SUMMARY: The chapter on Real Numbers in Class 10 Mathematics explores the fundamental properties and applications of real numbers, including their representation and operations. KEY TOPICS: Euclid's Division Lemma, Fundamental Theorem of Arithmetic, prime factorization, HCF and LCM, irrational numbers, decimal representation of rational numbers, revisiting rational numbers and their decimal expansions.
Which of the following statements is true according to Euclid's Division Lemma?
AIf a divides b, then b is a multiple of a.
BEvery integer can be expressed as a product of prime numbers.
CThe HCF of two numbers is always greater than their LCM.
DIrrational numbers can be expressed as fractions.
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Correct answer: Option 1 — If a divides b, then b is a multiple of a.
Q21 Mark
What is the HCF of 24 and 36?
A6
B12
C18
D24
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Correct answer: Option 1 — 6
Q31 Mark
Which of the following is an example of an irrational number?
A1/3
B√2
C0.75
D7
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Correct answer: Option 2 — √2
Q41 Mark
According to the Fundamental Theorem of Arithmetic, how can every integer greater than 1 be expressed?
AAs a sum of two prime numbers.
BAs a product of prime numbers in a unique way.
CAs a difference of two squares.
DAs a product of its factors.
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Correct answer: Option 2 — As a product of prime numbers in a unique way.
Q51 Mark
What is the decimal representation of the rational number 1/7?
A0.142857
B0.3333
C0.25
D0.6666
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Correct answer: Option 1 — 0.142857
Short Answer Questions5 questions
Q63 Marks
State Euclid's Division Lemma and provide an example to illustrate it.
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Euclid's Division Lemma states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. For example, if a = 17 and b = 5, then 17 = 5 × 3 + 2, where q = 3 and r = 2.
Q73 Marks
Explain the Fundamental Theorem of Arithmetic.
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The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed uniquely as a product of prime numbers, up to the order of the factors. For instance, the number 60 can be expressed as 2 × 2 × 3 × 5 or 2² × 3 × 5.
Q83 Marks
How do you find the HCF of two numbers using prime factorization?
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To find the HCF of two numbers using prime factorization, first express both numbers as products of their prime factors. Then, identify the common prime factors and take the lowest power of each common factor. For example, for 36 (2² × 3²) and 48 (2⁴ × 3¹), the HCF is 2² × 3¹ = 12.
Q93 Marks
What is the decimal representation of the rational number 1/8?
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The decimal representation of the rational number 1/8 is 0.125. This is obtained by performing the division of 1 by 8, which results in a terminating decimal.
Q103 Marks
Prove that √2 is an irrational number.
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To prove that √2 is irrational, assume the contrary that √2 can be expressed as a fraction p/q, where p and q are coprime integers. Squaring both sides gives 2 = p²/q², leading to p² = 2q². This implies p² is even, hence p is even. Let p = 2k, substituting gives 2k² = q², implying q is also even. This contradicts the assumption that p and q are coprime, thus √2 is irrational.
Long Answer Questions5 questions
Q116 Marks
Using Euclid's Division Lemma, demonstrate how to find the HCF of two numbers, 56 and 98. Show each step of your calculation.
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To find the HCF of 56 and 98 using Euclid's Division Lemma, we start by dividing the larger number by the smaller number. We divide 98 by 56, which gives a quotient of 1 and a remainder of 42. Next, we apply the lemma again by dividing 56 by 42, resulting in a quotient of 1 and a remainder of 14. We then divide 42 by 14, yielding a quotient of 3 and a remainder of 0. Since the remainder is now 0, the last non-zero remainder, which is 14, is the HCF of 56 and 98.
Q126 Marks
Explain the Fundamental Theorem of Arithmetic and demonstrate its application by finding the prime factorization of 180.
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The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed uniquely as a product of prime numbers, up to the order of the factors. To find the prime factorization of 180, we start by dividing it by the smallest prime number, which is 2. Dividing 180 by 2 gives us 90. Continuing, we divide 90 by 2 again to get 45. Since 45 is not divisible by 2, we move to the next prime number, which is 3. Dividing 45 by 3 gives us 15, and dividing 15 by 3 gives us 5. Finally, 5 is a prime number. Therefore, the prime factorization of 180 is 2^2 × 3^2 × 5.
