If `[x]^(2)- 7 [x] +10 lt 0 and 4 [y] ^(2) -16[y] + 7 lt 0,` then `[x+y]` cannot be ([.]` denotes greatest integer function):

Options

A. 7
B. 8
C. 9
D. both (b) and (c )

Answer

To solve the problem step by step, we need to analyze the inequalities given for the greatest integer function (denoted as [x]).

### Step 1: Define the variables
Let:
- \([x] = p\)
- \([y] = q\)

We need to analyze the inequalities:
1. \(p^2 - 7p + 10 < 0\)
2. \(4q^2 - 16q + 7 < 0\)

### Step 2: Solve the first inequality
We start with the first inequality:
\[ p^2 - 7p + 10 < 0 \]

To find the roots, we can factor the quadratic:
\[ p^2 - 7p + 10 = (p - 2)(p - 5) \]
Setting this equal to zero gives us the roots \(p = 2\) and \(p = 5\).

The quadratic opens upwards (as the coefficient of \(p^2\) is positive), so it is negative between the roots:
\[ 2 < p < 5 \]

### Step 3: Solve the second inequality
Now, we solve the second inequality:
\[ 4q^2 - 16q + 7 < 0 \]

Dividing the entire inequality by 4 gives:
\[ q^2 - 4q + \frac{7}{4} < 0 \]

To find the roots, we use the quadratic formula:
\[ q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 1\), \(b = -4\), and \(c = \frac{7}{4}\):
\[ q = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot \frac{7}{4}}}{2 \cdot 1} \]
\[ q = \frac{4 \pm \sqrt{16 - 7}}{2} = \frac{4 \pm \sqrt{9}}{2} = \frac{4 \pm 3}{2} \]

This gives us the roots:
\[ q = \frac{7}{2} \quad \text{and} \quad q = \frac{1}{2} \]

The quadratic opens upwards, so it is negative between the roots:
\[ \frac{1}{2} < q < \frac{7}{2} \]

### Step 4: Determine possible integer values for p and q
From the inequalities:
- For \(p\): \(2 < p < 5\) implies \(p\) can be \(3\) or \(4\).
- For \(q\): \(\frac{1}{2} < q < \frac{7}{2}\) implies \(q\) can be \(1\), \(2\), or \(3\).

### Step 5: Calculate possible values of \([x + y]\)
Now we need to find the possible values of \([x + y] = [p + q]\):
- If \(p = 3\) and \(q = 1\), then \(p + q = 4\).
- If \(p = 3\) and \(q = 2\), then \(p + q = 5\).
- If \(p = 3\) and \(q = 3\), then \(p + q = 6\).
- If \(p = 4\) and \(q = 1\), then \(p + q = 5\).
- If \(p = 4\) and \(q = 2\), then \(p + q = 6\).
- If \(p = 4\) and \(q = 3\), then \(p + q = 7\).

### Step 6: Determine which values cannot be achieved
The possible values for \([x + y]\) are \(4, 5, 6, 7\). The options given are \(7, 8, 9\).

Since \(8\) and \(9\) cannot be achieved from the combinations of \(p\) and \(q\), we conclude that:
- \([x + y]\) cannot be \(8\) or \(9\).

However, since the question asks which of the provided options cannot be \([x + y]\), we find that:
- The correct answer is \(9\).

### Final Answer
\([x + y]\) cannot be \(9\).