Part CCSIR NET June 2024reaching-a-positive-value-is-not-definiteness

Reaching a positive value is not definiteness

Let $q_{1} (x_{1}, x_{2})$ and $q_{2} (y_{1}, y_{2})$ be real quadratic forms such that there exist $(u_{1}, u_{2}), (v_{1}, v_{2}) \in R^{2}$ such that $q_{1} (u_{1}, u_{2}) = 1 = q_{2} (v_{1}, v_{2})$ . Define $q (x_{1}, x_{2}, y_{1}, y_{2}) = q_{1} (x_{1}, x_{2}) - q_{2} (y_{1}, y_{2})$ . Which of the following statements are necessarily true?

A.q is a quadratic form in $x_{1}, x_{2}, y_{1}, y_{2}$ ✓
B.There exists $(t_{1}, t_{2}) \in R^{2}$ such that $q_{1} (t_{1}, t_{2}) = 5$ ✓
C.There does not exist $(s_{1}, s_{2}) \in R^{2}$ such that $q_{2} (s_{1}, s_{2}) = - 5$
D.Given $α \in R$ , there exists a vector $w \in R^{4}$ such that $q (w) = α$ ✓

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.

See pricing

50 are analysed free — try those first.

Reaching a positive value is not definiteness

More on this topic