<p><strong>24.</strong> If \(x, 2y, 3z\) are in A.P., where the distinct numbers \(x, y, z\) are in G.P., then the common ratio of the G.P. is</p>
Step-by-Step Solution
Key Concept: Since x, y, z are in G.P., write y = xr and z = xr² for common ratio r. Then use the A.P. condition on x, 2y, 3z to form an equation in r.
<p><strong>Step 1:</strong> Let x, y, z be in G.P. with common ratio r. Then: y = xr and z = xr²</p><p><strong>Step 2:</strong> Given that x, 2y, 3z are in A.P., the common difference condition gives:</p><p>2y − x = 3z − 2y</p><p>⟹ 4y = x + 3z</p><p><strong>Step 3:</strong> Substitute y = xr and z = xr²:</p><p>4(xr) = x + 3(xr²)</p><p>4xr = x + 3xr²</p><p><strong>Step 4:</strong> Divide by x (x ≠ 0):</p><p>4r = 1 + 3r²</p><p>3r² − 4r + 1 = 0</p><p><strong>Step 5:</strong> Factor: (3r − 1)(r − 1) = 0</p><p>r = 1/3 or r = 1</p><p><strong>Step 6:</strong> Since x, y, z are <strong>distinct</strong>, r ≠ 1. Therefore r = 1/3</p><p>∴ Answer: <strong>B (1/3)</strong></p>
Correct Answer: B