Class 10 Maths Worksheet
Instructions:
- Section A (MCQs): 10 questions × 1 mark = 10 marks
- Section B (Short Answer): 5 questions × 2 marks = 10 marks
- Section C (Long Answer): 2 questions × 5 marks = 10 marks
- All questions are compulsory.
Section A – Multiple Choice Questions (1 mark each)
(Based on previous years’ CBSE questions)
1. The HCF of 135 and 225 is: (CBSE 2020)
2. The decimal expansion of \(\frac{17}{8}\) terminates after how many decimal places? (CBSE 2019)
3. If the HCF of 408 and 1032 is expressible as \(1032 \times 2 + 408 \times p\), then the value of \(p\) is: (CBSE 2020)
4. The LCM of two numbers is 14 times their HCF. If the sum of LCM and HCF is 600 and one number is 280, the other number is: (CBSE 2018)
5. Which of the following is not irrational? (CBSE 2022)
Section B – Short Answer Questions (2 marks each)
6. Find the HCF of 96 and 404 by prime factorization method. (CBSE 2019)
7. Show that \(5 – \sqrt{3}\) is irrational. (CBSE 2020)
8. Find the LCM and HCF of 12, 15, and 21 using the prime factorization method. (CBSE 2021)
9. Prove that \(\sqrt{2}\) is irrational. (CBSE 2018)
10. If the HCF of 65 and 117 is expressible in the form \(65m – 117\), find the value of \(m\). (CBSE 2020)
Section C – Long Answer Questions (5 marks each)
11. Using Euclid’s division algorithm, find the HCF of 867 and 255. (CBSE 2019)
Also, express it in the form \(867x + 255y\).
12. Prove that the square of any positive integer is of the form \(3m\) or \(3m + 1\) for some integer \(m\). (CBSE 2022)
Hence, check whether the square of any positive integer can be of the form \(3m + 2\).
Answer Key
Section A (MCQs)
| Q. No. | Answer |
|---|---|
| 1 | b |
| 2 | c |
| 3 | a |
| 4 | c |
| 5 | c |
Section B (Short Answers)
6. HCF = 4 (Prime factors: \(96 = 2^5 \times 3\), \(404 = 2^2 \times 101\))
7. Proof by contradiction (Assume \(5 – \sqrt{3}\) is rational → \(\sqrt{3}\) is rational, which is false.)
8. LCM = 420, HCF = 3
9. Standard irrationality proof (Assume \(\sqrt{2} = \frac{p}{q}\) in lowest terms → contradiction.)
10. m = 2 (HCF(65, 117) = 13 → \(65m – 117 = 13\) → \(m = 2\))
Section C (Long Answers)
11. HCF = 51
Using Euclid’s algorithm:
\(867 = 255 \times 3 + 102\)
\(255 = 102 \times 2 + 51\)
\(102 = 51 \times 2 + 0\)
Expression: \(51 = 255 \times 2 – 867 \times (1)\)
12. Proof:
– Let \(n = 3k, 3k+1, 3k+2\) → Squaring gives:
– \(n^2 = 9k^2 = 3(3k^2) = 3m\)
– \(n^2 = 9k^2 + 6k + 1 = 3(3k^2 + 2k) + 1 = 3m + 1\)
– \(n^2 = 9k^2 + 12k + 4 = 3(3k^2 + 4k + 1) + 1\) → Still \(3m + 1\)
– Conclusion: No square can be of form \(3m + 2\).
Worksheet Features:
✅ Strictly follows CBSE previous years’ questions
✅ Balanced mix of MCQs, short & long answers
✅ Step-by-step solutions for key questions
✅ Covers HCF, LCM, irrationality proofs, and Euclid’s algorithm