Basic Mathematics & Logarithm
Logarithmic properties and applications
Grade 11
Question:
<p>If <i>p, q</i> ∈ ℕ satisfy the equation <i>x</i><sup><i>x</i></sup> = (<i>x</i><sup>1/<i>x</i></sup>)<sup><i>x</i></sup> and <i>q</i> ≠ <i>p</i>, then <i>q</i> is a perfect number.</p>
<p>(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1</p>
<p>(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1</p>
<p>(c) Statement-1 is true, Statement-2 is false</p>
<p>(d) Statement-1 is false, Statement-2 is true</p>
Step-by-Step Solution
Key Concept: Taking logarithm of exponential equations and analyzing the resulting algebraic conditions. A perfect number equals the sum of its proper divisors (e.g., 6 = 1+2+3).
<p><strong>Solution:</strong></p><p>Given: <i>x</i><sup><i>x</i></sup> = (<i>x</i><sup>1/<i>x</i></sup>)<sup><i>x</i></sup></p><p>Taking logarithm on both sides on base <i>e</i>:</p><p>$$\ln(x^x) = \ln\left((x^{1/x})^x\right)$$</p><p>$$x\ln x = x \cdot \ln(x^{1/x})$$</p><p>$$x\ln x = x \cdot \frac{1}{x}\ln x$$</p><p>$$x\ln x = \ln x$$</p><p>$$\ln x\left(x - 1\right) = 0$$</p><p>This gives <i>x</i> = 1 or <i>x</i> = <i>e</i>. Since we need natural numbers satisfying both <i>p</i> and <i>q</i> with <i>q</i> ≠ <i>p</i>, the analysis of perfect numbers must be checked separately. Statement-2 correctly defines a perfect number, but the original equation does not directly lead to Statement-1 being true.</p><p>∴ Answer is (d).</p>
Correct Answer: D