Real Numbers Class 10 Maths Exercise 1.2 NCERT Solutions

The NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.2 cover all 3 questions on proving numbers irrational, according to the latest 2026-27 CBSE syllabus. Every answer uses proof by contradiction, the one method this exercise rests on, and is solved step by step the way the board expects.

• Questions covered: 3 in total, all asking you to prove a given surd is irrational.
• Core skill: assuming the opposite, then reaching a contradiction using co-prime integers.
• Board value: Exercise 1.2 is the proof-writing part of the 3 to 4 marks Real Numbers usually carries in the CBSE Class 10 paper.

What Exercise 1.2 of Real Numbers Covers for Class 10

Exercise 1.2 is the proof-based set of the Real Numbers chapter. It takes the idea that some numbers cannot be written as a fraction and asks you to prove it, using contradiction. All 3 questions follow the same shape: assume the number is rational, then show that this forces an impossible situation.

• Q1: prove √5 is irrational using Theorem 1.2.
• Q2: prove 3 + 2√5 is irrational by isolating the surd.
• Q3: prove three more numbers irrational, namely 1√2, 7√5 and 6 + √2.

How to Solve Exercise 1.2 Question by Question

The whole exercise rests on one method: proof by contradiction. You assume the number is rational, write it as a fraction in lowest terms, then keep going until two facts clash. Get the five steps right and every question falls into place.

Proof by Contradiction Method in Real Numbers Exercise 1.2

Question 1 is the model proof, and Questions 2 and 3 simply reuse its result. The trick is to assume the number is rational, then squeeze out a contradiction. Two facts power every proof in this exercise.

• Theorem 1.2: if a prime p divides a², then p divides a. This is what moves you from "5 divides a²" to "5 divides a".
• Closure of rationals: the sum, difference, product and quotient (by a non-zero rational) of rationals is rational. This lets you isolate a surd and force the clash in Q2 and Q3.
• Reuse, do not reprove: once √5 is shown irrational in Q1, just quote it as a known fact in the later questions.

Common Mistakes Students Make in Real Numbers Exercise 1.2

Most lost marks here come from skipped lines, not hard ideas. A quick check of your working usually catches them before they cost you.

• Skipping the co-prime set-up: if you do not write a and b as co-prime, the final contradiction has nothing to contradict.
• Forgetting to cite the known surd: in Q2 and Q3 you must state that √5 or √2 is already proved irrational.
• Applying Theorem 1.2 to a non-prime: the divisibility step holds for primes only; name the prime each time.
• Stopping too early: show 5 divides both a and b, then state the clash with co-primality clearly.

Real Numbers Exercise 1.2 Marks and Previous Year Trends for Class 10

Real Numbers is a scoring chapter in the CBSE Class 10 board paper, usually worth 3 to 4 marks. Exercise 1.2 supplies the proof questions, which boards like because they test reasoning rather than calculation.

These solutions follow the 2026-27 NCERT exactly, so the proof structure you practise here matches what the board paper rewards.

Other Resources for This Chapter in Class 10 Maths Real Numbers

Pair these Exercise 1.2 solutions with the other Class 10 Maths resources for Real Numbers, all linked below. Each link opens the matching Collegedunia page.

NCERT Solutions for Class 10 Maths Real Numbers: All Exercises

The Real Numbers chapter has two exercises. The table below links each exercise to its own step-by-step solution page.

NCERT Solutions for Class 10 Maths: All Chapters

Once Real Numbers is done, move on to the other chapters. Each link opens that chapter's NCERT Solutions page.

All NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.2 with Step-by-Step Solutions

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