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State true or false:

If $5$ is an irrational number then $3 + 5$ is also an irrational number:

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True

False

answer is A.

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Detailed Solution

Given statement if

5

is an irrational number then

3 + 5

is also an irrational number.
A rational number can be expressed in terms of

p q

form
Let us consider that

5

is rational number, then

5 = p q ⇒ 5 q 2 = p 2

As we have to prove

5, 3 + 5

are irrational numbers.
By Euclid's division lemma

p

is a factor of

5

this means

p = 5 m ⇒ p 2 = 25 m 2

From both the equations we get

q 2 = 5 m 2

⇒ q

is a multiple of

5

.
But p,q are co-primes hence it contradicts, therefore,

5

is irrational number.
As proved

5

irrational.
And sum of a rational and an irrational number is always an irrational number. Therefore

5 + 3

is also irrational.
So,

5 and 5 + 3

are irrational number.
Hence option 1 is correct and the statement is true.

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