Class 10 Maths Chapter 1 Test Paper with Solution

Class 10 Maths Chapter 1 Test Paper: If you’re a Class 10 student preparing for your upcoming Maths exam, specifically for Chapter 1, which typically covers Real Numbers, you’ve come to the right place.In this article, we’ll provide you with a well organized Class 10 Maths Chapter 1 test paper complete with solutions, tips, and tricks to help you excel in your exam.

We’ll cover all the important concepts, question types, and practical strategies to ensure you’re thoroughly prepared.

Chapter 1: Real Numbers – Overview

Before diving into the test paper and solutions, let’s briefly review what Chapter 1 on Real Numbers entails. This chapter is a critical foundation for various higher-level topics in mathematics and consists of the following key concepts:

• Euclid’s Division Lemma: The principle that any two positive integers can be expressed as a divisor and a quotient, with the remainder being less than the divisor.
• The Fundamental Theorem of Arithmetic: Every composite number can be factorized uniquely as a product of prime numbers.
• The Decimal Expansion of Rational Numbers: Understanding how to convert fractions into decimal form and how to determine if the decimal expansion is terminating or non terminating.
• Real Numbers and Irrational Numbers: The distinction between rational numbers (fractions) and irrational numbers (non repeating, non terminating decimals) and their properties.

These concepts are crucial for both the theoretical understanding and practical problem-solving required in the Class 10 exam.

Class 10 Maths Chapter 1 Test Paper

Now that we’ve covered the basics, let’s dive into a Class 10 Maths Chapter 1 Test Paper on Real Numbers. This will give you a feel for the types of questions you might encounter in your exam. We’ll also include detailed solutions to help reinforce your understanding.

 Sample Test Paper: Class 10 Maths Chapter 1 – Real Numbers

Section A: Very Short Answer (1 Mark Each)

1. Find the HCF of 56 and 72 using Euclid’s Division Lemma.
2. Write the prime factorization of 84.
3. State whether 0.75 is a rational number or not. Justify your answer.
4. Find the least number which is divisible by 5, 7, and 9. Section B: Short Answer (2 Marks Each)
5. Explain the Fundamental Theorem of Arithmetic with an example.
6. Find the HCF and LCM of 24 and 36 using the prime factorization method.
7. Write the decimal expansion of the rational number 5/12. Is it terminating or non-terminating? Section C: Long Answer (3 Marks Each)
8. Prove that ( \sqrt{2} ) is an irrational number.
9. Show that the product of two irrational numbers may be a rational number, giving an example.
10. Find the HCF and LCM of 48 and 180 using Euclid’s Division Lemma. Section D: Very Long Answer (5 Marks)
11. The decimal expansion of a rational number is given as 0.6. Find its fractional form and state whether the decimal expansion is terminating or non-terminating.
12.  Using Euclid’s Division Lemma, find the HCF of 132 and 225, and verify your result by the LCM HCF relation.

 Solutions to Class 10 Maths Chapter 1 Test Paper

Section A: Very Short Answer

1. Find the HCF of 56 and 72 using Euclid’s Division Lemma.

• Step 1: Divide 72 by 56: ( 72 = 56 \times 1 + 16 )
• Step 2: Divide 56 by 16: ( 56 = 16 \times 3 + 8 )
• Step 3: Divide 16 by 8: ( 16 = 8 \times 2 + 0 ) The remainder is now 0, so the HCF is 8.

1. Prime factorization of 84. ( 84 = 2 \times 2 \times 3 \times 7 = 2^2 \times 3 \times 7 )
2. Is 0.75 a rational number? Yes, 0.75 is a rational number because it can be expressed as a fraction:
   ( 0.75 = \frac{75}{100} = \frac{3}{4} ).
3. Find the least number divisible by 5, 7, and 9. Use the LCM of 5, 7, and 9:
   ( \text{LCM}(5, 7, 9) = 5 \times 7 \times 9 = 315 ). Hence, the least number divisible by 5, 7, and 9 is 315. Section B: Short Answer
4. Fundamental Theorem of Arithmetic. The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be factored uniquely into prime numbers, except for the order of factors.
   Example: The prime factorization of 30 is ( 2 \times 3 \times 5 ), and no other factorization exists.
5. HCF and LCM of 24 and 36 using prime factorization.

• Prime factorization of 24: ( 2^3 \times 3 )
• Prime factorization of 36: ( 2^2 \times 3^2 )
• HCF: ( 2^2 \times 3 = 12 )
• LCM: ( 2^3 \times 3^2 = 72 )

1. Decimal expansion of 5/12. Dividing 5 by 12, we get ( 5 \div 12 = 0.416666… ), which is a non terminating, repeating decimal. Section C: Long Answer
2. Prove that ( \sqrt{2} ) is an irrational number.

• Assume ( \sqrt{2} ) is rational, i.e., ( \sqrt{2} = \frac{p}{q} ) where p and q are coprime integers.
• Squaring both sides: ( 2 = \frac{p^2}{q^2} ), so ( p^2 = 2q^2 ).
• This implies ( p^2 ) is even, so p must be even.
• Let ( p = 2k ). Substituting into the equation, we get ( 4k^2 = 2q^2 ) or ( q^2 = 2k^2 ), which means q is also even.
• But this contradicts the assumption that p and q are coprime. Hence, ( \sqrt{2} ) is irrational.

1. Product of two irrational numbers may be rational. Example: Consider ( \sqrt{2} \times \sqrt{8} = \sqrt{16} = 4 ), which is a rational number.
2. HCF and LCM of 48 and 180 using Euclid’s Division Lemma. Follow the Euclidean algorithm as demonstrated in Section A for the given numbers. The result will be HCF = 12, and LCM = 720. Section D: Very Long Answer
3. Decimal expansion of 0.6 as a fraction. ( 0.6 = \frac{6}{10} = \frac{3}{5} ). The decimal expansion is terminating because it ends after one digit.
4. Find the HCF of 132 and 225 using Euclid’s Division Lemma. By applying the division lemma method, the HCF of 132 and 225 is found to be 3. Verify the result using the LCM-HCF relationship:
   ( \text{LCM} \times \text{HCF} = \text{Product of the numbers} ), i.e.,
   ( \text{LCM}(132, 225) = \frac{132 \times 225}{3} = 9900 ).

Conclusion

Class 10 Maths Chapter 1 Test Paper: By practicing these types of questions and solutions, you can enhance your understanding of Real Numbers and perform exceptionally well in your Class 10 Maths exam. Make sure to focus on key concepts such as Euclid’s Division Lemma, prime factorization, and the Fundamental Theorem of Arithmetic A, as they form the backbone of this chapter.

Regular practice of problems, along with understanding the concepts behind them, will ensure that you’re fully prepared for your exam.