10. Five mark example problems - II

1. Prove that the equation \(x^2(p^2 + q^2) + 2x(pr + qs) + r^2 + s^2 = 0\) has no real roots. If \(ps = qr\), then show that the roots are real and equal.

 

Answer:

 

Here, \(a =\) , \(b =\) , \(c =\)

 

\(\Delta = b^2 - 4ac =\)

 

Since \(\Delta =\) , the given equation has .

 

If \(ps = qr\), then \(\Delta =\) .

 

Therefore,  if \(ps = qr\) and so the roots will be .

 

Hence, we proved.

 

2. If the difference between the roots of the equation \(x^2 - 13x + k = 0\) is \(17\), then find the value of \(k\).

 

Answer:

 

The value of \(k\) is