Question
Download Solution PDFयदि \({\left( {\sqrt x - \frac{k}{{{x^2}}}} \right)^{10}}\) के प्रसार में अचर पद 405 है, तो k के मान क्या हो सकते हैं?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
(a + b)n के द्विपद प्रसरण में वह पद जो किसी चर में शामिल नहीं है, उसे स्वतंत्र पद कहा जाता है।
(a + b)n के द्विपद प्रसरण में सामान्य पद निम्न द्वारा ज्ञात किया जाता है: \({T_{r + 1}} = {\;^n}{C_r} \times {a^{n - r}} \times {b^r}\)
गणना:
दिया गया है: \({\left( {\sqrt x - \frac{k}{{{x^2}}}} \right)^{10}}\) के प्रसरण में स्थिरांक पद 405 है
अर्थात्\({\left( {\sqrt x - \frac{k}{{{x^2}}}} \right)^{10}}\) का स्वतंत्र पद 405 है।
माना कि (r + 1)वां पद स्वतंत्र पद है।\(\Rightarrow {T_{r + 1}} = {\;^{10}}{C_r} \times {\left( {\sqrt x } \right)^{10 - r}} \times {\left( {\frac{{ - k}}{{{x^2}}}} \right)^r} = \;{\;^{10}}{C_r} \times {\left( { - k} \right)^r} \times {\left( x \right)^{\frac{{10 - 5r}}{2}}}\;\)
∵ (r + 1)वां पद स्वतंत्र पद है।
\(\Rightarrow \frac{{10 - 5r}}{2} = 0 \Rightarrow r = 2\)
\(\Rightarrow {\;^{10}}{C_2} \times {\left( { - k} \right)^2} = 405 \Rightarrow 45{k^2} = 405\)
⇒ k = ± 3Last updated on Jul 7, 2025
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