Q136 Marks
Define irrational numbers and provide examples. Explain how the decimal representation of an irrational number differs from that of a rational number.
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Irrational numbers are numbers that cannot be expressed as a fraction of two integers, meaning they cannot be represented in the form p/q where p and q are integers and q is not zero. Examples of irrational numbers include √2, π, and e. The decimal representation of an irrational number is non-terminating and non-repeating, meaning it goes on forever without repeating any sequence of digits. In contrast, the decimal representation of a rational number either terminates (like 0.75) or repeats (like 0.333...). This fundamental difference highlights the unique nature of irrational numbers within the real number system.
Q146 Marks
Calculate the LCM and HCF of the numbers 24 and 36 using their prime factorizations. Show your work clearly.
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To calculate the LCM and HCF of 24 and 36, we first find their prime factorizations. The prime factorization of 24 is 2^3 × 3^1, and for 36, it is 2^2 × 3^2. To find the HCF, we take the lowest power of each prime factor present in both factorizations. For 2, the lowest power is 2^2, and for 3, it is 3^1. Therefore, HCF = 2^2 × 3^1 = 4 × 3 = 12. To find the LCM, we take the highest power of each prime factor. For 2, the highest power is 2^3, and for 3, it is 3^2. Therefore, LCM = 2^3 × 3^2 = 8 × 9 = 72.
Q156 Marks
Discuss the concept of decimal representation of rational numbers. Provide examples of both terminating and non-terminating repeating decimals and explain how to convert a fraction into its decimal form.
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The decimal representation of rational numbers can either be terminating or non-terminating repeating decimals. A terminating decimal is one that ends after a certain number of digits, such as 0.75 or 2.5. A non-terminating repeating decimal, on the other hand, continues indefinitely but has a repeating pattern, such as 1/3 = 0.333... or 2/11 = 0.181818.... To convert a fraction into its decimal form, we can perform long division. For example, to convert 1/4 into decimal form, we divide 1 by 4, which gives us 0.25, a terminating decimal. For 1/3, dividing 1 by 3 results in 0.333..., a non-terminating repeating decimal.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): Euclid's Division Lemma states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b.
Reason (R): This lemma is used to find the HCF of two numbers using the division algorithm.
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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 Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed as a product of prime numbers in a unique way.
Reason (R): This theorem helps in finding the HCF and LCM of two numbers.
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Correct answer: Option 2 —
Both A and R are true, but R is not the correct explanation of A.
Q181 Mark
Assertion (A): Irrational numbers can be expressed as a fraction of two integers.
Reason (R): Irrational numbers have non-repeating, non-terminating decimal expansions.
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Correct answer: Option 3 —
A is true, but R is false.
Q191 Mark
Assertion (A): The decimal representation of a rational number is either terminating or non-terminating repeating.
Reason (R): This property does not apply to irrational numbers.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q201 Mark
Assertion (A): The HCF of two numbers can be found by multiplying their LCM with their product.
Reason (R): This statement is true only if both numbers are prime.
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Correct answer: Option 3 —
A is true, but R is false.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: Euclid's Division Lemma states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b.
Statement 2: The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed as a product of prime numbers in more than one way.
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Correct answer: Option 2 —
Only Statement 1 is true.
Q221 Mark
Statement 1: The HCF of two numbers is the largest number that divides both of them without leaving a remainder.
Statement 2: The LCM of two numbers is the smallest number that is a multiple of both of them.
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Correct answer: Option 1 —
Both statements are true.
Q231 Mark
Statement 1: Irrational numbers can be expressed as a fraction of two integers.
Statement 2: The decimal representation of a rational number is either terminating or repeating.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q241 Mark
Statement 1: The prime factorization of a number is unique, except for the order of the factors.
Statement 2: The HCF of two prime numbers is always 1.
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Correct answer: Option 1 —
Both statements are true.
Q251 Mark
Statement 1: Every rational number has a decimal representation that is either terminating or non-terminating.
Statement 2: The LCM of two numbers can be found using the formula: LCM(a, b) = (a * b) / HCF(a, b).
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Correct answer: Option 1 —
Both statements are true.
