Sure! Here's an exam paper on Number System for Class 9. It covers various concepts related to rational and irrational numbers, real numbers, and operations on numbers. You can modify the difficulty and format based on your needs.

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Class 9 Mathematics: Number System Exam Paper

Time Allowed: 2 hours

Maximum Marks: 50

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General Instructions:

1. All questions are compulsory.
2. The paper consists of 4 sections – A, B, C, and D.
3. Section A contains 2 marks questions.
4. Section B contains 3 marks questions.
5. Section C contains 5 marks questions.
6. Section D contains questions related to higher-order thinking skills (HOTS).
7. Show all the working steps wherever applicable.

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Section A: (2 Marks each)

1. Define a rational number and give two examples.
2. Simplify the following expression: $\sqrt{50} + 2\sqrt{2}$.
3. If $x = \frac{7}{3}$ and $y = \frac{4}{5}$, find $x + y$.
4. Convert the decimal $0. \overline{3}$ into a fraction.
5. Find the HCF and LCM of 12 and 18 using prime factorization.

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Section B: (3 Marks each)

1. Prove that $\sqrt{2}$ is an irrational number.
2. Simplify: $\frac{5}{\sqrt{3}} + \frac{3}{\sqrt{3}}$ and rationalize the denominator.
3. Express $\frac{5}{6}$ as a decimal. Is it terminating or non-terminating?
4. What is the smallest number that must be multiplied with 72 to make it a perfect square?
5. Find the value of $\left( \sqrt{5} + \sqrt{7} \right)^2$.

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Section C: (5 Marks each)

1. Prove that the sum of a rational number and an irrational number is irrational.
2. If $x$ and $y$ are two irrational numbers such that $x + y$ is rational, prove that both $x$ and $y$ are rational.
3. Simplify and express in the simplest form:
   $\frac{4}{\sqrt{3} + \sqrt{2}}.$
4. Show that the number $\pi$ is irrational. Discuss its significance in mathematics.
5. Find the sum and product of the roots of the quadratic equation formed by the following:
   $x^2 - 4x + 4 = 0$

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Section D: Higher-Order Thinking Skills (HOTS): (5 Marks each)

1. Is the number $\frac{9}{7} + \sqrt{3}$ rational or irrational? Justify your answer.
2. Show that $\frac{1}{\sqrt{5}}$ is an irrational number. Discuss the reasoning behind it.
3. Prove that the difference between a rational number and an irrational number is irrational.
4. A number is expressed as $x + \sqrt{y}$, where $x$ and $y$ are rational. Discuss the possible values of $x$ and $y$ based on rational and irrational conditions.
5. If $p = \sqrt{2} + \sqrt{3}$ and $q = \sqrt{2} - \sqrt{3}$, find $p \times q$.

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End of Paper

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Additional Notes:

• Be sure to follow through the algebraic steps to simplify expressions.
• Practice rationalization techniques for dealing with square roots in the denominator.

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5 Related Questions to Explore Further:

1. What are the properties of rational and irrational numbers when added, subtracted, or multiplied?
2. How do you identify whether a given decimal is terminating or non-terminating?
3. How do we determine if a number is a perfect square or not?
4. What is the significance of irrational numbers in real-world applications?
5. Can a sum of two rational numbers ever be irrational? Why or why not?

Tip: When working with square roots and simplifying expressions, always check for possible perfect square factors to make calculations easier.

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Let me know if you'd like any more questions added or if you need clarification on any topic!