JEE Mains · Maths · STD 11 - 4.2 Quadratic equations and inequations
If the set of all \(\mathrm{a} \in \mathbf{R}\), for which the equation \(2 x^2+(a-5) x+15=3 \mathrm{a}\) has no real root, is the interval \((\alpha, \beta)\), and \(X=\{x \in Z: \alpha \lt x \lt \beta\}\), then \(\sum_{x \in X} x^2\) is equal to :
- A 2109
- B 2129
- C 2119
- D 2139
Answer & Solution
Correct Answer
(D) 2139
Step-by-step Solution
Detailed explanation
\begin{aligned} & (a-5)^2-8(15-3 a) < 0 \\ & a^2+14 a+25-120 < 0 \\ & a^2+14 a-95 < 0 \\ & (a+19)(a-5) < 0 \\ & a \in(-19,5) \\ & \therefore-19 < x < 5 \\ & \therefore \sum_{x \in X} x^2=\left(1^2+2^2+\ldots .+4^2\right)+\left(1^2+2^2+\ldots+18^2\right) \\ & =\frac{4 \times 5…
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