JEE Advanced 2024 Mathematics - Class 12 JEEAdv24Ma250d Solution: Let denote . Let Then which of the following statements is…

QJEEAdv24Ma250d: JEE Advanced 2024 Mathematics - Class 12 JEEAdv24Ma250d Solution: Let denote . Let Then which of the following statements is…

Let $\mathbb{R}^2$ denote $\mathbb{R} \times \mathbb{R}$ . Let

$S=\left\{(a, b, c): a, b, c \in \mathbb{R} \text { and } a x^2+2 b x y+c y^2>0 \text { for all }(x, y) \in \mathbb{R}^2-\{(0,0)\}\right\} .$

Then which of the following statements is (are) TRUE?

Multiple Choice Options

A $\left(2, \frac{7}{2}, 6\right) \in S$
B If $\left(3, b, \frac{1}{12}\right) \in S$ , then $|2 b|<1$ .
C For any given $(a, b, c) \in S$ , the system of linear equations $\begin{aligned} & a x+b y=1 \\ & b x+c y=-1 \end{aligned}$ has a unique solution.
D For any given $(a, b, c) \in S$ , the system of linear equations $\begin{aligned} & (a+1) x+b y=0 \\ & b x+(c+1) y=0 \end{aligned}$ has a unique solution.

Step-by-Step Verified Solution

$\begin{aligned}\text{(A)}\quad & \mathrm{ax}^2+2 \mathrm{bxy}+\mathrm{cy}^2>0 \forall(\mathrm{x}, \mathrm{y}) \in \mathbb{R}^2-\{(0,0)\} \\& \Rightarrow \mathrm{ax}+2 \mathrm{bxy}+\mathrm{cy}^2 \text { must represent pair of imaginary lines and } \mathrm{a}, \mathrm{c}>0 . \\& \Rightarrow \mathrm{b}^2<\mathrm{ac}\end{aligned}$

$\mathrm{(B)}\quad\mathrm{b}^2<3 \times \frac{1}{12} \Rightarrow|2 \mathrm{~b}|<1$

$\mathrm{(C)}\quad$ since $\mathrm{b}^2 \neq \mathrm{ac}$

$\Rightarrow \mathrm{ax}+\mathrm{by}=1 \text { and } \mathrm{bx}+\mathrm{cy}=-1$

are not parallel lines.

$\mathrm{(D)}\quad\mathrm{ac}+\mathrm{a}+\mathrm{c}>\mathrm{b}^2 \Rightarrow$ lines are not parallel.

$\Rightarrow$ Options (B), (C), (D) are Correct.

QJEEAdv24Ma250d: JEE Advanced 2024 Mathematics - Class 12 JEEAdv24Ma250d Solution: Let denote . Let Then which of the following statements is…

Multiple Choice Options

Step-by-Step Verified Solution

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