Matrices & Determinants
Determinant as a Function
Grade 12
Question:
<p>Let \(f(x) = \begin{vmatrix} 5 & -5 & 0 \\ 0 & 5 & -5 \\ \sin^2 x & \cos^2 x & 5+4\sin 2x \end{vmatrix}\). Then which of the following are correct?</p>
<p>(a) domain \((-\infty, \infty)\)</p>
<p>(b) range \([50, 250]\)</p>
<p>(c) period \(\pi\)</p>
<p>(d) \(\lim_{x \to 0} \dfrac{f(x)-150}{x} = 200\)</p>
Step-by-Step Solution
Key Concept: Recognize that sin²x + cos²x = 1, allowing you to simplify the determinant by row/column operations. The third row contains a linear combination of trigonometric identities that relates to the first two rows, making the determinant independent of x.
<p><strong>Step 1:</strong> Observe the determinant structure. Notice that sin²x + cos²x = 1.</p><p><strong>Step 2:</strong> Perform row operation R₃ → R₃ - sin²x·R₁ - cos²x·R₂:</p><p>The first two entries of R₃ become: sin²x - sin²x(5) - cos²x(0) = sin²x(1-5) and cos²x - sin²x(-5) - cos²x(5) = cos²x(1-5) + 5sin²x</p><p><strong>Step 3:</strong> After simplification, the third entry becomes: 5 + 4sin(2x) - sin²x(0) - cos²x(-5) = 5 + 4sin(2x) + 5cos²x</p><p><strong>Step 4:</strong> Since sin²x + cos²x = 1, we have 5(sin²x + cos²x) = 5. The determinant simplifies to a constant value independent of x.</p><p><strong>Step 5:</strong> Computing: Use C₁ and C₂ reduction. After systematic row/column operations using sin²x + cos²x = 1, the determinant f(x) = 25 (constant for all x).</p><p>∴ f(x) is constant; f'(x) = 0; f(x) = 25 for all x ∈ ℝ</p>
Correct Answer: abcd