Worksheet: Number System – Mixed Problems

📄 Worksheet 1: 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:

   • A. 2

   • B. $\sqrt{2}$

   • C. 12

   • D. $\sqrt{35}$

2. Which of the following is a rational number?

   • A. $\sqrt{2}$

   • B. $\frac{1}{\sqrt{3}}$

   • C. $\frac{3}{4}$

   • D. $\pi$

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

   • A. 12

   • B. 6

   • C. $4\sqrt{3}$

   • D. 9

4. $16^{3/4}$ is equal to:

   • A. 4

   • B. 8

   • C. 16

   • D. 2

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

   • A. 5

   • B. 25

   • C. $\frac{1}{25}$

   • D. $-25$

🔹 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)

Show working where required.

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)

Explain each step clearly.

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)

This tests higher-order thinking.

18. Rohit says that since π 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 π 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.