Question

15. Prove that √11 is irrational and hence show that 3 + √11 is also irrational.​

Answer 1

SOLUTION

TO PROVE

1. √11 is irrational

2. Hence show that 3 + √11 is also irrational.

PROOF

ANSWER TO QUESTION : 1

Let us assume that √11 is a rational number.

then, as we know a rational number should be in the form of p/q

where p and q are co- prime number.

So,

√11 = p/q { where p and q are co- prime}

√11q = p

Now, by squaring both the side

we get,

(√11q)² = p²

⇒ 11q² = p² - - - - - - (i)

So, 11 is the factor of p²

Then 11 is also a factor of p - - - - - - (ii)

⇒ Let p = 11m ( where m is any integer )

Squaring both sides

p² = (11m)²

p² = 121m²

Putting the value of p² in equation ( i )

11q² = p²

⇒ 11q² = 121m²

⇒ q² = 11m²

So, 11 is factor of q²

Then , 11 is also factor of q

Thus 11 is factor of p & q both

So, our assumption that p & q are co- prime is wrong

Hence √11 is an irrational number

ANSWER TO QUESTION : 2

If possible let 3 + √11 is also rational

Now 3 is rational

So ( 3 + √11 - 3 ) is also rational

⇒ √11 is also rational

━━━━ Which is a contradiction

Hence 3 + √11 is also irrational

Hence the proof follows

━━━━━━━━━━━━━━━━

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