Which of the following expression gives rational values ?

Answer

To solve the problem of finding which expressions give rational values, we need to analyze the given equation and determine the integer pairs (a, b) that satisfy it.

### Step-by-Step Solution:

1. **Understand the Equation**:
We are given the equation \( 2a^2 + 3b^2 = 35 \), where \( a \) and \( b \) are integers.

2. **Rearranging the Equation**:
We can rearrange the equation to isolate one variable. However, since we are looking for integer solutions, we can directly substitute integer values for \( a \) and solve for \( b \).

3. **Substituting Integer Values for \( a \)**:
We will substitute integer values for \( a \) and check if \( b \) remains an integer:
- For \( a = 0 \):
\[ 2(0)^2 + 3b^2 = 35 \implies 3b^2 = 35 \implies b^2 = \frac{35}{3} \quad \text{(not an integer)} \]
- For \( a = 1 \):
\[ 2(1)^2 + 3b^2 = 35 \implies 2 + 3b^2 = 35 \implies 3b^2 = 33 \implies b^2 = 11 \quad \text{(not an integer)} \]
- For \( a = 2 \):
\[ 2(2)^2 + 3b^2 = 35 \implies 8 + 3b^2 = 35 \implies 3b^2 = 27 \implies b^2 = 9 \implies b = \pm 3 \quad \text{(integer solutions: (2, 3), (2, -3))} \]
- For \( a = 3 \):
\[ 2(3)^2 + 3b^2 = 35 \implies 18 + 3b^2 = 35 \implies 3b^2 = 17 \implies b^2 = \frac{17}{3} \quad \text{(not an integer)} \]
- For \( a = 4 \):
\[ 2(4)^2 + 3b^2 = 35 \implies 32 + 3b^2 = 35 \implies 3b^2 = 3 \implies b^2 = 1 \implies b = \pm 1 \quad \text{(integer solutions: (4, 1), (4, -1))} \]
- For \( a = -2 \):
\[ 2(-2)^2 + 3b^2 = 35 \implies 8 + 3b^2 = 35 \implies 3b^2 = 27 \implies b^2 = 9 \implies b = \pm 3 \quad \text{(integer solutions: (-2, 3), (-2, -3))} \]
- For \( a = -4 \):
\[ 2(-4)^2 + 3b^2 = 35 \implies 32 + 3b^2 = 35 \implies 3b^2 = 3 \implies b^2 = 1 \implies b = \pm 1 \quad \text{(integer solutions: (-4, 1), (-4, -1))} \]

4. **Collecting All Solutions**:
The integer pairs (a, b) that satisfy the equation are:
- (2, 3)
- (2, -3)
- (4, 1)
- (4, -1)
- (-2, 3)
- (-2, -3)
- (-4, 1)
- (-4, -1)

5. **Counting the Solutions**:
We have found a total of 8 integer pairs.

### Final Answer:
Thus, the number of elements that give rational values is **8**.