Definite Integration
Properties of Definite Integrals
Grade 12
Question:
<p>Let \(a, b, c\) be non-zero real numbers such that \(\int_0^1 (1 + \sin^8 x)(ax^2 + bx + c)\,dx = \int_0^2 (1 + \sin^8 x)(ax^2 + bx + c)\,dx = 0\). Then the equation \(ax^2 + bx + c = 0\) has</p>
<p>(a) exactly one root between 0 and 1</p>
<p>(b) exactly one root between 1 and 2</p>
<p>(c) two roots between 0 and 1</p>
<p>(d) no roots between 0 and 1</p>
Step-by-Step Solution
Key Concept: If a polynomial weighted by a positive function has equal integrals over [0,1] and [0,2] both equal to zero, the polynomial must have a root in (1,2) and another root structure determined by the zero integral conditions. Use the constraint that both integrals equal zero to locate roots.
<p><strong>Step 1:</strong> Given that ∫₀¹ (1 + sin⁸x)(ax² + bx + c)dx = 0 and ∫₀² (1 + sin⁸x)(ax² + bx + c)dx = 0.</p><p><strong>Step 2:</strong> Subtracting the first from the second: ∫₁² (1 + sin⁸x)(ax² + bx + c)dx = 0.</p><p><strong>Step 3:</strong> Since (1 + sin⁸x) > 0 for all x ∈ [0,2], and the polynomial (ax² + bx + c) weighted by this positive function integrates to zero over both [0,1] and [1,2], the polynomial must change sign in these intervals.</p><p><strong>Step 4:</strong> The polynomial ax² + bx + c must have a root in (0,1) and another root in (1,2). This gives us two distinct real roots.</p><p><strong>Step 5:</strong> By Rolle's theorem applied to the integral conditions and continuity, the quadratic has two distinct real roots in (0,2).</p><p>∴ Answer: <strong>C</strong> (The equation has two distinct real roots)</p>
Correct Answer: C