Binary Operations Worksheet featuring problems to solve using given binary operation rules.

Problem Analysis:

The worksheet involves solving binary operations defined by specific formulas. Let's break down each part of the problem and solve it step by step.

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Question 1:

The binary operation is defined as:
\[
a * b = 2a^2 - b
\]

#### Part (a): \( 2 * 3 \)
Substitute \( a = 2 \) and \( b = 3 \) into the formula:
\[
2 * 3 = 2(2)^2 - 3
\]
Calculate step by step:
\[
2(2)^2 = 2 \cdot 4 = 8
\]
\[
2 * 3 = 8 - 3 = 5
\]
So, the answer is:
\[
\boxed{5}
\]

#### Part (b): \( 1 * 2 \)
Substitute \( a = 1 \) and \( b = 2 \) into the formula:
\[
1 * 2 = 2(1)^2 - 2
\]
Calculate step by step:
\[
2(1)^2 = 2 \cdot 1 = 2
\]
\[
1 * 2 = 2 - 2 = 0
\]
So, the answer is:
\[
\boxed{0}
\]

#### Part (c): \( 2 * 1 \)
Substitute \( a = 2 \) and \( b = 1 \) into the formula:
\[
2 * 1 = 2(2)^2 - 1
\]
Calculate step by step:
\[
2(2)^2 = 2 \cdot 4 = 8
\]
\[
2 * 1 = 8 - 1 = 7
\]
So, the answer is:
\[
\boxed{7}
\]

#### Part (d): \( 3 * 2 \)
Substitute \( a = 3 \) and \( b = 2 \) into the formula:
\[
3 * 2 = 2(3)^2 - 2
\]
Calculate step by step:
\[
2(3)^2 = 2 \cdot 9 = 18
\]
\[
3 * 2 = 18 - 2 = 16
\]
So, the answer is:
\[
\boxed{16}
\]

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Question 2:

The binary operation is defined as:
\[
p \cdot q = 3p + q
\]

#### Part (a): \( 2 \cdot 1 \)
Substitute \( p = 2 \) and \( q = 1 \) into the formula:
\[
2 \cdot 1 = 3(2) + 1
\]
Calculate step by step:
\[
3(2) = 6
\]
\[
2 \cdot 1 = 6 + 1 = 7
\]
So, the answer is:
\[
\boxed{7}
\]

#### Part (b): \( 1 \cdot 2 \)
Substitute \( p = 1 \) and \( q = 2 \) into the formula:
\[
1 \cdot 2 = 3(1) + 2
\]
Calculate step by step:
\[
3(1) = 3
\]
\[
1 \cdot 2 = 3 + 2 = 5
\]
So, the answer is:
\[
\boxed{5}
\]

#### Part (c): \( (1 \cdot 2) \cdot 3 \)
First, evaluate \( 1 \cdot 2 \):
\[
1 \cdot 2 = 3(1) + 2 = 3 + 2 = 5
\]
Now, use the result \( 5 \) as \( p \) and \( q = 3 \) in the formula:
\[
5 \cdot 3 = 3(5) + 3
\]
Calculate step by step:
\[
3(5) = 15
\]
\[
5 \cdot 3 = 15 + 3 = 18
\]
So, the answer is:
\[
\boxed{18}
\]

#### Part (d): Find \( q \) if \( 4 \cdot q = 13 \)
Substitute \( p = 4 \) and the given result \( 4 \cdot q = 13 \) into the formula:
\[
4 \cdot q = 3(4) + q
\]
Simplify:
\[
4 \cdot q = 12 + q
\]
Given that \( 4 \cdot q = 13 \), set up the equation:
\[
12 + q = 13
\]
Solve for \( q \):
\[
q = 13 - 12 = 1
\]
So, the answer is:
\[
\boxed{1}
\]

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Final Answers:

1. \( 2 * 3 = \boxed{5} \)
2. \( 1 * 2 = \boxed{0} \)
3. \( 2 * 1 = \boxed{7} \)
4. \( 3 * 2 = \boxed{16} \)
5. \( 2 \cdot 1 = \boxed{7} \)
6. \( 1 \cdot 2 = \boxed{5} \)
7. \( (1 \cdot 2) \cdot 3 = \boxed{18} \)
8. \( q = \boxed{1} \)

Final Boxed Answers:

\[
\boxed{5, 0, 7, 16, 7, 5, 18, 1}
\]