Question

3 Prove that 213 + 15 is an irrational number. Also,
check whether (213 + V5) - (213 - 15) is rational or
irrational.
CBSE 2012​

Answer 1

$\huge\star{\green{\underline{\mathfrak{Answer :-}}}}$

Let us assume that (2√3 + √5) is a rational number.

So, let us square the number given.

(2√3 + √5)²

= (2√3)² + (√5)² + 2(2√3)(√5)

= (4*3) + (5) + 4√15

= 12 + 5 + 4√15

= 17 + 4√15

Now, as as we have assumed this as a rational number.

So, we can write it as,

$\hookrightarrow\tt{17 + 4\sqrt{15} = \dfrac{p}{q}}$

$\hookrightarrow\tt{4\sqrt{15} = \dfrac{p-17q}{q}}$

$\hookrightarrow\tt{\sqrt{15} = \dfrac{p-17q}{4q}}$

Now,we can clearly see that the left hand side is irrational and the right hand side is a rational number.

So, our assumption was wrong.

Hence, the given number is an irrational number.

Now, let us solve the other part of the question.

(2√3+√5)(2√3-√5)

= 12 - 2√15 + 2√15 - 5

= 7

As, the result we got after solving is a rational number.

So, we can say that the given number is a rational one, because it can be expressed in p by q form.

$\rule{200}2$

Answer 2

$\huge\bold\green{Question}$

Prove that 213 + 15 is an irrational number. Also,

check whether (213 + V5) - (213 - 15) is rational or

irrational.

$\huge\bold\blue{AnsWer :}$

A rational number is a number that can be expressed as the quotient or fraction p/q of two integers

Let us assume that (2√3 + √5) is a rational number.

Now , squaring the number (2√3 + √5)² we get ,

By using identity [ (a+b)² = a² + b² + 2ab ]

→ (2√3)² + (√5)² + 2(2√3)(√5)

→ (4×3) + (5) + 4√15

→12 + 5 + 4√15

→ 17 + 4√15

Hence, as as we have assumed that this as a rational number. So, we can write this number as

→$\bold\red{17 + 4\sqrt{15} = \dfrac{p}{q}}$

→$\bold\red{4\sqrt{15} = \dfrac{p-17q}{q}}$

→$\bold\red{\sqrt{15} = \dfrac{p-17q}{4q}}$

So, It can be clearly see that the L.H.S is irrational and the R.H.S is a rational number.

Hence, our assumption was wrong

So, the given number is an irrational number.

Now, we solve the other part of the question.

→ (2√3+√5) (2√3-√5)

→ 12 - 2√15 + 2√15 - 5

→ 7

Hence , it is a rational number.

So, we can say that the given number is a rational one, because it can be expressed in p by q form.