PRACTICE TEST - 2024

Class X - Mathematics (Standard/Basic)

Chapter 1: Real Numbers

Time: 1 Hour Maximum Marks: 30

General Instructions:

1. All questions are compulsory.
2. Section A consists of 5 MCQs of 1 mark each.
3. Section B consists of 5 Short Answer questions of 2 marks each.
4. Section C consists of 2 Long Answer questions of 5 marks each.
5. Section D consists of 1 Case Study question of 5 marks.

Section A (1 Mark Each) 5 Marks

1. The HCF of the smallest prime number and the smallest composite number is:

• 1
• 2
• 4
• 0

2. If \(p\) and \(q\) are two coprime numbers, then their LCM is:

• 1
• \(p + q\)
• \(pq\)
• \(p/q\)

3. The prime factorisation of 140 is:

• \(2^2 \times 5 \times 7\)
• \(2 \times 5^2 \times 7\)
• \(2 \times 5 \times 7^2\)
• \(2^3 \times 5 \times 7\)

4. If \(\text{HCF}(306, 657) = 9\), then their LCM is:

• 22338
• 23338
• 22332
• 22383

5. For what value of \(n\), \(6^n\) can end with the digit 0?

• \(n = 5\)
• \(n = 2\)
• No value of \(n\)
• Any even \(n\)

Section B (2 Marks Each) 10 Marks

6. Explain why \(7 \times 11 \times 13 + 13\) is a composite number.

7. Find the HCF and LCM of 12, 15 and 21 by applying the prime factorisation method.

8. If \(\text{HCF}(a, b) = 12\) and \(a \times b = 1800\), find the \(\text{LCM}(a, b)\).

9. Check whether \(4^n\) can end with the digit zero for any natural number \(n\).

10. Show that \(5 - \sqrt{3}\) is irrational, given that \(\sqrt{3}\) is irrational.

Section C (5 Marks Each) 10 Marks

11. Prove that \(\sqrt{5}\) is irrational.

12. Find the HCF and LCM of 404 and 96 and verify that \(\text{HCF} \times \text{LCM} = \text{Product of the two numbers}\).

Section D (Case Study - 5 Marks) 5 Marks

Case Study: The Sports Field
There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction.

Based on the above information, answer the following (1 mark each):

1. What concept is used to find after how many minutes they meet again? (HCF or LCM?)
2. After how many minutes will they meet again at the starting point?
3. What is the prime factorisation of 18?
4. If Sonia takes 20 minutes instead of 18, after how many minutes would they meet Ravi (who still takes 12 mins)?
5. Is the number of minutes they meet a composite number or prime number?

Answer Key (for Evaluation)

Section A: 1(b), 2(c), 3(a), 4(a), 5(c)

Section B Hints:
6. \(13(7 \times 11 + 1) = 13 \times 78\). Has factors other than 1 and itself.
7. HCF=3, LCM=420.
8. \(LCM = 1800 / 12 = 150\).
9. \(4^n = (2^2)^n\). Only prime factor is 2. For ending in 0, 5 must be a factor.
10. Assume rational \(5 - \sqrt{3} = a/b \Rightarrow \sqrt{3} = 5 - a/b\). Rational ≠ Irrational.

Section C Hints:
11. Proof by contradiction: Assume \(\sqrt{5} = p/q\). Show \(p\) and \(q\) both divisible by 5.
12. \(404 = 2^2 \times 101\), \(96 = 2^5 \times 3\). HCF=4, LCM=9696. \(4 \times 9696 = 38784\), \(404 \times 96 = 38784\).

Section D: (i) LCM, (ii) 36 mins, (iii) \(2 \times 3^2\), (iv) 60 mins, (v) Composite.