Class-10th Math Complete Exercise-1.2 Solution

Class-10th
Chapter-1 (Real Numbers)
Exercise- 1.2

Que-1 Prove that √5 is irrational.

Solution-

Let us assume the contrary that √5 is Rational.
√5 = a/b [a and b are co-prime]
√5b = a
(√5b)² = a² [Squaring both side]
5b² = a² [ a² is divisible by 5, therefore a is also divisible by 5]---------Eq(1)
Let
a = 5c [c is an integer]
5b² = (5c)²
5b² = 25c²
b² = 5c² [b² is divisible by 5, therefore b is also divisible by 5]---------Eq(2)
a and b having common factor 5.
So, Our assumption is wrong.
Therefore √5 is irrational number.

Que-2 Prove that 3+2√5 is irrational.

Solution-  

Let us assume the contrary that 3+2√5 is Rational.
3+2√5 = a/b [a and b are co-prime]
2√5 = a - 3
           b   1
2√5 = a - 3b
              b
√5 = a - 3b
           2b
Since, a and b are integers.
Therefore, a - 3b is rational.
                   2b
But we know that √5 is irrational.
So, Our assumption is wrong.
Therefore, 3+2√5 is irrational.

Que-3 Prove the following are irrational :-

(i) 1
   √2

Solution-