Q: How do I prove a number is irrational?

Use proof by contradiction: assume the number is rational (p/q in lowest terms), then show this leads to a contradiction — typically that both p and q share a common factor, which contradicts HCF(p,q)=1. This technique is used for proving √2, √3, √5 and expressions like 3+2√5 are irrational.

Q: What is the difference between terminating and non-terminating decimals?

A rational number p/q (in lowest terms) has a terminating decimal if and only if q has no prime factors other than 2 and 5. For example, 17/8 = 17/2³ terminates (= 2.125), but 1/3 does not (= 0.333...). Non-terminating repeating decimals are still rational; non-terminating non-repeating decimals are irrational.

Question 1

Where can I find NCERT Class 10 Maths solutions?

Accepted Answer

Learnizo provides free step-by-step NCERT Class 10 Maths solutions for all 15 chapters. You can search by chapter, exercise number, or keyword directly on this page.

Question 2

Are the NCERT Class 10 Maths solutions on this page accurate?

Accepted Answer

Yes. Solutions are written step-by-step by Learnizo's academic team and cover all exercises in the NCERT Class 10 Mathematics textbook. Each step is explained so students understand the method, not just the answer.

Question 3

How do I use Euclid's algorithm to find HCF?

Accepted Answer

Euclid's Division Lemma states: for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. To find HCF, repeatedly apply this: divide the larger number by the smaller, then replace the larger with the smaller and the smaller with the remainder. Stop when the remainder is 0. The last non-zero remainder is the HCF.

Question 4

How do I prove a number is irrational?

Accepted Answer

Use proof by contradiction: assume the number is rational (p/q in lowest terms), then show this leads to a contradiction — typically that both p and q share a common factor, which contradicts HCF(p,q)=1. This technique is used for proving √2, √3, √5 and expressions like 3+2√5 are irrational.

Question 5

What is the difference between terminating and non-terminating decimals?

Accepted Answer

A rational number p/q (in lowest terms) has a terminating decimal if and only if q has no prime factors other than 2 and 5. For example, 17/8 = 17/2³ terminates (= 2.125), but 1/3 does not (= 0.333...). Non-terminating repeating decimals are still rational; non-terminating non-repeating decimals are irrational.

NCERT Class 10 Maths Solutions

Chapter 1: Real Numbers

Exercise 1.1

Exercise 1.2

Exercise 1.3

Exercise 1.4

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