Let α, β are the roots of 375 x 2-25 x -2=0 and y= displaystyle∑ r =110( t - r )2, where t ∈ R If a, b and c also represent the sides of a triangle, then the complete set of α2 is

Question

Let α, β are the roots of 375 x 2-25 x -2=0 and y= displaystyle∑ r =110( t - r )2, where t ∈ R If a, b and c also represent the sides of a triangle, then the complete set of α2 is (A) ((1/3), 3) (B) (2,3) (C) [(1/3), 2] (D) ((√5+3/2), 3)

Tardigrade Admin · Accepted Answer

Let b=a r, c=a r2, and r>0 As the sum of two sides is more than the third side, we have r ∈((√5-1/2), (√5+1/2))- 1 ⇒ r+(1/r)-1 ∈(1, √5-1) As α2=(r2+r+1/r2-r+1)=1+(2/r+(1)r-1) ∴ α2 ∈((√5+3/2), 3)

Q. Let $α, β$ are the roots of $375 x^{2} - 25 x - 2 = 0$ and $y = r = 1 \sum 10 (t - r)^{2}$ , where $t \in R$
If $a, b$ and $c$ also represent the sides of a triangle, then the complete set of $α^{2}$ is

$(\frac{1}{3}, 3)$

$(2, 3)$

$[\frac{1}{3}, 2]$

$(\frac{5 + 3}{2}, 3)$

Q. Let α,β are the roots of 375x2−25x−2=0 and y=r=1∑10​(t−r)2, where t∈R If a,b and c also represent the sides of a triangle, then the complete set of α2 is

(31​,3)

(2,3)

[31​,2]

(25​+3​,3)

Let b=ar,c=ar2, and r>0 As the sum of two sides is more than the third side, we have r∈(25​−1​,25​+1​)−{1} ⇒r+r1​−1∈(1,5​−1) As α2=r2−r+1r2+r+1​=1+r+r1​−12​ ∴α2∈(25​+3​,3)

Q. Let $α, β$ are the roots of $375 x^{2} - 25 x - 2 = 0$ and $y = r = 1 \sum 10 (t - r)^{2}$ , where $t \in R$
If $a, b$ and $c$ also represent the sides of a triangle, then the complete set of $α^{2}$ is

$(\frac{1}{3}, 3)$

$(2, 3)$

$[\frac{1}{3}, 2]$

$(\frac{5 + 3}{2}, 3)$