Mastering the fundamentals of mathematics is crucial for board exam success, and it all begins with your very first textbook chapter. Real Numbers sets the mathematical tone for the entire academic year, carrying significant weight in the board examinations. To help you ace your preparation, these comprehensive ncert solutions class 10 maths chapter 1 are designed to break down complex theorems and textbook problems into simple, digestible steps.
Whether you are revising concepts or trying to figure out how to structure your answers for maximum marks, this guide provides complete, step-by-step solutions aligned with the latest CBSE guidelines. Let us dive straight into the core concepts and exercises that will make you exam-ready.
Understanding Real Numbers in Class 10
Before jumping into the exercises, let us quickly recap what this chapter entails. Real numbers encompass both rational and irrational numbers. In Class 10, you build upon your previous knowledge by exploring fundamental topics like the Fundamental Theorem of Arithmetic and the proofs of irrationality of numbers like $\sqrt{2}$ and $\sqrt{3}$.
If you want to strengthen your foundational concepts before solving questions, you can always Browse all chapter notes available on our portal. For a broader library of answers covering other subjects, check out our dedicated NCERT solutions page.
Key Theorems and Concepts
To solve the exercise problems efficiently, you need to be familiar with two primary concepts:
- Euclid's Division Lemma: Given positive integers $a$ and $b$, there exist unique integers $q$ and $r$ satisfying $a = bq + r$, where $0 \le r < b$. This is primarily used to compute the Highest Common Factor (HCF) of two positive integers.
- The Fundamental Theorem of Arithmetic: Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.
For reliable curriculum updates, you can cross-verify topics directly from the official NCERT website.
Step-by-Step Exercise Solutions
Let us solve some of the most important textbook questions that frequently appear in school assessments and board exams.
Exercise 1.1: HCF Using Euclid's Division Algorithm
Question: Use Euclid's division algorithm to find the HCF of 135 and 225.
Solution:
Step 1: Compare the two numbers. Since $225 > 135$, we apply Euclid's division lemma to 225 and 135 to get: $225 = 135 \times 1 + 90$
Step 2: Since the remainder $90 \neq 0$, we apply the division lemma to the divisor 135 and the remainder 90: $135 = 90 \times 1 + 45$
Step 3: Since the remainder $45 \neq 0$, we apply the division lemma to 90 and 45: $90 = 45 \times 2 + 0$
Step 4: The remainder has now become zero, so our current divisor is the HCF.
Final Answer: The HCF of 135 and 225 is 45.
Exercise 1.2: Prime Factorization and LCM/HCF Relationship
Question: Find the LCM and HCF of 26 and 91 and verify that $\text{LCM} \times \text{HCF} = \text{Product of the two numbers}$.
Solution:
Step 1: Find the prime factors of both numbers: $26 = 2 \times 13$ $91 = 7 \times 13$
Step 2: Calculate HCF (product of the smallest power of each common prime factor): $\text{HCF}(26, 91) = 13$
Step 3: Calculate LCM (product of the greatest power of each prime factor involved): $\text{LCM}(26, 91) = 2 \times 7 \times 13 = 182$
Step 4: Verify the relationship: Product of numbers $= 26 \times 91 = 2366$ $\text{LCM} \times \text{HCF} = 182 \times 13 = 2366$
Final Answer: HCF = 13, LCM = 182, and the relationship is successfully verified.
Exercise 1.3: Proving Irrationality
Question: Prove that $\sqrt{5}$ is irrational.
Solution:
Step 1 (Assumption): Let us assume to the contrary that $\sqrt{5}$ is rational. Therefore, we can find two coprime integers $a$ and $b$ ($b \neq 0$) such that: $\sqrt{5} = \frac{a}{b}$
Step 2: Rearrange and square both sides: $a = \sqrt{5}b \implies a^2 = 5b^2$ This means $5$ divides $a^2$, and consequently, $5$ divides $a$ as well.
Step 3: Therefore, we can write $a = 5c$ for some integer $c$. Substituting this back: $(5c)^2 = 5b^2 \implies 25c^2 = 5b^2 \implies b^2 = 5c^2$ This means $5$ divides $b^2$, which implies $5$ divides $b$.
Step 4 (Contradiction): From our steps, $5$ is a common factor of both $a$ and $b$. However, this contradicts our initial assumption that $a$ and $b$ are coprime.
Final Answer: Our assumption was incorrect. Hence, $\sqrt{5}$ is irrational.
To check your grasp of these concepts after practicing, try taking a few Practice tests to evaluate your speed and accuracy.
Tips to Score Full Marks in Chapter 1
Scoring a perfect 100 in Class 10 Maths requires more than just getting the right answer; presentation matters immensely in board evaluations. Follow these expert tips:
- Show Every Step: Never skip intermediate calculation steps, especially when applying Euclid's division algorithm or prime factorization. Examiners award step-marking.
- Memorize Proof Formats: Questions asking to prove irrationality (like $\sqrt{2}$, $\sqrt{3}$, or $\sqrt{5}$) follow a standard template. Practice writing out these contradiction proofs multiple times.
- Double-Check LCM and HCF Rules: Always remember the formula $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$. Note that this rule specifically applies to two positive integers, not three!
- Keep Your Work Neat: Use sharp pencils for rough work columns and keep your final answer clearly boxed or underlined.
If you ever wish to discover more ways to boost your overall academic performance, feel free to Explore CircleOfLearning to find comprehensive study guides and resources tailored for students.
Frequently Asked Questions
What is the weightage of Real Numbers in the Class 10 Board Exam?
Real Numbers generally carries around 6 marks in the CBSE Class 10 Mathematics board question paper. It typically features a mix of short-answer questions based on HCF/LCM and a mandatory 3-mark question on proving the irrationality of a number.
How do I use Euclid's Division Lemma effectively?
Euclid's Division Lemma states $a = bq + r$ where $0 \le r < b$. To find the HCF of two numbers, you keep dividing the larger number by the smaller one, and then the divisor by the successive remainders until the remainder drops down to zero. The final non-zero remainder is your HCF.
Can the LCM of two numbers be smaller than their HCF?
No, the LCM of two or more numbers is always greater than or equal to their HCF. In fact, the HCF is always a factor of the LCM for any given set of numbers.
Are NCERT solutions enough to score 95+ in Class 10 Maths?
NCERT textbooks and solutions form the core bedrock of the CBSE curriculum. Solving every in-text question and exercise example thoroughly is mandatory. However, to score 95+, you should supplement your practice with previous years' question papers and sample mock tests.
Is decimal expansion of rational numbers included in the current syllabus?
No, according to the rationalized NCERT curriculum, topics involving long division to determine terminating or non-terminating repeating decimal expansions have been removed from the current academic session. Focus primarily on prime factorization, LCM, HCF, and irrationality proofs.
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