Answer :
Answer:
yes
Explanation:
Sure, here are the solutions to the questions:
1. **Represent on the number line:**
- (i) \(\frac{5}{7}\)
- (ii) \(-\frac{6}{7}\)
2. **Find rational numbers between:**
- (i) \(\frac{2}{3}\) and \(\frac{3}{4}\)
- (ii) \(\frac{1}{9}\) and \(\frac{2}{9}\)
For (i):
\[
\frac{2}{3} = 0.666\ldots \quad \text{and} \quad \frac{3}{4} = 0.75
\]
Rational numbers between them: 0.67, 0.68, 0.69, 0.7, 0.71, 0.72, 0.73, 0.74 (e.g., \(\frac{67}{100}\), \(\frac{68}{100}\))
For (ii):
\[
\frac{1}{9} = 0.111\ldots \quad \text{and} \quad \frac{2}{9} = 0.222\ldots
\]
Rational numbers between them: 0.12, 0.13, 0.14, 0.15, 0.16, 0.17, 0.18, 0.19, 0.2, 0.21 (e.g., \(\frac{12}{100}\), \(\frac{13}{100}\))
3. **Find three rational numbers between \(\frac{3}{5}\) and \(\frac{4}{7}\):**
Convert to decimals:
\[
\frac{3}{5} = 0.6 \quad \text{and} \quad \frac{4}{7} = 0.571\ldots
\]
Rational numbers: 0.58, 0.59, 0.595
4. **Find five rational numbers between \(\frac{3}{8}\) and \(\frac{3}{4}\):**
\[
\frac{3}{8} = 0.375 \quad \text{and} \quad \frac{3}{4} = 0.75
\]
Rational numbers: 0.4, 0.45, 0.5, 0.6, 0.7
5. **Find six rational numbers between 3 and 4:**
Rational numbers: 3.1, 3.2, 3.3, 3.4, 3.5, 3.6
6. **Represent \(\sqrt{2}\) on the number line:**
Use the geometric method to draw the square root on the number line.
7. **Represent \(\sqrt{3}\) on the number line:**
Use the geometric method to draw the square root on the number line.
8. **Represent \(\sqrt{5}\) on the number line:**
Use the geometric method to draw the square root on the number line.
9. **Write in decimal form and identify the type of decimal expansion:**
- (i) \(\frac{36}{100} = 0.36\) (Terminating)
- (ii) \(\frac{1}{11} \approx 0.0909\ldots\) (Non-terminating, repeating)
- (iii) \(\frac{4}{8} = 0.5\) (Terminating)
- (iv) \(\frac{9}{10} = 0.9\) (Terminating)
- (v) \(\frac{3}{7} \approx 0.428571\ldots\) (Non-terminating, repeating)
- (vi) \(\frac{7}{5} = 1.4\) (Terminating)
- (vii) \(\frac{1}{7} \approx 0.142857\ldots\) (Non-terminating, repeating)
- (viii) \(\frac{229}{400} = 0.5725\) (Terminating)
10. **Find three irrational numbers between 5 and 7:**
\(\sqrt{26}\), \(\sqrt{27}\), \(\sqrt{28}\)
11. **Express in the form of \(p/q\):**
- (i) \(0.6 = \frac{6}{10} = \frac{3}{5}\)
- (ii) \(0.47 = \frac{47}{100}\)
- (iii) \(18.47 = 18 + \frac{47}{100} = \frac{1847}{100}\)
- (iv) \(0.235 = \frac{235}{1000} = \frac{47}{200}\)
- (v) \(0.053 = \frac{53}{1000}\)
- (vi) \(0.2 = \frac{2}{10} = \frac{1}{5}\)
- (vii) \(0.53 = \frac{53}{100}\)
- (viii) \(0.2 = \frac{2}{10} = \frac{1}{5}\)
12. **Express \(0.\overline{9999}\) in the form of \(p/q\):**
Let \(x = 0.\overline{9999}\),
\[
10x = 9.\overline{9999}
\]
\[
10x - x = 9.9999 - 0.9999
\]
\[
9x = 9
\]
\[
x = 1
\]
13. **Classify the following as rational or irrational:**
- (i) \(\sqrt{23}\) (Irrational)
- (ii) \(\sqrt{225} = 15\) (Rational)
- (iii) 0.35 (Rational)
- (iv) 7.4747 (Rational)
- (v) \(1.010010001\ldots\) (Irrational)
14. **Visualize 3.765 on the number line:**
Between 3.7 and 3.8, closer to 3.8.
15. **Visualize 4.127 on the number line:**
Between 4.1 and 4.2, closer to 4.1.
16. **Find two rational and two irrational numbers between 0.5 and 0.55:**
Rational: 0.51, 0.52
Irrational: 0.515515515\ldots, 0.525525525\ldots
17. **Add:**
\[
(2\sqrt{3} - 5\sqrt{2}) + (\sqrt{3} + 2\sqrt{2}) = 3\sqrt{3} - 3\sqrt{2}
\]
18. **Multiply:**
- (i) \(\sqrt{3} \times \sqrt{2} = \sqrt{6}\)
- (ii) \(3\sqrt{2} \times \sqrt{7} = 3\sqrt{14}\)
19. **Divide:**
- (i) \(\frac{16\sqrt{5}}{4\sqrt{2}} = \frac{16}{4} \times \frac{\sqrt{5}}{\sqrt{2}} = 4\sqrt{\frac{5}{2}}\)
- (ii) \(\frac{12\sqrt{5}}{4\sqrt{3}} = \frac{12}{4} \times \frac{\sqrt{5}}{\sqrt{3}} = 3\sqrt{\frac{5}{3}}\)
20. **Simplify:**
\[
(3 + \sqrt{2})(3 - \sqrt{2}) = 3^2 - (\sqrt{2})^2 = 9 - 2 = 7
\]
