Indefinite Integration
Primitives and initial conditions
Grade 12
Question:
<p><strong>Assertion (A):</strong> If the primitive of <span class="math">\(f(x) = \sin x + 2x - 4\)</span> has the value 3 for <span class="math">\(x = 1\)</span>, then there are exactly two values of <span class="math">\(x\)</span> for which the primitive of <span class="math">\(f(x)\)</span> vanishes.</p><p><strong>Reason (R):</strong> <span class="math">\(\cos x\)</span> has period <span class="math">\(2\pi\)</span>.</p>
<p>(A) Both A and R are true and R is the correct explanation of A</p>
<p>(B) Both A and R are true but R is NOT the correct explanation of A</p>
<p>(C) A is true but R is false</p>
<p>(D) A is false but R is true</p>
Step-by-Step Solution
Key Concept: Integrate f(x) to find the primitive, use the initial condition to find the constant, then analyze the zeros of the resulting equation
<p><strong>Step 1:</strong> The primitive of <span class="math">$f(x) = \sin x + 2x - 4$</span> is <span class="math">$F(x) = -\cos x + x^2 - 4x + C$</span></p><p><strong>Step 2:</strong> Given <span class="math">$F(1) = 3$</span>: <span class="math">$-\cos(1) + 1 - 4 + C = 3$</span>, so <span class="math">$C = 6 + \cos(1)$</span></p><p><strong>Step 3:</strong> For <span class="math">$F(x) = 0$</span>: <span class="math">$-\cos x + x^2 - 4x + 6 + \cos(1) = 0$</span></p><p><strong>Step 4:</strong> Analysis shows the equation may have more or fewer than exactly two solutions.</p><p><strong>Step 5:</strong> However, <span class="math">$\cos x$</span> does have period <span class="math">$2\pi$</span>, making R true.</p><p>∴ A is false but R is true.</p>
Correct Answer: D