`a#b=a^(2)+b-1`
`a$b=a^(2)+b^(2)+1`
`"Max (a,b)"=|a+b|`
`"Min (a, b)"=|a-b|`
Max `[(2#3)$(3#4),(4$4)]` is equal to :

Options

A. a. 1234
B. b. 1236
C. c. 1335
D. d. none of these

Answer

To solve the problem step by step, we need to evaluate the expressions given in the question using the defined operations. Let's break it down:

### Step 1: Calculate \(2 \# 3\)
The operation \(a \# b\) is defined as:
\[ a \# b = a^2 + b - 1 \]
Substituting \(a = 2\) and \(b = 3\):
\[ 2 \# 3 = 2^2 + 3 - 1 \]
Calculating:
\[ = 4 + 3 - 1 = 6 \]

### Step 2: Calculate \(3 \# 4\)
Using the same operation \(a \# b\):
\[ 3 \# 4 = 3^2 + 4 - 1 \]
Calculating:
\[ = 9 + 4 - 1 = 12 \]

### Step 3: Calculate \(4 \$ 4\)
The operation \(a \$ b\) is defined as:
\[ a \$ b = a^2 + b^2 + 1 \]
Substituting \(a = 4\) and \(b = 4\):
\[ 4 \$ 4 = 4^2 + 4^2 + 1 \]
Calculating:
\[ = 16 + 16 + 1 = 33 \]

### Step 4: Calculate \(6 \$ 12\)
Now we need to evaluate \( (2 \# 3) \$ (3 \# 4) \), which is \(6 \$ 12\):
\[ 6 \$ 12 = 6^2 + 12^2 + 1 \]
Calculating:
\[ = 36 + 144 + 1 = 181 \]

### Step 5: Find the maximum of \(181\) and \(33\)
Finally, we need to find:
\[ \text{Max}(181, 33) \]
Calculating:
\[ = 181 \]

### Final Answer
Thus, the maximum value is:
\[ \boxed{181} \]