Basic Mathematics & Logarithm
Greatest Integer Function
Grade 11
Question:
<p>The number of solutions of equation <strong>8[x² - x] + 4[x] = 13 + 12[sin x]</strong>, where <strong>[·]</strong> denotes the Greatest Integer Function (GIF).</p>
<p>(a) 0</p>
<p>(b) 2</p>
<p>(c) 4</p>
<p>(d) 6</p>
Step-by-Step Solution
Key Concept: The LHS produces only even values while RHS produces only odd values, so no solution exists.
<p><strong>Solution:</strong> The LHS is <strong>8[x² - x] + 4[x]</strong>, which is always even since both terms are multiples of even numbers (8 and 4).</p><p>The RHS is <strong>13 + 12[sin x]</strong>. Since <strong>12[sin x]</strong> is even, the RHS is always <strong>13 + even = odd</strong>.</p><p>Since LHS is always even and RHS is always odd, they can never be equal.</p><p><strong>∴ Answer is (a) 0.</strong></p>
Correct Answer: A