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Matrices and Determinants question

2024 · Shift 1 · Q23

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JEE Advanced
Mathematics
Matrices and Determinants
2024 · Shift 1 · Q23

Matrices and Determinants question

2024 · Shift 1 · Q23

JEE AdvancedMathematicsMatrices and DeterminantsMultiple correct+4 / −2

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\gt 0 \text { for all }(x, y) \in \mathbb{R}^2-\{(0,0)\}\right\} .

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

A
$\left(2, \frac{7}{2}, 6\right) \in S$
B
If $\left(3, b, \frac{1}{12}\right) \in S$ , then $|2 b|\lt 1$ .
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.

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