Question

if p and q are two distinct irrational numbers ,then which of the following is always an irrational number and why?
(1) p/q (2) pq

(3) (p+q)² (4)p²q+qp/pq[tex][/tex]

Answer 1

ans is option 4
because other options are rational in my solving system..

Answer 2

As  given p and q are two distinct irrational numbers ,

let, p= 2 + √3, and q= 2 -√3

then 1., p/q= $\frac{2 + \sqrt3}{ 2 -\sqrt3}=\frac{4+3-4\sqrt{3}}{2 + \sqrt3}=\frac{7-4\sqrt3}{2 + \sqrt3}$

2. p× q=(2 +√3)(2-√3)=4-3=1

3. (p+q)²=4²=16

4. p²q+qp/pq=p×q[p  +1]÷ p q=p+1= always an irrational number.Because sum of rational and irrational is always irrational.

But if you take p=√3 and q=-√3, only 4. p²q+qp/pq=p×q[p  +1]÷ p q=p+1 is irrational.

(4.) p²q+qp/pq is always irrational.