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Activity on Real Numbers

In a classroom activity on real numbers, students have to pick a number card from a pile and frame a question about it if it is not a rational number for the rest of the class. The number cards picked up by the first 5 students and their questions on the numbers for the rest of the class are as shown below. Answer them.

(i) Suraj picked up [tex]\sqrt{8}[/tex] and his question was: Which of the following is true about [tex]\sqrt{8}[/tex]?
(a) It is a natural number
(b) It is an irrational number
(c) It is a rational number
(d) None of these

(ii) Shreya picked up 'BONUS' and her question was: Which of the following is not irrational?
(a) [tex]3-4 \sqrt{5}[/tex]
(b) [tex]\sqrt{7}-6[/tex]
(c) [tex]2+2 \sqrt{9}[/tex]
(d) [tex]4 \sqrt{11}-6[/tex]

(iii) Ananya picked up [tex]\sqrt{15}-\sqrt{10}[/tex] and her question was: [tex]\sqrt{15}-\sqrt{10}[/tex] is a ______ number.
(a) natural
(b) irrational
(c) whole
(d) rational

(iv) Suman picked up [tex]\frac{1}{\sqrt{5}}[/tex] and her question was: [tex]\frac{1}{\sqrt{5}}[/tex] is a ______ number.
(a) whole
(b) rational
(c) irrational
(d) natural

(v) Preethi picked up [tex]\sqrt{6}[/tex] and her question was: Which of the following is not irrational?
(a) [tex]15+3 \sqrt{6}[/tex]
(b) [tex]\sqrt{24}-9[/tex]
(c) [tex]5 \sqrt{150}[/tex]
(d) None of these

Sagot :

GinnyAnswer
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.
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