Let R^2 denote R × R. Let S = \ (a,b,c) : a,b,c ∈ R and ax^2 + 2bxy + cy^2 > 0 for all (x,y) ∈ R^2 - \(0,0)\ \. Then which of the following statements is (are) TRUE?

If ( 3, b, 1/12 ) ∈ S, then |2b| < 1.

For any given (a,b,c) ∈ S, the system of linear equations ax+by=1, bx+cy=-1 has a unique solution.

For any given (a,b,c) ∈ S, the system of linear equations (a+1)x+by=0, bx+(c+1)y=0 has a unique solution.

Let R^2 denote R × R. Let S = \ (a,b,c) : a,b,c ∈ R and… — JEE Advanced 2024 Mathematics PYQ

JEE Advanced 2024 · Paper 1 · one or more correct

Let

\mathbb{R}^2

denote

\mathbb{R} \times \mathbb{R}

. Let

S = \left\{ (a,b,c) : a,b,c \in \mathbb{R} \text{ and } ax^2 + 2bxy + cy^2 > 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 $|2b| < 1$ .
C.
For any given $(a,b,c) \in S$ , the system of linear equations $ax+by=1$ , $bx+cy=-1$ has a unique solution.
D.
For any given $(a,b,c) \in S$ , the system of linear equations $(a+1)x+by=0$ , $bx+(c+1)y=0$ has a unique solution.

Show answer

Correct options: B, C, D

[JEE Advanced 2024 · Paper 1]

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