Let's evaluate the answers provided for the questions surrounding irrational and rational numbers.
-
Irrational number between 9 and 11:
- Responses:
- \( \sqrt{100} \) (This is actually 10, which is rational)
- \( \sqrt{99} \) (This is irrational and is between 9 and 11)
- \( \sqrt{20} \) (This is approximately 4.47, which is not between 9 and 11)
- \( \sqrt{10} \) (This is approximately 3.16, which is also not between 9 and 11)
- Correct answer: \( \sqrt{99} \)
- Responses:
-
Least integer whose square root is an irrational number between 5 and 7:
- The correct answer should be the least integer \(n\) such that \(5 < \sqrt{n} < 7\).
- The smallest integer for which \( \sqrt{n} \) is irrational and greater than 5 is 26, as \( \sqrt{25} = 5\) (rational),
- Therefore, the answers were:
- Responses:
- 26 (Correct, as \( \sqrt{26} \) is irrational and between 5 and 7)
- 6 (Too small)
- 25 (Not suitable, as 5 is rational)
- 13 (Too small)
- Correct answer: 26
- The correct answer should be the least integer \(n\) such that \(5 < \sqrt{n} < 7\).
-
Rational number that is between \( \sqrt{10} \) and \( \sqrt{11} \):
- \( \sqrt{10} \) is approximately 3.16, and \( \sqrt{11} \) is approximately 3.32.
- We need a rational number between these values.
- Responses:
- 3.1 (Valid, as it falls between \( \sqrt{10} \) and \( \sqrt{11} \))
- \( \frac{16}{5} \) (This is 3.2, which is also valid)
- 3.4 (Valid, as it is also between \( \sqrt{10} \) and \( \sqrt{11} \))
- 103 (Not valid)
- Correct answers: 3.1, \( \frac{16}{5} \), and 3.4.
-
Irrational number that is between \( \sqrt{14} \) and \( \sqrt{18} \):
- The approximate values are \( \sqrt{14} \approx 3.74 \) and \( \sqrt{18} \approx 4.24 \).
- Responses:
- \( \sqrt{13} \) (Approximately 3.61, valid)
- \( \sqrt{10} \) (Approximately 3.16, not valid)
- \( \sqrt{15} \) (Approximately 3.87, valid)
- \( \sqrt{19} \) (Approximately 4.36, not valid)
- Correct answers: \( \sqrt{13} \) and \( \sqrt{15} \).
-
How many irrational numbers are there between \( \sqrt{11} \) and \( \sqrt{17} \):
- There are indeed an infinite number of irrational numbers between any two distinct real numbers.
- Responses:
- an infinite number (Correct)
- four numbers (Incorrect)
- six numbers (Incorrect)
- twenty-eight numbers (Incorrect)
Overall, the responses contained correct answers but also included incorrect options for several questions. The confirmed accurate answers for specific questions you asked about are provided above.
1 answer