GATE 2025 MA – Question 61
Linear Algebra · Inner Products and Quadratic Forms · 2 marks · Numerical answer
Under ⟨f,g⟩=∫₋₁¹f(x)g(x)dx, polynomial p(x)=α+βx²−30x⁴ is orthogonal to every polynomial of degree at most three. Find α+5β (integer).
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Correct answer: 126 to 126
Explanation
Odd test monomials vanish automatically by parity. Orthogonality to 1 gives α+β/3−6=0; to x² gives α/3+β/5−30/7=0. Solving yields β=180/7,α=−18/7. Therefore α+5β=(−18+900)/7=**126**.
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