<p><strong>36.</strong> If \((1-p)(1+3x+9x^2+27x^3+81x^4+243x^5) = 1 - p^6\), \(p \neq 1\), then the value of \(\dfrac{p}{x}\) is</p>
Step-by-Step Solution
Key Concept: Recognize that 1+3x+9x²+27x³+81x⁴+243x⁵ is a geometric series with first term 1, common ratio 3x, and 6 terms. This can be written as (1-(3x)⁶)/(1-3x), and the given equation factors as a difference of sixth powers.
<p><strong>Step 1:</strong> Recognize that 1+3x+9x²+27x³+81x⁴+243x⁵ is a geometric series with first term a=1, common ratio r=3x, and n=6 terms.</p><p><strong>Step 2:</strong> Use the geometric series formula: 1+3x+9x²+...+243x⁵ = (1-(3x)⁶)/(1-3x) = (1-729x⁶)/(1-3x)</p><p><strong>Step 3:</strong> Substitute into the given equation:<br>(1-p)·(1-729x⁶)/(1-3x) = 1-p⁶</p><p><strong>Step 4:</strong> Factor both sides using difference of nth powers:<br>1-p⁶ = (1-p)(1+p+p²+p³+p⁴+p⁵)<br>1-729x⁶ = (1-3x)(1+3x+9x²+27x³+81x⁴+243x⁵)</p><p><strong>Step 5:</strong> The equation becomes:<br>(1-p)·(1-3x)(1+3x+9x²+27x³+81x⁴+243x⁵)/(1-3x) = (1-p)(1+p+p²+p³+p⁴+p⁵)</p><p><strong>Step 6:</strong> Cancel (1-p) and (1-3x):<br>1+3x+9x²+27x³+81x⁴+243x⁵ = 1+p+p²+p³+p⁴+p⁵</p><p><strong>Step 7:</strong> Compare coefficients of corresponding terms: p = 3x</p><p><strong>Step 8:</strong> Therefore, p/x = 3</p><p>∴ Answer: B</p>
Correct Answer: B