Answer :
Sure, let's solve these problems step-by-step:
### Problem 1:
The sum of the squares of two consecutive natural numbers is 113. What is the difference of the squares of these two numbers?
Let's denote the consecutive natural numbers by [tex]\( x \)[/tex] and [tex]\( x+1 \)[/tex].
Their squares are [tex]\( x^2 \)[/tex] and [tex]\( (x+1)^2 \)[/tex].
The sum of their squares is given:
[tex]\[ x^2 + (x+1)^2 = 113 \][/tex]
Expanding and simplifying this equation, we get:
[tex]\[ x^2 + x^2 + 2x + 1 = 113 \][/tex]
[tex]\[ 2x^2 + 2x + 1 = 113 \][/tex]
[tex]\[ 2x^2 + 2x - 112 = 0 \][/tex]
[tex]\[ x^2 + x - 56 = 0 \][/tex]
Solving this quadratic equation for [tex]\( x \)[/tex], we find the value of [tex]\( x \)[/tex]:
[tex]\[ x = 7 \][/tex]
The consecutive numbers are [tex]\( 7 \)[/tex] and [tex]\( 8 \)[/tex].
The difference of their squares:
[tex]\[ (8)^2 - (7)^2 = 64 - 49 = 15 \][/tex]
Thus, the correct answer is 15 (Option B).
### Problem 2:
If two chords, [tex]\(\overline{ AB }\)[/tex] and [tex]\(\overline{ CD }\)[/tex], of a circle intersect at right angles at a point inside the circle and [tex]\( m(\angle BAC )=30^{\circ} \)[/tex], then which one of the following is equal to [tex]\( m(\angle ABD ) \)[/tex]?
Since the chords intersect at right angles, and [tex]\( \angle BAC = 30^{\circ} \)[/tex], we can deduce:
[tex]\[ m(\angle BAD) + m(\angle DAC) = 90^{\circ} \][/tex]
Considering the right angle property of the intersecting chords:
[tex]\[ \angle ABD = 90^{\circ} - \angle BAC = 90^{\circ} - 30^{\circ} = 60^{\circ} \][/tex]
Thus, the correct answer is 60° (Option C).
### Problem 3:
Let [tex]\( S = \{x \mid 1 \leq x \leq \sqrt{17} \)[/tex] and [tex]\( x \)[/tex] is prime \}. Which one of the following is true?
The approximate value of [tex]\( \sqrt{17} \)[/tex] is slightly more than 4. Therefore, we are considering prime numbers between 1 and just over 4:
The prime numbers within this range are 2 and 3.
Therefore, [tex]\( S = \{2, 3\} \)[/tex] and the number of elements in [tex]\( S \)[/tex] is 2.
Thus, the correct answer is [tex]\( n(S) = 2 \)[/tex] (Option B).
### Problem 4:
Which one of the following is the simplified form of [tex]\( \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} \)[/tex]?
To rationalize the denominator, multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{(\sqrt{3} - \sqrt{2})(\sqrt{3} - \sqrt{2})}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})} \][/tex]
[tex]\[ = \frac{(\sqrt{3})^2 - 2\sqrt{3}\sqrt{2} + (\sqrt{2})^2}{(\sqrt{3})^2 - (\sqrt{2})^2} \][/tex]
[tex]\[ = \frac{3 - 2\sqrt{6} + 2}{3 - 2} \][/tex]
[tex]\[ = \frac{5 - 2\sqrt{6}}{1} \][/tex]
[tex]\[ = 5 - 2\sqrt{6} \][/tex]
Thus, the correct answer is [tex]\( 5 - 2\sqrt{6} \)[/tex] (Option D).
### Problem 5:
The Greatest Common Divisor (GCD) of two numbers is 10 and the Least Common Multiple (LCM) of these two numbers is 53,300. If one of the two numbers is 650, which one of the following is the other number?
Using the relationship between the GCD and LCM:
[tex]\[ \text{GCD} \times \text{LCM} = \text{Product of the Numbers} \][/tex]
[tex]\[ 10 \times 53300 = 650 \times \text{Other Number} \][/tex]
Solving for the other number:
[tex]\[ \text{Other Number} = \frac{10 \times 53300}{650} \][/tex]
[tex]\[ \text{Other Number} = \frac{533000}{650} \][/tex]
[tex]\[ \text{Other Number} = 820 \][/tex]
Thus, the correct answer is 820 (Option A).
