CBSE MATHEMATICS WORKSHEET

Class: X | Chapter: Real Numbers | Worksheet No: 01

Difficulty Level: Mixed | Time: 3 Hours | Total Marks: 80

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Chapter Introduction

In this chapter, we explore the properties of positive integers based on the NCERT curriculum. We focus on The Fundamental Theorem of Arithmetic, which states that every composite number can be uniquely expressed as a product of prime numbers. This theorem is essential for determining the HCF and LCM of integers and provides the foundation for proving the irrationality of numbers like \(\sqrt{2}\), \(\sqrt{3}\), and \(\sqrt{5}\). We also explore the relationship between prime factorisation and decimal expansions of rational numbers.

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Section A: Multiple Choice Questions (10 × 1 Mark)

1. The exponent of \(2\) in the prime factorisation of \(144\) is:
   a) \(2\) | b) \(4\) | c) \(3\) | d) \(5\)
2. If \(p\) and \(q\) are two coprime numbers, then \(HCF(p, q)\) is:
   a) \(p\) | b) \(q\) | c) \(1\) | d) \(pq\)
3. The product of a non-zero rational and an irrational number is:
   a) Always rational | b) Always irrational | c) One | d) Zero
4. If \(a = x^3y^2\) and \(b = xy^3\) (\(x, y\) are primes), then \(LCM(a, b)\) is:
   a) \(xy\) | b) \(x^2y^2\) | c) \(x^3y^3\) | d) \(x^3y^2\)
5. Which of the following is an irrational number?
   a) \(3.14\) | b) \(\sqrt{16}\) | c) \(5 - \sqrt{3}\) | d) \(22/7\)
6. The prime factorisation of a natural number is unique except for the:
   a) Value of factors | b) Number of factors | c) Order of factors | d) Sum of factors
7. If \(p\) is a prime and \(p\) divides \(a^2\), then \(p\) divides:
   a) \(2a\) | b) \(a\) | c) \(a^3\) | d) None
8. The \(HCF\) of the smallest prime number and the smallest composite number is:
   a) \(1\) | b) \(2\) | c) \(4\) | d) \(0\)
9. The product of \(HCF(6, 20)\) and \(LCM(6, 20)\) is:
   a) \(120\) | b) \(60\) | c) \(20\) | d) \(2\)
10. The prime factorisation of \(5005\) contains how many distinct prime factors?
    a) \(2\) | b) \(3\) | c) \(4\) | d) \(5\)

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Section B: Very Short Answer Questions (10 × 1 Mark)

1. State the Fundamental Theorem of Arithmetic.
2. Express \(156\) as a product of its prime factors.
3. Find the \(HCF\) of \(96\) and \(404\) using prime factorisation.
4. If \(HCF(306, 657) = 9\), find their \(LCM\).
5. Why can \(4^n\) never end with the digit \(0\) for any natural number \(n\)?
6. Check if \(7 \times 11 \times 13 + 13\) is a composite number.
7. What is the \(LCM\) of two prime numbers \(p\) and \(q\)?
8. Write the prime factorisation of \(3825\).
9. If \(HCF(a, b) = 1\), what are \(a\) and \(b\) called?
10. Define an irrational number.

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Section C: Short Answer Questions (10 × 2 Marks)

1. Find the \(HCF\) and \(LCM\) of \(12, 15,\) and \(21\) using prime factorisation.
2. Prove that \(3\sqrt{2}\) is irrational.
3. Verify the relationship \(HCF(26, 91) \times LCM(26, 91) = 26 \times 91\).
4. Find the prime factors of \(5005\) using a factor tree.
5. Show that \(5 - \sqrt{3}\) is irrational, given that \(\sqrt{3}\) is irrational.
6. Check whether \(6^n\) can end with the digit zero for any natural number \(n\).
7. Explain why \(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5\) is a composite number.
8. Find the largest number that divides \(70\) and \(125\), leaving remainders \(5\) and \(8\) respectively.
9. If \(p\) is a prime number, prove that \(\sqrt{p}\) is irrational.
10. Find the \(HCF\) and \(LCM\) of \(6, 72,\) and \(120\) using prime factorisation.

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Section D: Long/Application-Based Questions (10 × 3 Marks)

1. Prove that \(\sqrt{3}\) is irrational using the method of contradiction.
2. Three bells toll at intervals of \(9, 12,\) and \(15\) minutes respectively. If they start together, after how many minutes will they next toll together?
3. Sonia takes \(18\) minutes to drive one round of a circular track, while Ravi takes \(12\) minutes. If they start at the same point and time, after how many minutes will they meet at the start?
4. Using the prime factorisation method, find the \(HCF\) and \(LCM\) of \(120, 72,\) and \(6\).
5. Prove that \(3 + 2\sqrt{5}\) is irrational.
6. A rectangular courtyard is \(18\text{m } 72\text{cm}\) long and \(13\text{m } 20\text{cm}\) broad. Find the least number of square tiles required to pave it.
7. Find the smallest number which when increased by \(17\) is exactly divisible by both \(468\) and \(520\).
8. Show that any number of the form \(8^n\) can never end with the digit \(0\).
9. Prove that \(\sqrt{2} + \sqrt{3}\) is irrational.
10. Use the factor tree method to factorise \(32760\) into powers of primes.

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Section E: HOTS (Higher Order Thinking Skills) (5 × 4 Marks)

1. If the \(HCF\) of \(210\) and \(55\) is expressible in the form \(210 \times 5 + 55y\), find \(y\).
2. Prove that for any prime number \(p\), \(\sqrt{p}\) is irrational.
3. Show that \(n^2 - n\) is divisible by \(2\) for every positive integer \(n\).
4. Find the \(HCF\) of \(2^{100} - 1\) and \(2^{120} - 1\).
5. Show that there is no positive integer \(n\) for which \(\sqrt{n-1} + \sqrt{n+1}\) is rational.

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Section F: Case Study Questions (3 × 4 Marks)

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Section G: Assertion–Reason Questions (5 × 1 Mark)

(a) Both A and R are true, R explains A. (b) Both true, R doesn't explain A. (c) A is true, R false. (d) A is false, R true.

1. A: \(HCF(11, 17) = 1\).
   R: \(11\) and \(17\) are prime numbers.
2. A: \(2 + \sqrt{3}\) is irrational.
   R: The sum of a rational and an irrational number is always irrational.
3. A: \(4^n\) ends in \(0\) for some \(n\).
   R: Prime factorisation of \(4^n\) contains only the prime \(2\).
4. A: The prime factorisation of \(32760\) is unique.
   R: The Fundamental Theorem of Arithmetic guarantees uniqueness apart from order.
5. A: \(HCF(a, b) \times LCM(a, b) = a \times b \times c\).
   R: The product rule only applies to two integers.

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Important Formulas

• \(HCF(a, b) \times LCM(a, b) = a \times b\)
• \(LCM(p, q, r) = \frac{p \cdot q \cdot r \cdot HCF(p, q, r)}{HCF(p, q) \cdot HCF(q, r) \cdot HCF(p, r)}\)
• Theorem 1.2: If \(p \mid a^2\), then \(p \mid a\) (for prime \(p\)).

Common Mistakes

• Using \(HCF \times LCM = a \times b \times c\) for three numbers. (This is false!)
• Forgetting to define \(a, b\) as coprime integers in irrationality proofs.
• Confusing prime factors with all factors of a number.

Exam Tips

• Always write the Factor Tree steps clearly for full marks.
• In irrationality proofs, explicitly state the contradiction to conclude the proof.
• For word problems: Use \(HCF\) for "maximum/greatest" and \(LCM\) for "minimum/next time".