Ex-1.5 Real Numbers, Class 10, Maths RD Sharma Solutions | Extra Documents, Videos & Tests for Class 10 PDF Download

Q.1: Show that the following numbers are irrational.

(i) 7√5

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

7√5  =  a/b

√5  =  a/7b

We know that  √5   is an irrational number

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

(ii) 6 + √2

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

6 + √2  =  a/b

√2  =  a/b  − 6

√2  =  

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) 3 − √5

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

3 − √5  =  a/b

√5  =  3 − a/b

√5  =  

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.

Q.2: Prove that the following numbers are irrationals.

Sol: (i) 2/√7

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

2√7  =  a/b

√7  =  2b/a

√7 is rational number which is a contradiction

Hence 2√7 is an irrational number

(ii) 325√

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

3/2√5  =  a/b

√5  =  3b/2a

√5 is rational number which is a contradiction

Hence 3/2√5 is irrational.

(iii) 4 + √2

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

4 + √2  =  a/b

√2  =  a/b − 4

√2  =  

√2 is rational number which is a contradiction

Hence 4 + √2 is irrational.

(iv) 5√2

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

5√2  =  ab

√2  =  a/b − 5

√2  =  

√2 is rational number which is a contradiction

Hence 5√2  is irrational

Q.3: Show that 2 − √3 is an irrational number.

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

2 − √3  =  a/b

√3  =  2 − a/b

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

Hence 2 − √3  is irrational

Q.4: Show that 3 + √2 is an irrational number.

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

3 + √2  =  a/b

√2  =  a/b − 3

√2  =  

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

Hence 3 + √2  is irrational

Q.5: Prove that 4 − 5√2 is an irrational number.

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

4 − 5√2  =  a/b

5√2  =  a/b − 4

This contradicts the fact that √2 is an irrational number

Hence 4 − 5√2 is irrational

Q.6: Show that 5 − 2√3 is an irrational number.

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

5 − 2√3  =  ab

2√3  =  a/b − 5

This contradicts the fact that √3 is an irrational number

Hence 5 − 2√3 is irrational

Q.7: Prove that 2√3 − 1 is an irrational number.

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

2√3 − 1  =  a/b

2√3   =  a/b + 1

This contradicts the fact that √3 is an irrational number

Hence 5 − 2√3 is irrational

Q.8: Prove that 2 − 3√5 is an irrational number.

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

2 − 3√5  =  a/b

3√5  =  a/b − 2

This contradicts the fact that √5 is an irrational number

Hence 2 − 3√5 is irrational

Q.9: Prove that √5 + √3 is irrational.

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

√5 + √3  =  a/b

√5 = a/b − √3

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

Q.10: Prove that √3 + √4 is irrational.

Sol: 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

Q.11: Prove that for any prime positive integer p, √p is an irrational number.

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

√p  =  a/b

p  =  (a/b)²

⇒ p  = a²/b²

⇒ pb² = a²

⇒ p|a²

⇒ p|a

⇒ a = pcforsomepositiveintegerc

⇒ b²p  =  a²

⇒ b²p  =  p²c² ( ∵ a = pc )

⇒ p|b² (since p|c²p)

⇒ p|b

⇒ p|a and p|b

This contradicts the fact that a and b are co primes

Hence √p is irrational

Q.12: If p, q are prime positive integers, prove that √p + √q is an irrational number.

Sol: 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