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Question VI: True-False Combinations

There are $10$ true-false questions. Each question has 2 options (true or false). The total number of combinations is: $2^{10} = 1024$ Answer: d) 1024 ways

Question VII: Statements on Simple Graphs

• A path is a sequence of edges connecting distinct vertices with no repetition.
• A trail is a sequence of edges where vertices can repeat, but edges do not.

Correct statement: c) Every trail is a path as well as every path is a trail.

Question VIII: Edges in a Complete Graph

The formula for the number of edges in a complete graph with $n$ vertices is: $\text{Edges} = \frac{n(n-1)}{2}$ Answer: b) $\frac{n(n-1)}{2}$

Question IX: Inverse Laplace Transform

The inverse Laplace transform of $\frac{s}{s^2 + a^2}$ is: $\sin(at)$ Answer: a) Sinat

Question X: Conditions for Laplace Transform

A Laplace transform exists if the function:

1. Is piecewise continuous.
2. Is of exponential order.

Answer: a) The function is piecewise continuous.

Section 2: Answer Any Five

(i) Subset Proof

Provide an example to prove $P(B) \subseteq P(A)$ is false when $A$ is a proper subset of $B$: Example: Let $A = \{1\}$ and $B = \{1, 2\}$. Here $P(A) = \{\{\}, \{1\}\}$, but $P(B) = \{\{\}, \{1\}, \{2\}, \{1, 2\}\}$. Clearly, $P(B) \not\subseteq P(A)$.

(ii) Complete Graph on Three Vertices

A complete graph $K_3$ on three vertices has: $\text{Edges} = \frac{3(3-1)}{2} = 3$ And it forms a triangle.

(iii) Four-Digit Numbers

To form a 4-digit number with digits $0, 1, 2, ..., 9$ without repetition:

• First digit: $9$ choices (no 0).
• Second digit: $9$ choices.
• Third digit: $8$ choices.
• Fourth digit: $7$ choices.

Total numbers: $9 \times 9 \times 8 \times 7 = 4536$

Would you like further clarification on any of these solutions or detailed explanations? Here are related follow-up questions:

1. How can we generalize the number of ways for true-false questions with $n$ questions?
2. What is the difference between a Eulerian trail and Hamiltonian path?
3. Can we prove the formula for the edges in a complete graph using induction?
4. What are other forms of Laplace transform results for trigonometric functions?
5. Why is piecewise continuity a necessary condition for the Laplace transform?

Tip: When dealing with subsets and power sets, visualize using Venn diagrams or small examples for clarity.