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)
a and b are integers.
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 2n × 5m, where m and n are non-negative integers.
(i)
q=3125=5×5×5×5×5=55
Here, denominator is of the form 2n × 5m, where m=5 and n=0.
It means rational number has a terminating decimal expansion.
(ii)
q=8=2×2×2=23
Here, denominator is of the form 2n × 5m, where m=0 and n=3.
It means rational number has a terminating decimal expansion.
(iii)
q=455=5×91
Here, denominator is not of the form 2n × 5m, where m and n are non-negative integers.
It means rational number has a non-terminating repeating decimal expansion.
(iv) 
q=320=2×2×2×2×2×2×5 = 26 × 5
Here, denominator is of the form 2n × 5m, where m=1 and n=6.
It means rational number has a terminating decimal expansion.
(v)
q=343=7×7×7
Here, denominator is not of the form 2n × 5m, where m and n are non-negative integers.
It means rational number has non-terminating repeating decimal expansion.
(vi)
q=23 × 52
Here, denominator is of the form 2n × 5m, where m=2 and n=3 are non-negative integers.
It means rational number has terminating decimal expansion.
(vii)
q=22 × 57 × 75
Here, denominator is not of the form 2n × 5m, where m and n are non-negative integers.
It means rational number has non-terminating repeating decimal expansion.
(viii) 
q=5=51
Here, denominator is of the form 2n × 5m, where m=1 and n=0.
It means rational number has terminating decimal expansion.
(ix) 
q=10=2×5=21 × 51
Here, denominator is of the form 2n × 5m, where m=1 and n= 1.
It means rational number has terminating decimal expansion.
(x) 
q=30=5×3×2
Here, denominator is not of the form 2n × 5m, where m and n are non-negative integers.
It means rational number has non-terminating repeating decimal expansion.