Let S= α: log 2(92 α-4+13)- log 2((5/2) ⋅ 32 α-4+1)=2 . Then the maximum value of β for which the equation x2-2( displaystyle∑α ∈ s α)2 x+ displaystyle∑α ∈ s(α+1)2 β=0 has real roots, is

Question

Let S= α: log 2(92 α-4+13)- log 2((5/2) ⋅ 32 α-4+1)=2 . Then the maximum value of β for which the equation x2-2( displaystyle∑α ∈ s α)2 x+ displaystyle∑α ∈ s(α+1)2 β=0 has real roots, is  (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

log 2(92 α-4+13)- log 2((5/2) ⋅ 32 α-4+1)=2 ⇒ (92 α-4+13/(5)2 32 α-4+1)=4 ⇒ α=2 text or 3 displaystyle∑α ∈ S α=5 text and displaystyle∑α ∈ S(α+1)2=25 ⇒ x2-50 x+25 β=0 text has real roots ⇒ β ≤ 25 ⇒ β max =25

Q. Let S={α:log2​(92α−4+13)−log2​(25​⋅32α−4+1)=2}. Then the maximum value of β for which the equation x2−2(α∈s∑​α)2x+α∈s∑​(α+1)2β=0 has real roots, is _____

log2​(92α−4+13)−log2​(25​⋅32α−4+1)=2 ⇒25​32α−4+192α−4+13​=4 ⇒α=2 or 3 α∈S∑​α=5 and α∈S∑​(α+1)2=25 ⇒x2−50x+25β=0 has real roots ⇒β≤25 ⇒βmax​=25

Q. Let $S = {α : lo g_{2} (9^{2 α - 4} + 13) - lo g_{2} (\frac{5}{2} \cdot 3^{2 α - 4} + 1) = 2}$ . Then the maximum value of $β$ for which the equation $x^{2} - 2 (α \in s \sum α)^{2} x + α \in s \sum (α + 1)^{2} β = 0$ has real roots, is _____