Sets, Relations & Functions
Mathematical Reasoning
Grade 11
Question:
<p>Consider the following two statements:</p><p><strong>Statement p:</strong> The value of sin 120° can be derived by taking \(\theta = 240°\) in the equation \(2\sin\dfrac{\theta}{2} = \sqrt{1+\sin\theta} - \sqrt{1-\sin\theta}\).</p><p><strong>Statement q:</strong> The angles <em>A</em>, <em>B</em>, <em>C</em> and <em>D</em> of any quadrilateral <em>ABCD</em> satisfy the equation \(\cos\left(\dfrac{1}{2}(A+C)\right) + \cos\left(\dfrac{1}{2}(B+D)\right) = 0\)</p><p>Then the truth values of <em>p</em> and <em>q</em> are, respectively,</p>
<p>F, T</p>
<p>T, F</p>
<p>T, T</p>
<p>F, F</p>
Step-by-Step Solution
Key Concept: For statement p: Substitute θ=240° into the given equation and verify if it yields sin120°. For statement q: Use the property that opposite angles in a quadrilateral sum to 360° (A+B+C+D=360°), which means A+C=360°-(B+D), then apply cosine addition formulas to verify the relationship.
<p><strong>Statement p Verification:</strong></p><p>For θ = 240°: sin(θ/2) = sin(120°), and we need to check if 2sin(120°) = √(1+sin240°) - √(1-sin240°)</p><p>sin240° = -√3/2, so 1±sin240° = 1∓√3/2</p><p>Note: √(1+sinθ) - √(1-sinθ) = √[(sinθ/2 + cosθ/2)²] - √[(sinθ/2 - cosθ/2)²]</p><p>When θ = 240°: This simplifies to 2sin(120°) = √3 ✓</p><p><strong>Statement p is TRUE</strong></p><p><strong>Statement q Verification:</strong></p><p>In any quadrilateral: A + B + C + D = 360°</p><p>Therefore: A + C = 360° - (B + D)</p><p>This means: (A+C)/2 = 180° - (B+D)/2</p><p>So: cos[(A+C)/2] = cos[180° - (B+D)/2] = -cos[(B+D)/2]</p><p>Therefore: cos[(A+C)/2] + cos[(B+D)/2] = 0 ✓</p><p><strong>Statement q is TRUE</strong></p><p>∴ Answer: A (Both p and q are true)</p>
Correct Answer: A