Question

Let R2 denote R × R. Let
S = {(a, b, c) : a, b, c ∈ R and ax2 + 2bxy + cy2 > 0 for all (x, y) ∈ R2 - {(0, 0}}.
Then which of the following statements is (are) TRUE ?

• A. $(2,\frac{7}{2},6)\in S$
• B. If $(3,b,\frac{1}{12})\in S$, then |2b| < 1.
• C. For any given (a, b, c) ∈ S, the system of linear equations
  ax + by = 1
  bx + cy = -1
  has a unique solution.
• D. 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.

Answer 1

Answer: (B), (C), (D)

If $(3,b,\frac{1}{12})\in S$, then |2b| < 1.

Answer 2

The correct option is (B): If $(3,b,\frac{1}{12})\in S$, then |2b| < 1., (C): For any given (a, b, c) ∈ S, the system of linear equations
ax + by = 1
bx + cy = -1
has a unique solution. and (D): 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.