JEE Mains · Maths · STD 11 - 7. binomial theoram
If the coefficient of \(x ^{15}\) in the expansion of \(\left(a x^3+\frac{1}{b x^{\frac{1}{3}}}\right)^{15}\) is equal to the coefficient of \(x^{-15}\) in the expansion of \(\left(a x^{\frac{1}{3}}-\frac{1}{b x^3}\right)^{15}\), where \(a\) and \(b\) are positive real numbers, then for each such ordered pair \((a, b) :\)
- A \(a=b\)
- B \(ab =1\)
- C \(a=3 b\)
- D \(a b=3\)
Answer & Solution
Correct Answer
(B) \(ab =1\)
Step-by-step Solution
Detailed explanation
\(\text { Coefficient Of } x^{15} \text { in }\left(a x^3+\frac{1}{b x^{1 / 3}}\right)^{15}\) \(T_{r+1}={ }^{15} C_r\left(a x^3\right)^{15-r}\left(\frac{1}{b x^{1 / 3}}\right)^r\) \(45-3 r-\frac{r}{3}=15\) \(30=\frac{10 r}{3}\) \(r=9\)…
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