Math, asked by gangadhar78, 11 months ago

pls solve all problems
step by step explanation required

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Answered by AARUSH0907
2

■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....

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