SUMMARY: The chapter "Number Systems" in Class 9 Mathematics introduces students to different types of numbers and their properties. KEY TOPICS: Real numbers, irrational numbers, rational numbers, decimal expansions, operations on real numbers, number line representation, laws of exponents for real numbers, surds, rationalization.
Which of the following is an example of an irrational number?
A√2
B1/3
C0.75
D-5
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Correct answer: Option 1 — √2
Q21 Mark
What is the decimal expansion of the rational number 1/8?
A0.125
B0.12
C0.1
D0.15
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Correct answer: Option 1 — 0.125
Q31 Mark
Which of the following statements about real numbers is true?
AAll real numbers are rational.
BIrrational numbers cannot be represented on the number line.
CEvery rational number has a terminating or repeating decimal expansion.
DReal numbers do not include integers.
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Correct answer: Option 3 — Every rational number has a terminating or repeating decimal expansion.
Q41 Mark
If a = 2 and b = 3, what is the value of a^b?
A5
B6
C8
D9
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Correct answer: Option 3 — 8
Q51 Mark
Rationalizing the denominator of the expression 1/(√5) results in which of the following?
A√5/5
B5/√5
C1/5
D√5
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Correct answer: Option 1 — √5/5
Short Answer Questions5 questions
Q63 Marks
Define rational numbers and provide two examples. How can they be represented on a number line?
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Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. Examples include 1/2 and -3/4. They can be represented on a number line by locating their positions based on their values relative to whole numbers.
Q73 Marks
What are irrational numbers? Give two examples and explain how they differ from rational numbers.
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Irrational numbers are numbers that cannot be expressed as a fraction of two integers. Examples include √2 and π. They differ from rational numbers in that their decimal expansions are non-terminating and non-repeating.
Q83 Marks
Explain the process of rationalizing the denominator of a fraction. Provide an example to illustrate your explanation.
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Rationalizing the denominator involves eliminating any irrational numbers from the denominator of a fraction. For example, to rationalize 1/√3, multiply the numerator and denominator by √3 to get √3/3, which has a rational denominator.
Q93 Marks
What is the law of exponents for real numbers? State and explain one law with an example.
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The law of exponents states that when multiplying two powers with the same base, you add the exponents. For example, a^m × a^n = a^(m+n). If a = 2, m = 3, and n = 2, then 2^3 × 2^2 = 2^(3+2) = 2^5 = 32.
Q103 Marks
Describe the decimal expansion of rational numbers and how it differs from that of irrational numbers.
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The decimal expansion of rational numbers either terminates or repeats after a certain point. For example, 1/4 = 0.25 (terminating) and 1/3 = 0.333... (repeating). In contrast, the decimal expansion of irrational numbers is non-terminating and non-repeating, such as π = 3.14159...
Long Answer Questions5 questions
Q116 Marks
Explain the difference between rational and irrational numbers. Provide examples of each and describe how they can be represented on the number line.
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Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. Examples include 1/2, -3, and 0.75. Irrational numbers, on the other hand, cannot be expressed as a simple fraction; their decimal expansions are non-repeating and non-terminating. Examples include √2 and π. On the number line, rational numbers can be precisely located, while irrational numbers are represented as points that cannot be exactly pinpointed, lying between rational numbers.
Q126 Marks
Discuss the laws of exponents for real numbers and provide examples to illustrate each law.
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The laws of exponents for real numbers include several key rules: 1) a^m × a^n = a^(m+n), which states that when multiplying like bases, you add the exponents; 2) a^m ÷ a^n = a^(m-n), which states that when dividing like bases, you subtract the exponents; 3) (a^m)^n = a^(mn), which states that when raising a power to another power, you multiply the exponents. For example, using the first law, 2^3 × 2^2 = 2^(3+2) = 2^5 = 32. These laws are essential for simplifying expressions involving powers.
Q136 Marks
What is a surd? Explain how to rationalize a surd with an example. Why is rationalization useful?
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A surd is an expression containing a root, such as a square root, that cannot be simplified to remove the root. For example, √3 is a surd. Rationalizing a surd involves eliminating the surd from the denominator of a fraction. For instance, to rationalize 1/√2, we multiply the numerator and denominator by √2, resulting in √2/2. Rationalization is useful because it simplifies calculations and provides a clearer representation of the number, especially in further mathematical operations.
Q146 Marks
Describe the decimal expansions of rational and irrational numbers. How can the nature of a number be determined based on its decimal expansion?
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Rational numbers have decimal expansions that either terminate (like 0.75) or repeat (like 0.333...). In contrast, irrational numbers have decimal expansions that are non-terminating and non-repeating (like 0.14159... for π). To determine the nature of a number based on its decimal expansion, one can observe whether the digits after the decimal point settle into a repeating pattern or come to an end. If they do not, the number is classified as irrational.
Q156 Marks
Illustrate the representation of real numbers on the number line. How can you locate both rational and irrational numbers on it?
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Real numbers can be represented on a number line, which is a straight line where each point corresponds to a real number. To locate rational numbers, one can mark fractions and whole numbers at equal intervals. For irrational numbers, such as √2, one can approximate its value (about 1.414) and find its position between 1 and 2 on the number line. By using a compass or ruler, one can also construct segments to represent irrational numbers accurately. This representation helps in visualizing the density of real numbers, where between any two rational numbers, there exist infinitely many irrational numbers.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): Every rational number can be expressed as a terminating or repeating decimal.
Reason (R): Irrational numbers have non-terminating, non-repeating decimal expansions.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q171 Mark
Assertion (A): The square root of 2 is a rational number.
Reason (R): Rational numbers can be expressed as the ratio of two integers.
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Correct answer: Option 3 —
A is true, but R is false.
Q181 Mark
Assertion (A): All integers are real numbers.
Reason (R): Real numbers include both rational and irrational numbers.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q191 Mark
Assertion (A): The decimal expansion of 1/3 is non-terminating and repeating.
Reason (R): All non-terminating decimals are irrational numbers.
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Correct answer: Option 3 —
A is true, but R is false.
Q201 Mark
Assertion (A): Rationalizing the denominator of a fraction can simplify the expression.
Reason (R): Rationalization involves eliminating surds from the denominator.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: All rational numbers can be expressed as terminating or repeating decimals.
Statement 2: Irrational numbers can be expressed as fractions of integers.
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Correct answer: Option 2 —
Only Statement 1 is true.
Q221 Mark
Statement 1: The square root of 2 is a rational number.
Statement 2: The decimal expansion of 1/3 is a non-terminating repeating decimal.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q231 Mark
Statement 1: Every real number can be represented on the number line.
Statement 2: The sum of two irrational numbers is always irrational.
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Correct answer: Option 2 —
Only Statement 1 is true.
Q241 Mark
Statement 1: Rationalization is used to eliminate surds from the denominator.
Statement 2: The exponent laws state that a^m * a^n = a^(m+n) for any real numbers a, m, and n.
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Correct answer: Option 1 —
Both statements are true.
Q251 Mark
Statement 1: The decimal expansion of a rational number is always terminating.
Statement 2: The number 0.101001000100001... is a rational number.
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Correct answer: Option 4 —
Both statements are false.
