equation.

### Step 7: Finding Possible Values for \(x\)
For \(x\) to be an integer, it must be divisible by 6 (as the fraction \(\frac{11x}{6}\) must yield an integer). The possible values for \(x\) in the range \(0\) to \(30\) are:

- \(0, 6, 12, 18, 24, 30\)

### Step 8: Finding Possible Values for \(y\)
For \(y\) to be an integer, it must be divisible by 4 (as the fraction \(\frac{5y}{4}\) must yield an integer). The possible values for \(y\) in the range \(0\) to \(30\) are:

- \(0, 4, 8, 12, 16, 20, 24, 28\)

### Step 9: Counting Ordered Pairs
Now, we can count the number of ordered pairs \((x, y)\):

- For each \(x\) value (6 options: \(6, 12, 18, 24\)), we can pair it with each \(y\) value (8 options: \(0, 4, 8, 12, 16, 20, 24, 28\)).

### Step 10: Total Ordered Pairs
The total number of ordered pairs \((x, y)\) is:

\[ \text{Total pairs} = \text{Number of valid } x \text{ values} \times \text{Number of valid } y \text{ values} = 4 \times 8 = 32 \]

### Conclusion
Since \(32\) is not listed in the options provided, the answer is:

**(D) None of these**