Question

Consider the following statements:
I. If the roots of the equation $a{x}^2+bx+c=0$ are negative reciprocal of each other, than a + c = 0.
II. A quadratic equation can have at all most two roots.
III. If α, β are the roots of $x^2-22x+105=0,$ then α + β = 22 and α − β = 8.

Of these statements:
(a) I and II are true and III is false.
(b) I and III are true and II is false.
(c) II and III are true I is false.
(d) I, II and III are all true.

Answer 1

(d) I, II and III are all true.

$$\begin{gathered} \left(\text{I}\right)\text{Let the roots be}a\text{and}-\frac{1}{a}. \\ \text{Then,} \\ a\times \left(\frac{-1}{a}\right)=\frac{c}{a} \\ \begin{array}{l}\Rightarrow c=-a\\ \Rightarrow a+c=0\end{array} \\ \therefore \text{I is true}\text{.} \\ \left(\text{II}\right)\text{Clearly, II}\text{}\text{is true}\text{.} \\ \left(\text{III}\right)\alpha \mathit{}+\beta =22\text{and}\alpha \beta \text{=}(15\times 7)=105 \\ \mathrm{On}\mathrm{solving}\mathrm{them},\mathrm{we}\mathrm{get}: \\ \alpha =15\text{and}\beta \text{= 7} \\ \alpha -\beta =15-7=8 \\ \therefore \mathrm{III}\mathrm{is}\mathrm{true}. \\ \therefore \text{I, II and III}\text{}\text{are all true}\text{.} \end{gathered}$$