Let be the roots of the quadratic equation 12 x 2 - 20 x + 3 = 0 . If 1 2 3 2 , then the sum of all possible values of is . [2026]

Question

Let $α, β$ be the roots of the quadratic equation $12 x^{2} - 20 x + 3 λ = 0, λ \in Z .$ If $\frac{1}{2} \leq | β - α | \leq \frac{3}{2},$ then the sum of all possible values of $λ is$ . [2026]

Smart Achievers · Accepted Answer

(2)

$\frac{1}{2} \leq | α - β | \leq \frac{3}{2}$

$\frac{1}{4} \leq | α - β |^{2} \leq \frac{9}{4}$

$\frac{1}{4} \leq {(α + β)}^{2} - 4 α β \leq \frac{9}{4}$

$\frac{1}{4} \leq \frac{25}{9} - 4 \times \frac{λ}{4} \leq \frac{9}{4}$

$- \frac{91}{36} \leq - λ \leq \frac{- 19}{36}$

$\frac{19}{36} \leq λ \leq \frac{91}{36}$

$λ = 1, 2$

$Sum = 3$

Solution

Q.
Let $α, β$ be the roots of the quadratic equation $12 x^{2} - 20 x + 3 λ = 0, λ \in Z .$ If $\frac{1}{2} \leq | β - α | \leq \frac{3}{2},$ then the sum of all possible values of $λ is$ . [2026]

1 1

2 3

3 6

4 4

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Solution

Q. Let α, β be the roots of the quadratic equation 12x2-20x+3λ=0, λ∈Z. If 12≤|β-α|≤32, then the sum of all possible values of λ is. [2026]

1 1

2 3

3 6

4 4

Ans. (2) 12≤|α-β|≤32 14≤|α-β|2≤94 14≤(α+β)2-4αβ≤94 14≤259-4×λ4≤94 -9136≤-λ≤-1936 1936≤λ≤9136 λ=1, 2 Sum=3

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Q.
Let $α, β$ be the roots of the quadratic equation $12 x^{2} - 20 x + 3 λ = 0, λ \in Z .$ If $\frac{1}{2} \leq | β - α | \leq \frac{3}{2},$ then the sum of all possible values of $λ is$ . [2026]