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

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.