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Part BCSIR NET June 2024cross-terms-do-not-imply-indefinite

Cross terms do not imply indefinite

For a quadratic form f(x,y,z)∈R[x,y,z]f(x, y, z) \in \mathbb{R}[x, y, z], we say that (a,b,c)∈R3(a, b, c) \in \mathbb{R}^{3} is a zero of f if f(a, b, c) = 0. Which of the following quadratic forms has at least one zero different from (0, 0, 0)?

  1. A.x2+2y2+3z2x^{2} + 2y^{2} + 3z^{2}
  2. B.x2+2y2+3z2−2x^{2} + 2y^{2} + 3z^{2} - 2xy
  3. C.x2+2y2+3z2−2x^{2} + 2y^{2} + 3z^{2} - 2xy − 2yz
  4. D.x2+2y2−3z2x^{2} + 2y^{2} - 3z^{2}✓ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

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50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: Every real symmetric matrix is positive definite if det > 0

More on this topic

The chapter behind this: Quadratic forms, signature and definiteness — free to read

From Inner Product Spaces and Forms › Quadratic forms, positive definiteness, Sylvester's law

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