In this chapter, we provide RD Sharma Class 10 Ex 1.5 Solutions Chapter 1 Real Numbers for English medium students, Which will very helpful for every student in their exams. Students can download the latest RD Sharma Class 10 Ex 1.5 Solutions Chapter 1 Real Numbers pdf, Now you will get step by step solution to each question.

Textbook | NCERT |

Class | Class 10 |

Subject | Maths |

Chapter | Chapter 1 |

Chapter Name | Real Numbers |

Exercise | 1.5 |

Category | RD Sharma Solutions |

**RD Sharma Solutions for Class 10 Chapter** **1 Real Numbers Ex 1.5 Download PDF**

**Chapter 1: Real Numbers Exercise – 1.5**

**Question: 1**

Show that the following numbers are irrational.

(i) 7 √5

(ii) 6 + √2

(iii) 3 – √5

**Solution:**

(i) Let us assume that 7 √5 is rational. Then, there exist positive co primes a and b such that

We know that √5 is an irrational number

Here we see that √5 is a rational number which is a contradiction.

(ii) Let us assume that 6+√2 is rational. Then, there exist positive co primes a and b such that

Here we see that √2 is a rational number which is a contradiction as we know that √2 is an irrational number

Hence 6 + √2 is an irrational number

(iii) Let us assume that 3 – √5 is rational. Then, there exist positive co primes a and b such that

Here we see that √5 is a rational number which is a contradiction as we know that √5 is an irrational number

Hence 3 – √5 is an irrational number.

**Question: 2**

Prove that the following numbers are irrationals.

(i) 2√7

(ii) 3/(2√5)

(iii) 4 + √2

(iv) 5√2

**Solution:**

(i) Let us assume that 2√7 is rational. Then, there exist positive co primes a and b such that

√7 is rational number which is a contradiction

Hence 2√7 is an irrational number

(ii) Let us assume that 3/(2√5) is rational. Then, there exist positive co primes a and b such that

√5 is rational number which is a contradiction

Hence 3/(2√5) is irrational.

(iii) Let us assume that 3/(2√5) is rational. Then, there exist positive co primes a and b such that

√2 is rational number which is a contradiction

Hence 4+ √2 is irrational.

(iv) Let us assume that 5√2 is rational. Then, there exist positive co primes a and b such that

√2 is rational number which is a contradiction

Hence 5√2 is irrational

**Question: 3**

Show that 2 – √3 is an irrational number.

**Solution:**

Let us assume that 2 – √3 is rational. Then, there exist positive co primes a and b such that

Here we see that √3 is a rational number which is a contradiction

Hence 2- √3 is irrational

**Question: 4**

Show that 3 + √2 is an irrational number.

**Solution:**

Let us assume that 3 + √2 is rational. Then, there exist positive co primes a and b such that

Here we see that √2 is a irrational number which is a contradiction

Hence 3 + √2 is irrational

**Question: 5**

Prove that 4 – 5√2 is an irrational number.

**Solution:**

Let us assume that 4 – 5√2 is rational. Then, there exist positive co primes a and b such that

This contradicts the fact that √2 is an irrational number

Hence 4 – 5√2 is irrational

**Question: 6**

Show that 5 – 2√3 is an irrational number.

**Solution:**

Let us assume that 5 -2√3 is rational. Then, there exist positive co primes a and b such that

This contradicts the fact that √3 is an irrational number

Hence 5 – 2√3 is irrational

**Question: 7**

Prove that 2√3 – 1 is an irrational number.

**Solution:**

Let us assume that 2√3 – 1 is rational. Then, there exist positive co primes a and b such that

This contradicts the fact that √3 is an irrational number

Hence 5 – 2√3 is irrational

**Question: 8**

Prove that 2 – 3√5 is an irrational number.

**Solution:**

Let us assume that 2 – 3√5 is rational. Then, there exist positive co primes a and b such that

This contradicts the fact that √5 is an irrational number

Hence 2 – 3√5 is irrational

**Question: 9**

Prove that √5 + √3 is irrational.

**Solution:**

Let us assume that √5 + √3 is rational. Then, there exist positive co primes a and b such that

Here we see that √3 is a rational number which is a contradiction as we know that √3 is an irrational number

Hence √5 + √3 is an irrational number

**Question: 10**

Prove that √3 + √4 is irrational.

**Solution:**

Let us assume that √3 + √4 is rational. Then, there exist positive co primes a and b such that

Here we see that √3 is a rational number which is a contradiction as we know that √3 is an irrational number

Hence √3 + √4 is an irrational number

**Question: 11**

Prove that for any prime positive integer p, √p is an irrational number.

**Solution:**

Let us assume that √p is rational. Then, there exist positive co primes a and b such that

⇒ pb^{2} = a^{2}

⇒ p|a^{2} ⇒ p|a^{2}

⇒ p|a ⇒ p|a

⇒ a = pc for some positive integer c ⇒ a = pc for some positive integer c

b^{2}p = a^{2}

⇒ b^{2}p = p^{2}c^{2} ( ∵ a = pc)

⇒ p|b^{2 }(since p|c^{2}p) ⇒ p|b^{2}(since p|c^{2}p)

⇒ p|b ⇒ p|a and p|b

This contradicts the fact that a and b are co primes

Hence √p is irrational

**Question: 12**

If p, q are prime positive integers, prove that √p + √q is an irrational number.

**Solution:**

Let us assume that √p + √q is rational. Then, there exist positive co primes a and b such that

Here we see that √q is a rational number which is a contradiction as we know that √q is an irrational number

Hence √p + √q is an irrational number

**All Chapter RD Sharma Solutions For Class10 Maths**

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