pls solve all problems
step by step explanation required
Answers
■QUES NO. (7).
Let √2 is rational no.
then, √2 = a / b wher a,b are integers b ≠ 0.
● We also suppose that a / b is written in the simplest form.
Now, √2 = a / b
》2 = a^2 / b^2
》 2 b^2 = a^2
∴ 2 b^2 is divisible by 2
⇒ a^2 is divisible by 2
⇒ a is divisible by 2
● Let a = 2c
》a^2 = 4c^2
⇒ 2 b^2 = 4c^2
⇒ b^2 = 2c^2
∴ 2c^2 is divisible by 2 and b^2 is divisible by 2...
Hence b is divisible by 2
● SO ,a and b both are divisible by 2......
●●This contradicts our supposition that a/b is written in the simplest form.
● Hence our supposition is wrong.. SO , √2 is irrational number.
▪▪ IN THE SIMILAR WAY , QUES. NO. 8 , 9 AND 10 CAN BE SOLVED..
○
○
■ QUES NO. (2).
Let a be any positive integer
We know by Euclid's algorithm, if a and b are two positive integers, there exist unique integers q and r satisfying,
● a = bq + r , where 0 ≤ r < b.
Take b = 4
a = 4q + r
Since 0 ≤ r < 4, the possible remainders are 0, 1, 2 and 3.
That is, a can be...
● 4q, 4q +1 , 4q + 2 or 4q + 3 where q is the quotient.
Since a is odd, it cannot be 4q or 4q + 2 as they are both divisible by 2.
● Therefore, any odd integer is of the form 4q + 1 or 4q + 3....
■■ IN THE SIMILAR WAY , QUES 1 AND 3 CAN BE DONE....