(D) Let $f(a, b, c) = \frac{a^2+b^2+c^2}{ab+bc+ca}$.Case $1$: If $ab+bc+ca > 0$,we know that $(a-b)^2 + (b-c)^2 + (c-a)^2 \geq 0$,which implies $2(a^2

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$(-\infty, -2] \cup [1, \infty)$

(D) Let $f(a, b, c) = \frac{a^2+b^2+c^2}{ab+bc+ca}$.Case $1$: If $ab+bc+ca > 0$,we know that $(a-b)^2 + (b-c)^2 + (c-a)^2 \geq 0$,which implies $2(a^2+b^2+c^2) - 2(ab+bc+ca) \geq 0$,so $a^2+b^2+c^2 \geq ab+bc+ca$.Thus,$\frac{a^2+b^2+c^2}{ab+bc+ca} \geq 1$.Case $2$: If $ab+bc+ca This implies $a^2+b^2+c^2 \geq -2(ab+bc+ca)$.Since $ab+bc+ca $\frac{a^2+b^2+c^2}{ab+bc+ca} \leq -2$.Combining both cases,the range of $S$ is $(-\infty, -2] \cup [1, \infty)$.

Class 11 Mathematics 2 Complex Numbers and Quadratic Equations: Quadratic Equations Mix Examples-Quadratic Equations and Inequations

Let $S = \left\{ \frac{a^2+b^2+c^2}{ab+bc+ca} : a, b, c \in \mathbb{R}, ab+bc+ca \neq 0 \right\}$ where $\mathbb{R}$ is the set of real numbers. Then,$S$ equals

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A
$(-\infty, -1] \cup [1, \infty)$
B
$(-\infty, 0) \cup (0, \infty)$
C
$(-\infty, -1] \cup [2, \infty)$
D
$(-\infty, -2] \cup [1, \infty)$

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2 Complex Numbers and Quadratic Equations: Quadratic Equations

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Mix Examples-Quadratic Equations and Inequations

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Let $\alpha_\theta$ and $\beta_\theta$ be the distinct roots of $2x^2 + (\cos \theta)x - 1 = 0$,where $\theta \in (0, 2\pi)$. If $m$ and $M$ are the minimum and the maximum values of $\alpha_\theta^4 + \beta_\theta^4$,then $16(M + m)$ equals:

DifficultJEE MAIN 2025

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Let $S = \left\{ \frac{a^2+b^2+c^2}{ab+bc+ca} : a, b, c \in \mathbb{R}, ab+bc+ca \neq 0 \right\}$ where $\mathbb{R}$ is the set of real numbers. Then,$S$ equals

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