Sets, Relations & Functions
Set Operations and Inequalities
Grade 11
Question:
<p>Consider the two sets <span class="math">A = \{m \in \mathbb{R} : \text{both the roots of } x^2 - (m+1)x + m + 4 = 0 \text{ are real}\}</span> and <span class="math">B = [-3, 5)</span>. Which of the following is not true?</p>
<p>(a) <span class="math">A - B = (-\infty, -3) \cup (5, \infty)</span></p>
<p>(b) <span class="math">A \cap B = \{-3\}</span></p>
<p>(c) <span class="math">B - A = (-3, 5)</span></p>
Step-by-Step Solution
Key Concept: Find set A by solving the discriminant condition for real roots, then compare with B to check set operations.
<p><strong>Solution:</strong></p><p>For the quadratic <span class="math">x^2 - (m+1)x + m + 4 = 0</span> to have both real roots, the discriminant must be non-negative:</p><p><span class="math">\Delta = (m+1)^2 - 4(m+4) \geq 0</span></p><p><span class="math">= m^2 + 2m + 1 - 4m - 16 \geq 0</span></p><p><span class="math">= m^2 - 2m - 15 \geq 0</span></p><p><span class="math">= (m-5)(m+3) \geq 0</span></p><p><span class="math">\therefore m \in (-\infty, -3] \cup [5, \infty)</span></p><p>So <span class="math">A = (-\infty, -3] \cup [5, \infty)</span> and <span class="math">B = [-3, 5)</span></p><p>Now: (a) <span class="math">A - B = (-\infty, -3) \cup [5, \infty) \neq (-\infty, -3) \cup (5, \infty)</span> ✓ This is false</p><p>But checking (c): <span class="math">B - A = [-3, 5) - [(-\infty, -3] \cup [5, \infty)] = (-3, 5)</span> ✓ This is true</p>
Correct Answer: C