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Part BCSIR NET June 2025a-quadratic-form-is-homogeneous-not-additive-the-cross-term-survives

A quadratic form is homogeneous not additive the cross term survives

Let V = {ax3+^{3} + bx2+^{2} + cx | a,b,c∈Ra, b, c \in \mathbb{R}}. For f ∈ V, define Q(f)=∫Q(f) = \int₋11(f′(t))2_{1}^{1} (f'(t))^{2} dt, where f′ denotes the derivative of f. Which of the following statements is FALSE?

  1. A.Q is a positive definite quadratic form on V.
  2. B.Q takes every positive real value.
  3. C.Q(x) = 2.
  4. D.For all f, g ∈ V, Q(f + g) = Q(f) + Q(g).✓ 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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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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