The correct option is B (−∞,13][−127,∞)
∵ x1,x2,x3 are in AP.
⇒ x1=a−d, x2=a, x3=a+d
where d is the common difference
Now, since x1,x2,x3 are the roots of the given equation
⇒ (a - d) + (a) + (a + d) = 1 . . . (i)
⇒ β = (a - d) a + a(a +d) + (a - d) (a + d) . .. (ii)
and ∑=αβγ=x1x2x3=−γ=(a−d)(a)(a+d) . . . (iii)
hence from (i) we get a=13
and from eq. (ii) we get
β=3a2−d2⇒ β=13−d2 (∵a=13)Thus β=13−d2≤13 [∵ d2≥0]⇒ β≤13∴ βϵ(−∞,13]
Again from eq. (iii)
a(a2−d2)=−γ⇒ (127)+(−d23)=−γ⇒ γ=d23−127⇒ γ≥−127 (∵ d2≥0)
Hence option (a) is correct.