Definite Integration
Grade 12
Question:
<p>The graph of f(x) = x<sup>2</sup> + ax + b intersects the x-axis at 2 distinct points A, B and y-axis at C. The centroid of <span class="math-tex">\(\triangle\)</span>ABC lie on the line y = x. If I = <span class="math-tex">\(\int_\limits{0}^{6}\)</span> f(x) dx, then the value of I cannot be :</p>
<p style="display:inline">-50</p>
<p style="display:inline">50</p>
<p style="display:inline">-100</p>
<p style="display:inline">100</p>
Step-by-Step Solution
Key Concept: Combine the centroid condition with the discriminant requirement for distinct roots to find the constrained range of the integral.
<p>Let A(<span class="math-tex">$\alpha$</span>, 0), B(<span class="math-tex">$\beta$</span>, 0) and (0, b) centroid = <span class="math-tex">$\left[\frac{\alpha+\beta}{3}, \frac{b}{3}\right]$</span><br />
<span class="math-tex">$\Rightarrow$</span> <span class="math-tex">$\alpha+\beta$</span> = b<br />
<span class="math-tex">$\Rightarrow$</span> -a + b<br />
Now, f(x) = x<sup>2</sup> - bx + b<br />
Since, y = f(x) has distinct roots.<br />
<span class="math-tex">$\Rightarrow$</span> b<sup>2</sup> - 4b > 0<br />
<span class="math-tex">$\Rightarrow$</span> b <span class="math-tex">$\in$</span> (<span class="math-tex">$-\infty$</span>, 0)<span class="math-tex">$\cup$</span>(4, <span class="math-tex">$\infty$</span>)<br />
Now, I = <span class="math-tex">$\int_{0}^{6}$</span> (x<sup>2</sup> - bx + b)dx = 72 - 12b<br />
But value of b cannot lie in [0, 4].<br />
So, value of l cannot lie in [24, 72].</p>
Correct Answer: B