Question
Download Solution PDFयदि \(\rm k^4+\frac{1}{k^4}=194\) है, तो \(\rm k^3+\frac{1}{k^3}\) का मान ज्ञात कीजिए।
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFदिया गया है:
\(\rm k^4+\frac{1}{k^4}=194\)
प्रयुक्त सूत्र:
\(\rm (a+\frac{1}{a})^2=\rm a^2+\frac{1}{a^2} + 2\)
यदि
\(\rm (a+\frac{1}{a})=b\)
तो,
\(\rm a^3+\frac{1}{a^3} =b^3-3b\)
गणना:
\(\rm k^4+\frac{1}{k^4}=194\)
⇒ \(\rm k^4+\frac{1}{k^4}+2=194+2\)
⇒ \(\rm k^4+\frac{1}{k^4}+2=196\)
⇒ \(\rm( k^2+\frac{1}{k^2})^2=14^2\)
⇒ \(\rm k^2+\frac{1}{k^2}=14\)
⇒ \(\rm k^2+\frac{1}{k^2}+2=14+2\)
⇒ \(\rm (k+\frac{1}{k})^2=16\)
⇒ \(\rm (k+\frac{1}{k})^2=4^2\)
⇒ \(\rm k+\frac{1}{k}=4\)
अब,
\(\rm k^3+\frac{1}{k^3}\) = 43 - 3 × 4
⇒ \(\rm k^3+\frac{1}{k^3}\) = 64 - 12
⇒ \(\rm k^3+\frac{1}{k^3}\) = 52
∴ अभीष्ट उत्तर 52 है।
Last updated on Jul 19, 2025
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