## CBSE Class 10 Maths Chapter-1 Real Numbers – Free PDF Download

Free PDF download of Important Questions with Answers for CBSE Class 10 Maths Chapter 1 – Real Numbers prepared by expert Maths teachers from latest edition of CBSE(NCERT) books only by CoolGyan to score more marks in CBSE board examination.

CBSE Class 10 Maths Chapter-1 Real Numbers Important Questions

**CBSE Class 10 Maths Important Questions Chapter 1 – Real Numbers**

**4 Mark Questions**

**1. Prove that the following are irrationals.**

**(i) **

**(ii) **

**(iii) **

**Ans .** **(i) **We can prove irrational by contradiction.

Let us suppose that is rational.

It means we have some co-prime integers *a* and *b* (*b*≠0) such that

=*ab*

⇒ … **(1)**

**R.H.S** of** (1)** is rational but we know that is irrational.

It is not possible which means our supposition is wrong.

Therefore, cannot be rational.

Hence, it is irrational.

**(ii) **We can prove irrational by contradiction.

Let us suppose that is rational.

It means we have some co-prime integers *a* and *b* (*b*≠0) such that

⇒ … **(1)**

**R.H.S** of** (1)** is rational but we know that is irrational.

It is not possible which means our supposition is wrong.

Therefore, cannot be rational.

Hence, it is irrational.

**(iii) **We will prove irrational by contradiction.

Let us suppose that () is rational.

It means that we have co-prime integers ** a and b **(

*b*≠0) such that

⇒

⇒

*…*

**(1)**

**are integers.**

*a*and*b*It means

**L.H.S**of

**(1)**is rational but we know that is irrational. It is not possible.

Therefore, our supposition is wrong. () cannot be rational.

Hence, () is irrational.

**2. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion.**

**(i) **

**(ii) **

**(iii) **

**(iv) **

**(v) **

**(vi) **

**(vii) **

**(viii) **

**(ix) **

**(x) **

**Ans.** According to Theorem, any given rational number of the form **where p and q are co-prime**, has a terminating decimal expansion if q is of the form 2

*× 5*

^{n}*, where m and n are non-negative integers.*

^{m}**(i)**

*q*=3125=5×5×5×5×5=5

^{5}

Here, denominator is of the form 2

*× 5*

^{n}*, where m=5 and n=0.*

^{m}It means rational number has a

**terminating**decimal expansion.

**(ii)**

*q*=8=2×2×2=2

^{3}

Here, denominator is of the form 2

*× 5*

^{n}*, where m=0 and n=3.*

^{m}It means rational number has a

**terminating**decimal expansion.

**(iii)**

*q*=455=5×91

Here, denominator is not of the form 2

*× 5*

^{n}*, where m and n are non-negative integers.*

^{m}It means rational number has a

**non-terminating repeating**decimal expansion.

**(iv)**

*q*=320=2×2×2×2×2×2×5 = 2

^{6}× 5

Here, denominator is of the form 2

*× 5*

^{n}*, where m=1 and n=6.*

^{m}It means rational number has a

**terminating**decimal expansion.

**(v)**

*q*=343=7×7×7

Here, denominator is not of the form 2

*× 5*

^{n}*, where m and n are non-negative integers.*

^{m}It means rational number has

**non-terminating repeating**decimal expansion.

**(vi)**

*q*=2

^{3}× 5

^{2}

Here, denominator is of the form 2

*× 5*

^{n}*, where m=2 and n=3 are non-negative integers.*

^{m}It means rational number has

**terminating**decimal expansion.

**(vii)**

*q*=2

^{2}× 5

^{7}× 7

^{5}

Here, denominator is not of the form 2

*× 5*

^{n}*, where m and n are non-negative integers.*

^{m}It means rational number has

**non-terminating repeating**decimal expansion.

**(viii)**

*q*=5=5

^{1}

Here, denominator is of the form 2

*× 5*

^{n}*, where m=1 and n=0.*

^{m}It means rational number has

**terminating**decimal expansion.

**(ix)**

*q*=10=2×5=2

^{1}× 5

^{1}

Here, denominator is of the form 2

*× 5*

^{n}*, where m=1 and n= 1.*

^{m}It means rational number has

**terminating decimal**expansion.

**(x)**

*q*=30=5×3×2

Here, denominator is not of the form 2

*× 5*

^{n}*, where m and n are non-negative integers.*

^{m}It means rational number has

**non-terminating repeating**decimal expansion.