Worksheet: Number System – Mixed Problems

📄 Worksheet: Number System – Mixed Problems

Class: 9 CBSE

Chapter: Number System

Topic: Simplification, Rational/Irrational, Rationalisation, Exponents

Total Marks: 40

Time: 60 Minutes

🔹 Section A: Multiple Choice Questions (1 mark each)

Choose the correct option and write the answer.

1. \( (\sqrt{7} - \sqrt{5})(\sqrt{7} + \sqrt{5}) \) equals:

• 2
• \( \sqrt{2} \)
• 12
• \( \sqrt{35} \)

2. Which of the following is a rational number?

• \( \sqrt{2} \)
• \( \frac{1}{\sqrt{3}} \)
• \( \frac{3}{4} \)
• \( \pi \)

3. \( (2\sqrt{3})^2 = \)

• 12
• 6
• \( 4\sqrt{3} \)
• 9

4. \( 16^{3/4} \) is equal to:

• 4
• 8
• 16
• 2

5. \( \left( \frac{1}{5^2} \right)^{-1} = \)

• 5
• 25
• \( \frac{1}{25} \)
• -25

🔹 Section B: Very Short Answer (2 marks each)

Answer in one step wherever possible.

6. Rationalise: \( \frac{1}{\sqrt{3} + 1} \)
7. Classify as rational or irrational: \( 3 - \sqrt{11} \)
8. Simplify: \( (5 + \sqrt{2})^2 \)
9. Represent \( \sqrt{2} \) geometrically on a number line. (Sketch or describe method)
10. Find:
    • a) \( 27^{1/3} \)
    • b) \( 81^{3/4} \)

🔹 Section C: Short Answer Questions (3 marks each)

11. Simplify and express in simplest form: \( \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}} \)
12. Evaluate:
    • a) \( (3^2 \cdot 3^3) \div 3^4 \)
    • b) \( (2^3)^2 \cdot 2^{-4} \)
13. Without actual calculation, state whether the following are rational or irrational. Justify:
    • a) \( \sqrt{49} \)
    • b) \( \frac{2}{\sqrt{7}} \)
    • c) \( \sqrt{3} + \sqrt{7} \)

🔹 Section D: Long Answer Questions (4 marks each)

14. Simplify the expression:
    \( (2 + \sqrt{3})^2 - (2 - \sqrt{3})^2 \)
15. Rationalise and simplify:
    \( \frac{3}{\sqrt{5} - \sqrt{2}} + \frac{2}{\sqrt{5} + \sqrt{2}} \)
16. Prove that \( \sqrt{2} \) is irrational. (Use contradiction method)
17. Evaluate:
    • \( \left( \frac{8}{27} \right)^{2/3} \)
    • Then use exponent laws to show: \( 8^{2/3} \div 2^{4/3} \)

🔹 Section E: Challenge Question (5 marks)

18. Rohit says that since \( \pi \) is defined as the ratio of circumference to diameter (C/d), it must be a rational number. Do you agree? Justify with reasoning. Then calculate an approximate value of \( \pi \) using a circle of radius 7 cm (Use \( \pi \approx \frac{22}{7} \)) and check the accuracy of this approximation.

✅ Instructions for Students:

• Solve each section step by step.
• For geometry-related questions, draw figures neatly.
• Highlight final answers.
• Use separate sheets if necessary.