`P ** q = p^2 - q^2`
`p % q = p^2 - q^2`
`p $ q = p^2 + q^2`
`p @ q = pq + p + q`
`p Delta q` = Remainder of `p/q`
`p © q` = greatest integer less than or equal to `p/q`.
If `p = 11 and q = 7` , then the value of `(p ** q) @ (p $ q)` is:

Options

A. 14641
B. 12482
C. 12243
D. None of these

Answer

To solve the problem step by step, we will first calculate the values of \( p ** q \) and \( p $ q \) using the definitions provided, and then we will use these results to find the final answer for \( (p ** q) @ (p $ q) \).

### Step 1: Calculate \( p ** q \)

The operator \( p ** q \) is defined as \( p^2 - q^2 \).

Given:
- \( p = 11 \)
- \( q = 7 \)

Now, calculate:
\[ p ** q = 11^2 - 7^2 \]
Calculating the squares:
\[ 11^2 = 121 \quad \text{and} \quad 7^2 = 49 \]
Now substitute these values:
\[ p ** q = 121 - 49 = 72 \]

### Step 2: Calculate \( p $ q \)

The operator \( p $ q \) is defined as \( p^2 + q^2 \).

Using the same values:
\[ p $ q = 11^2 + 7^2 \]
Again, substituting the squares:
\[ p $ q = 121 + 49 = 170 \]

### Step 3: Calculate \( (p ** q) @ (p $ q) \)

Now we need to calculate \( (p ** q) @ (p $ q) \), which is \( 72 @ 170 \).

The operator \( p @ q \) is defined as \( pq + p + q \).

Substituting the values:
\[ 72 @ 170 = 72 \times 170 + 72 + 170 \]

Calculating \( 72 \times 170 \):
\[ 72 \times 170 = 12240 \]

Now, add \( 72 + 170 \):
\[ 72 + 170 = 242 \]

Now combine these results:
\[ 72 @ 170 = 12240 + 242 = 12482 \]

### Final Answer

Thus, the value of \( (p ** q) @ (p $ q) \) is:
\[ \boxed{12482} \]