# Important Questions for CBSE Class 10 Maths Chapter 1 - Real Numbers 4 Mark Question

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

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CBSE Class 10 Maths Chapter-1 Real Numbers Important Questions

## 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.