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Question 31:

We need to find the difference $n(A \cup B) - n(A \cap B)$, given:

• $n(A \cup B) = 42$,
• $n(B) = 36$,
• $n(A \cap B) = 3$.

Using the formula for union of sets: $n(A \cup B) = n(A) + n(B) - n(A \cap B),$ we rearrange to find $n(A)$: $n(A) = n(A \cup B) - n(B) + n(A \cap B) = 42 - 36 + 3 = 9.$

Now calculate: $n(A \cup B) - n(A \cap B) = 42 - 3 = 39.$

Answer: C) 39

Question 32:

Simplify $(a - 5)(a + 5) + 4(a - 5) + (a - 5)$.

Factor out $(a - 5)$: $(a - 5)(a + 5) + 4(a - 5) + (a - 5) = (a - 5)[(a + 5) + 4 + 1].$ Simplify inside the brackets: $(a - 5)[a + 5 + 4 + 1] = (a - 5)(a + 10).$

The final answer: Answer: E) $(a - 5)(a + 10)$

Question 33:

Find $2^{15}$ divided by 3.

We know: $2^{15} = 32768.$

Divide by 3: $32768 \div 3 = 10922.666...$

The remainder is 2. This implies the division leaves us in the form: $\text{Remainder: } 2.$

Answer: E)

Question 34:

The table describes distances $x, y, z$ with corresponding times. We are to determine the relationships among $x, y, z$.

Given that the speed is constant: $\text{Time } \propto \text{Distance}.$

From the table:

• Distance $x = 150$, $y = 120$, $z = 90$.
• Time $t_x > t_y > t_z$.

Thus: Answer: D) $x > y > z$.

Let me know if you'd like further explanations or details for any of these problems!

Questions for further exploration:

1. How to apply set theory formulas in real-life problems?
2. How to factorize expressions efficiently?
3. What are properties of exponential remainders?
4. How does proportionality relate to distance and time?
5. Can you provide examples of similar logical reasoning in math?

Tip: Always check units and relationships between variables to confirm the logic of your solutions!