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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.

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Solution

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

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