Matrices & Determinants
Determinants with AP Sequences
Grade 12
Question:
<p>If <span>p</span>, <span>q</span>, <span>r</span> and <span>s</span> are in AP and <span>f(x) = \begin{vmatrix} p + \sin x & q + \sin x & p + r + \sin x \\ q + \sin x & r + \sin x & -1 + \sin x \\ r + \sin x & s + \sin x & s + q + \sin x \end{vmatrix}</span> such that <span>\int_0^1 f(x) dx = -2</span>, the common difference of the AP can be</p>
<p>(a) <span>-1</span></p>
<p>(b) <span>1/2</span></p>
<p>(c) <span>1</span></p>
<p>(d) <span>2</span></p>
Step-by-Step Solution
Key Concept: Use the AP property to express all terms in terms of the first term and common difference, then integrate the determinant.
<p><strong>Step 1:</strong> Since <span>p</span>, <span>q</span>, <span>r</span>, <span>s</span> are in AP with common difference <span>d</span>, write <span>q = p + d</span>, <span>r = p + 2d</span>, <span>s = p + 3d</span>.</p><p><strong>Step 2:</strong> Substitute these into the determinant and simplify using properties of determinants.</p><p><strong>Step 3:</strong> Evaluate <span>\int_0^1 f(x) dx = -2</span> to find the value of <span>d</span>.</p><p>∴ Answer is <strong>(a), (c)</strong>.</p>
Correct Answer: A