NCERT Solutions For Class 10 Math Chapter – 1 Exercise – 1.3
Q1. Prove that √5 is irrational.
Solution:-
Let take √5 as irrational number.
If a and b are two co – prime and b is not equal to 0.
We can write√5 = a/b
b√5 = a
To remove root , squaring on both sides , we get
5b2 = a2
Therefore , 5 divides a2 and according to theorem of rational number , for any prime number p which is divides a2 then it will divides a also.
That means 5 will divides a so we can write.
A = 5c
Putting value of a in equation (i) we get
5b2 = (5c)2
5b2 = 25c2
Divide by 25 we get
B2/5 = c2
Similarly , we get that b will divide by 5.
And we have already get that a is divisible by 5.
But a and b are co – prime number so it contradicts.
Hence √5 is not a rational number , it is irrational.
Q2. Prove that 3 +2√5 is irrational.
Solution:-
Let take that 3 + 2√5 is a irrational number.
So we can write this number as
3 + 2√5 = a/b
Here a and b are two co – prime number and b is not equal to 0.
Subtract 3 both sides we get
2√5 = a/b – 3
2√5 = (a – 3b)/b
Now divide by 2, we get
√5 = (a – 3b)/2b
Here a and b are integers so (a – 3b)/2b is a rational number so √5 should be a rational number. But √5 is a irrational number so it contradicts.
Hence , 3 + 2√5 is a irrational number.
Q3. Prove that the following are irrationals:
- 1/√2
Solution:-
Let take that 1/√2 is a rational number.
So we can write this number as
1/√2 = a/b
Here a and b are two co – prime number and b is not equal to 0.
Multiply by √2 both sides we get
1 = (a√2)/b
Now multiply by b.
B = a√2
Divide by a we get
B/a = √2
Here a and b are integer so b/a is a rational number so √2 should be a rational number But √2 is a irrational number so it contradicts.
Hence 1/√2 is a irrational number.
- Let take that 7√5 is rational number.
So we can write this number as
7/√5 = a/b
Here a and b are two co – prime number and b is not equal to 0.
Divide by 7 we get
√5 = a/(7b)
Here a and b are integer so a/7b is a rational numbers so √5 should be a rational number but √5 is irrational number so it contradicts.
Hence , 7√5 is a irrational number.
- Let take that 6 + √2 is a rational number.
So we can write this number as
6 + √2 = a/b
Here a and b are two co – prime number and b is not equal to 0.
Subtract 6 both sides we get
√2 = a/b – 6
√2 (a – 6b)/b
Here a and b are integer so ( a – 6b)/b is a rational number so √2 should be rational number.
But √2 is a irrational number so it contradicts.
Hence 6 + √2 is a irrational number.