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; &reals; 2 satisfying 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 is necessarily true?

Given α &in; &reals;, there exists a vector ω &in; &reals; 4 such that q(ω) = α

There does not exist (s 1 , s 2 ) &in; &reals; 2 such that q 2 (s 1 , s 2 ) = -5

There exists (t 1 , t 2 ) &in; &reals; 2 such that q 1 (t 1 , t 2 ) = 5

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; &reals; 2 satisfying 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 is necessarily true?

Given α &in; &reals;, there exists a vector ω &in; &reals; 4 such that q(ω) = α

There does not exist (s 1 , s 2 ) &in; &reals; 2 such that q 2 (s 1 , s 2 ) = -5

There exists (t 1 , t 2 ) &in; &reals; 2 such that q 1 (t 1 , t 2 ) = 5

Let q 1 (x 1 , x 2 ) and q 2 (y 1 , y 2 ) be real quadratic forms... | Mathematics MCQ [Updated November 2026] | GkGet

Mathematics · Linear Algebra

Let q₁(x₁, x₂) and q₂(y₁, y₂) be real quadratic forms such that there exist (u₁, u₂), (v₁, v₂) ∈ ℝ² satisfying q₁(u₁, u₂) = 1 = q₂(v₁, v₂). Define q(x₁, x₂, y₁, y₂) = q₁(x₁, x₂) - q₂(y₁, y₂). Which of the following statements is necessarily true?

Verified Solution4 OptionsUpdated: October 2026👁️1 View

1Given α &in; &reals;, there exists a vector ω &in; &reals; 4 such that q(ω) = α
2There does not exist (s 1 , s 2 ) &in; &reals; 2 such that q 2 (s 1 , s 2 ) = -5
3There exists (t 1 , t 2 ) &in; &reals; 2 such that q 1 (t 1 , t 2 ) = 5
4q is a quadratic form in x 1 , x 2 , y 1 , y 2✓

Correct answer

q is a quadratic form in x 1 , x 2 , y 1 , y 2

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