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Sagot :
Certainly! Let me walk you through each part of the question step-by-step.
### (i) Suraj's Question: About \(\sqrt{8}\)
Question: Which of the following is true about \(\sqrt{8}\)?
Options are:
1. (a) It is a natural number
2. (b) It is an irrational number
3. (c) It is a rational number
4. (d) None of these
Solution:
- A natural number is a positive integer: \(\sqrt{8}\) is not a natural number.
- A rational number can be expressed in the form \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\): \(\sqrt{8}\) cannot be expressed this way as it cannot be simplified to a fraction with integer components.
- Since \(\sqrt{8}\) is not rational, it must be irrational.
Answer: (b) It is an irrational number
### (ii) Shreya's Question: About Which is Not Irrational
Question: Which of the following is not irrational?
Options are:
1. (a) \(3 - 4\sqrt{5}\)
2. (b) \(\sqrt{7} - 6\)
3. (c) \(2 + 2\sqrt{9}\)
4. (d) \(4\sqrt{11} - 6\)
Solution:
- \(3 - 4\sqrt{5}\) involves the irrational number \(\sqrt{5}\) making it irrational.
- \(\sqrt{7} - 6\) involves the irrational number \(\sqrt{7}\) making it irrational.
- \(2 + 2\sqrt{9} = 2 + 2 \times 3 = 2 + 6 = 8\) which is a rational number.
- \(4\sqrt{11} - 6\) involves the irrational number \(\sqrt{11}\) making it irrational.
Answer: (c) \(2 + 2\sqrt{9}\)
### (iii) Ananya's Question: About \(\sqrt{15} - \sqrt{10}\)
Question: \(\sqrt{15} - \sqrt{10}\) is ______ number.
Options are:
1. (a) a natural
2. (b) an irrational
3. (c) a whole
4. (d) a rational
Solution:
- The difference between two irrational numbers is typically irrational unless they cancel out perfectly, which they do not in this case.
Answer: (b) an irrational
### (iv) Suman's Question: About \(\frac{1}{\sqrt{5}}\)
Question: \(\frac{1}{\sqrt{5}} \) is ______ number.
Options are:
1. (a) a whole
2. (b) a rational
3. (c) an irrational
4. (d) a natural
Solution:
- \(\sqrt{5}\) is an irrational number, and dividing 1 by an irrational number remains irrational.
Answer: (c) an irrational
### (v) Preethi's Question: About \(\sqrt{6}\)
Question: Which of the following is not irrational?
Options are:
1. (a) \(15 + 3\sqrt{6}\)
2. (b) \(\sqrt{24} - 9\)
3. (c) \(5\sqrt{150}\)
4. (d) None of these
Solution:
- Each of the given options involves an irrational number component, thus all mentioned expressions are irrational.
Answer: (d) None of these
Summarizing the answers:
1. (i) (b) It is an irrational number
2. (ii) (c) \(2 + 2\sqrt{9}\)
3. (iii) (b) an irrational
4. (iv) (c) an irrational
5. (v) (d) None of these
These are the correct solutions.
### (i) Suraj's Question: About \(\sqrt{8}\)
Question: Which of the following is true about \(\sqrt{8}\)?
Options are:
1. (a) It is a natural number
2. (b) It is an irrational number
3. (c) It is a rational number
4. (d) None of these
Solution:
- A natural number is a positive integer: \(\sqrt{8}\) is not a natural number.
- A rational number can be expressed in the form \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\): \(\sqrt{8}\) cannot be expressed this way as it cannot be simplified to a fraction with integer components.
- Since \(\sqrt{8}\) is not rational, it must be irrational.
Answer: (b) It is an irrational number
### (ii) Shreya's Question: About Which is Not Irrational
Question: Which of the following is not irrational?
Options are:
1. (a) \(3 - 4\sqrt{5}\)
2. (b) \(\sqrt{7} - 6\)
3. (c) \(2 + 2\sqrt{9}\)
4. (d) \(4\sqrt{11} - 6\)
Solution:
- \(3 - 4\sqrt{5}\) involves the irrational number \(\sqrt{5}\) making it irrational.
- \(\sqrt{7} - 6\) involves the irrational number \(\sqrt{7}\) making it irrational.
- \(2 + 2\sqrt{9} = 2 + 2 \times 3 = 2 + 6 = 8\) which is a rational number.
- \(4\sqrt{11} - 6\) involves the irrational number \(\sqrt{11}\) making it irrational.
Answer: (c) \(2 + 2\sqrt{9}\)
### (iii) Ananya's Question: About \(\sqrt{15} - \sqrt{10}\)
Question: \(\sqrt{15} - \sqrt{10}\) is ______ number.
Options are:
1. (a) a natural
2. (b) an irrational
3. (c) a whole
4. (d) a rational
Solution:
- The difference between two irrational numbers is typically irrational unless they cancel out perfectly, which they do not in this case.
Answer: (b) an irrational
### (iv) Suman's Question: About \(\frac{1}{\sqrt{5}}\)
Question: \(\frac{1}{\sqrt{5}} \) is ______ number.
Options are:
1. (a) a whole
2. (b) a rational
3. (c) an irrational
4. (d) a natural
Solution:
- \(\sqrt{5}\) is an irrational number, and dividing 1 by an irrational number remains irrational.
Answer: (c) an irrational
### (v) Preethi's Question: About \(\sqrt{6}\)
Question: Which of the following is not irrational?
Options are:
1. (a) \(15 + 3\sqrt{6}\)
2. (b) \(\sqrt{24} - 9\)
3. (c) \(5\sqrt{150}\)
4. (d) None of these
Solution:
- Each of the given options involves an irrational number component, thus all mentioned expressions are irrational.
Answer: (d) None of these
Summarizing the answers:
1. (i) (b) It is an irrational number
2. (ii) (c) \(2 + 2\sqrt{9}\)
3. (iii) (b) an irrational
4. (iv) (c) an irrational
5. (v) (d) None of these
These are the correct solutions.
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