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

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

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.

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A quadratic form is homogeneous not additive the cross term survives

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