Q. Prove that √2 is an irrational number.

• Ans. To prove that √2 is an irrational number, let us assume its opposite i.e.,√2 is a rational number.
• Also let it be a/b in its simplest form where a and b are co-prime numbers having an H.C.F. of 1. Also let b ≠ 0.

Hence, from (i) and (ii), we find that 2 is a common factor of both a and b. However,this contradicts the fact that a and b have only 1 as their common factor. Such a contradiction arises by considering √2 as a rational number. Hence, it is proved that √2 is an irrational number.

Q. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively:

Sol. (i) Let the zeroes of the polynomial f(x) be α and β.

(ii) Let the zeroes of the polynomial f(x) be α and β.