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Question 1:

• Given: $A = \{a, b, p\}, B = \{2, 3\}, C = \{p, q, r, s\}$ Find $n[(A \cup C) \times B]$.

• Solution:

  1. First, calculate $A \cup C$: $A \cup C = \{a, b, p\} \cup \{p, q, r, s\} = \{a, b, p, q, r, s\}.$ So, $|A \cup C| = 6$.

  2. The Cartesian product $(A \cup C) \times B$ means combining every element of $A \cup C$ with every element of $B$.

     • $|B| = 2$.
     • Total number of elements in $(A \cup C) \times B = |A \cup C| \times |B| = 6 \times 2 = 12$.

  Answer: (c) 12.

Question 2:

• Given: If $n(A) = p$ and $n(B) = q$, the total number of relations that exist from $A$ to $B$ is $2^{pq}$.

• Solution:

  1. A relation from $A$ to $B$ is any subset of $A \times B$.
  2. The total number of elements in $A \times B = p \times q$.
  3. The number of subsets of a set with $n$ elements is $2^n$.
     • Thus, total relations = $2^{pq}$.

  Answer: (e) $2^{pq}$.

Question 3:

• Given: $F_1 = 1, F_2 = 3$, and $F_n = F_{n-1} + F_{n-2}$. Find $F_5$.

• Solution:

  1. Compute the sequence:
     • $F_3 = F_2 + F_1 = 3 + 1 = 4$,
     • $F_4 = F_3 + F_2 = 4 + 3 = 7$,
     • $F_5 = F_4 + F_3 = 7 + 4 = 11$.

  Answer: (d) 11.

Question 4:

• Given: If the sequence $t_1, t_2, t_3, \dots$ is in Arithmetic Progression (A.P.), find the nature of the sequence $t_6, t_{12}, t_{18}, \dots$.

• Solution:

  1. An A.P. is defined by $t_n = t_1 + (n-1)d$, where $d$ is the common difference.
  2. For terms like $t_6, t_{12}, t_{18}, \dots$, the indices $6, 12, 18, \dots$ form an A.P. with a common difference of 6.
  3. Hence, the corresponding terms $t_6, t_{12}, t_{18}, \dots$ also form an A.P.

  Answer: (b) An Arithmetic Progression.

Let me know if you’d like detailed steps for any of these solutions!

Here are 5 related questions to expand your understanding:

1. What is the difference between a Cartesian product and a union of sets?
2. How many subsets does a set with $n$ elements have?
3. What are the first 10 terms of the Fibonacci sequence?
4. How can we identify whether a sequence is in A.P., G.P., or neither?
5. What are some applications of relations in mathematics?

Tip:

For sequences defined by recurrence relations like Fibonacci, compute a few terms explicitly to understand the pattern before generalizing.