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Question

Consider the following statements:
I. If the roots of the equation ax2+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 x2-22x+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.

(I) Let the roots be a and 1a.Then, a × (1a) = ca c = a a + c = 0 I is true.(II) Clearly, II is true.(III) α + β = 22 and αβ = (15 × 7) = 105On solving them, we get:α = 15 and β = 7α β = 15 - 7 = 8 III is true. I, II and III are all true.

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