### Problem 1:
The sum of the squares of two consecutive natural numbers is 113. What is the difference of the squares of these two numbers?
Let's denote the consecutive natural numbers by [tex]\( x \)[/tex] and [tex]\( x+1 \)[/tex].
Their squares are [tex]\( x^2 \)[/tex] and [tex]\( (x+1)^2 \)[/tex].
The sum of their squares is given:
[tex]\[ x^2 + (x+1)^2 = 113 \][/tex]
Expanding and simplifying this equation, we get:
[tex]\[ x^2 + x^2 + 2x + 1 = 113 \][/tex]
[tex]\[ 2x^2 + 2x + 1 = 113 \][/tex]
[tex]\[ 2x^2 + 2x - 112 = 0 \][/tex]
[tex]\[ x^2 + x - 56 = 0 \][/tex]
Solving this quadratic equation for [tex]\( x \)[/tex], we find the value of [tex]\( x \)[/tex]:
[tex]\[ x = 7 \][/tex]
The consecutive numbers are [tex]\( 7 \)[/tex] and [tex]\( 8 \)[/tex].
The difference of their squares:
[tex]\[ (8)^2 - (7)^2 = 64 - 49 = 15 \][/tex]
Thus, the correct answer is 15 (Option B).
### Problem 2:
If two chords, [tex]\(\overline{ AB }\)[/tex] and [tex]\(\overline{ CD }\)[/tex], of a circle intersect at right angles at a point inside the circle and [tex]\( m(\angle BAC )=30^{\circ} \)[/tex], then which one of the following is equal to [tex]\( m(\angle ABD ) \)[/tex]?
Since the chords intersect at right angles, and [tex]\( \angle BAC = 30^{\circ} \)[/tex], we can deduce:
[tex]\[ m(\angle BAD) + m(\angle DAC) = 90^{\circ} \][/tex]
Considering the right angle property of the intersecting chords:
[tex]\[ \angle ABD = 90^{\circ} - \angle BAC = 90^{\circ} - 30^{\circ} = 60^{\circ} \][/tex]
Thus, the correct answer is 60° (Option C).
### Problem 3:
Let [tex]\( S = \{x \mid 1 \leq x \leq \sqrt{17} \)[/tex] and [tex]\( x \)[/tex] is prime \}. Which one of the following is true?
The approximate value of [tex]\( \sqrt{17} \)[/tex] is slightly more than 4. Therefore, we are considering prime numbers between 1 and just over 4:
The prime numbers within this range are 2 and 3.
Therefore, [tex]\( S = \{2, 3\} \)[/tex] and the number of elements in [tex]\( S \)[/tex] is 2.
Thus, the correct answer is [tex]\( n(S) = 2 \)[/tex] (Option B).
### Problem 4:
Which one of the following is the simplified form of [tex]\( \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} \)[/tex]?
To rationalize the denominator, multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{(\sqrt{3} - \sqrt{2})(\sqrt{3} - \sqrt{2})}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})} \][/tex]
[tex]\[ = \frac{(\sqrt{3})^2 - 2\sqrt{3}\sqrt{2} + (\sqrt{2})^2}{(\sqrt{3})^2 - (\sqrt{2})^2} \][/tex]
[tex]\[ = \frac{3 - 2\sqrt{6} + 2}{3 - 2} \][/tex]
[tex]\[ = \frac{5 - 2\sqrt{6}}{1} \][/tex]
[tex]\[ = 5 - 2\sqrt{6} \][/tex]
Thus, the correct answer is [tex]\( 5 - 2\sqrt{6} \)[/tex] (Option D).
### Problem 5:
The Greatest Common Divisor (GCD) of two numbers is 10 and the Least Common Multiple (LCM) of these two numbers is 53,300. If one of the two numbers is 650, which one of the following is the other number?
Using the relationship between the GCD and LCM:
[tex]\[ \text{GCD} \times \text{LCM} = \text{Product of the Numbers} \][/tex]
[tex]\[ 10 \times 53300 = 650 \times \text{Other Number} \][/tex]
Solving for the other number:
[tex]\[ \text{Other Number} = \frac{10 \times 53300}{650} \][/tex]
[tex]\[ \text{Other Number} = \frac{533000}{650} \][/tex]
[tex]\[ \text{Other Number} = 820 \][/tex]
Thus, the correct answer is 820 (Option A).
