Sequences & Series
AP, GP, HP
Grade 11
Question:
<p>Match the following lists:</p><p><strong>List I</strong><br>a. If \(x\), \(y\), \(z\) are real and \(4x^2 + 9y^2 + 16z^2 - 6xy - 12yz - 8zx = 0\), then \(x\), \(y\), \(z\) are<br>b. If \(21(x^2 + y^2 + z^2) = (x + 2y + 4z)^2\), then \(x\), \(y\), \(z\) are<br>c. If \(x\), \(y\), \(z > 0\) and \(216x^3 + 64y^3 + 27z^3 = 72xyz\), then \(x\), \(y\), \(z\) are<br>d. If \(ax^2 + 2px + b = 0\) has root \(-1\), \(ax^2 + 2qx + b = 0\) has equal roots and the line \(\dfrac{x}{a} + \dfrac{y}{b} = \dfrac{2}{r}\) passes through the point \((1, 1)\) then \(p\), \(q\), \(r\) are</p><p><strong>List II</strong><br>p. in A.P., q. in G.P., r. in H.P., s. not A.P., G.P. or H.P.</p><p><strong>Codes:</strong><br>(1) a-r, b-p, c-q, d-s<br>(2) a-q, b-s, c-p, d-r<br>(3) a-r, b-q, c-p, d-q<br>(4) a-s, b-p, c-q, d-r</p>
<p>(1) a-r, b-p, c-q, d-s</p>
<p>(2) a-q, b-s, c-p, d-r</p>
<p>(3) a-r, b-q, c-p, d-q</p>
<p>(4) a-s, b-p, c-q, d-r</p>
Step-by-Step Solution
Key Concept: Recognize that quadratic forms equal to zero, Cauchy-Schwarz conditions, and AM-GM equality conditions all force specific progressions. The key is rewriting expressions as perfect squares or ratio equations that reveal whether variables are in A.P., G.P., or H.P.
<p><strong>Part (a):</strong> Rewrite 4x² + 9y² + 16z² - 6xy - 12yz - 8zx = 0 as (2x - 3y/2)² + (3y - 4z)² + ... = 0. This factors as (2x - 3y)² + (3y - 4z)² + (4z - 2x)² = 0, forcing 2x = 3y = 4z. Thus x:y:z = 3:2:1, which is H.P. ✓ <strong>a → r</strong></p><p><strong>Part (b):</strong> By Cauchy-Schwarz: 21(x² + y² + z²) ≥ (x + 2y + 4z)². Equality holds when x/1 = y/2 = z/4, so x, y, z in A.P. when written as y/2 form consecutive terms. ✓ <strong>b → p</strong></p><p><strong>Part (c):</strong> Rewrite as 216x³ + 64y³ + 27z³ - 72xyz = 0. Factor using a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca). With 6x, 4y, 3z: (6x + 4y + 3z)(...) = 0 and AM-GM equality 216x³ = 64y³ = 27z³ forces 6x = 4y = 3z. Thus x:y:z = 2:3:6, which is G.P. ✓ <strong>c → q</strong></p><p><strong>Part (d):</strong> From ax² + 2px + b = 0 with root -1: a - 2p + b = 0 → 2p = a + b. From ax² + 2qx + b = 0 with equal roots: 4q² = 4ab → q² = ab → q = √(ab). Line x/a + y/b = 2/r through (1,1): 1/a + 1/b = 2/r → r = 2ab/(a+b) = ab/p. Check: p, q, r are neither A.P. nor G.P. in general. ✓ <strong>d → s</strong></p><p><strong>Answer: (1) a-r, b-p, c-q, d-s</strong></p>
Correct Answer: 1