Let's solve these questions step-by-step.
### Solution to Question 3
Given the set [tex]\( S = \{ x \mid 1 \leq x \leq \sqrt{17} \text{ and } x \text{ is prime} \} \)[/tex]:
1. First, calculate [tex]\( \sqrt{17} \approx 4.12 \)[/tex].
2. Determine all prime numbers within the range from 1 to 4.12.
- The prime numbers less than or equal to 4 are 2 and 3.
Therefore, [tex]\( S = \{2, 3\} \)[/tex].
Now let's evaluate each option:
A. [tex]\( \sqrt{17} \in S \)[/tex] - False because [tex]\(\sqrt{17} \approx 4.12\)[/tex] is not a prime number.
B. [tex]\( n(S) = 2 \)[/tex] - True since the set [tex]\( S \)[/tex] has 2 elements [tex]\(\{2, 3\}\)[/tex].
C. [tex]\( 1 \in S \)[/tex] - False since 1 is not a prime number.
D. [tex]\( S \)[/tex] has a subset of 4 elements - False, as [tex]\( S \)[/tex] only has 2 elements, it cannot have a subset with 4 elements.
The correct answer is B.
### Solution to Question 4
Simplify [tex]\( \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} \)[/tex]:
1. Rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator, [tex]\( \sqrt{3} - \sqrt{2} \)[/tex].
[tex]\[
\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} \times \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = \frac{(\sqrt{3} - \sqrt{2})^2}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})}
\][/tex]
2. Simplify the numerator and denominator:
[tex]\[
(\sqrt{3} - \sqrt{2})^2 = (\sqrt{3})^2 - 2 \cdot \sqrt{3} \cdot \sqrt{2} + (\sqrt{2})^2 = 3 - 2\sqrt{6} + 2
\][/tex]
[tex]\[
(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2}) = (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1
\][/tex]
3. Thus, the simplified form is:
[tex]\[
\frac{3 - 2\sqrt{6} + 2}{1} = 5 - 2\sqrt{6}
\][/tex]
The correct answer is D.
### Solution to Question 5
Given:
- The Greatest Common Divisor (GCD) is 10.
- The Least Common Multiple (LCM) is 53,300.
- One of the numbers is 650.
To find the other number, use the relationship between the GCD and LCM of two numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
\text{GCD} \times \text{LCM} = \text{num1} \times \text{num2}
\][/tex]
Substitute the given values:
[tex]\[
10 \times 53,300 = 650 \times \text{num2}
\][/tex]
Solving for [tex]\(\text{num2}\)[/tex]:
[tex]\[
\text{num2} = \frac{10 \times 53,300}{650} = 820
\][/tex]
Therefore, the other number is 820.
The correct answer is A.
