If the difference in the roots of the equation x2 – px + q = 0 is un

1.	If the difference in the roots of the equation x2 – px + q = 0 is unity, then which one of the following is correct ?(a) p2 + 4q = 1 (b) p2 – 4q = 1 (c) p2 + 4q = – 1 (d) p2 – 4q = – 1
Answer» (b) p² - 4q = 1. Given, x² – px + q = 0 Let α, β be the roots of the given equation. Then, α + β = – \(\frac{(-p)}{1}\) = p ...(i), αβ = \(\frac{q}{1}\) = q ...(ii) Also, α - β = 1 (given) ...(iii) ∴ From (i) and (iii), 2α = p + 1 ⇒ α = \(\frac{p+1}{2}\) ∴ From (i) and (iii), 2β = p – 1 ⇒ b = \(\frac{p-1}{2}\) Substituting these values of a and b in (ii), we have \(\big(\frac{p+1}{2}\big)\)\(\big(\frac{p-1}{2}\big)\) = q ⇒ \(\frac{p^2-1}{4}\) = q ⇒ p² - 1 = 4q ⇒ p² - 4q = 1.

If the difference in the roots of the equation x2 – px + q = 0 is unity, then which one of the following is correct ?(a) p2 + 4q = 1 (b) p2 – 4q = 1 (c) p2 + 4q = – 1 (d) p2 – 4q = – 1

Answer»

(b) p² - 4q = 1.

Given, x² – px + q = 0

Let α, β be the roots of the given equation. Then,

α + β = – \(\frac{(-p)}{1}\) = p ...(i), αβ = \(\frac{q}{1}\) = q ...(ii)

Also, α - β = 1 (given) ...(iii)

∴ From (i) and (iii), 2α = p + 1 ⇒ α = \(\frac{p+1}{2}\)

∴ From (i) and (iii), 2β = p – 1 ⇒ b = \(\frac{p-1}{2}\)

Substituting these values of a and b in (ii), we have \(\big(\frac{p+1}{2}\big)\)\(\big(\frac{p-1}{2}\big)\) = q

⇒ \(\frac{p^2-1}{4}\) = q ⇒ p² - 1 = 4q ⇒ p² - 4q = 1.