First slide

Binomial theorem for positive integral Index

Question

Coefficient of constant term in the expansion of
${(x^{3} + 5 (3)^{- \log \sqrt{3}} \sqrt{x^{3}})}^{4}$ is

Moderate

A

125
B

$\log_{\sqrt{3}} 5$
C

$\log_{3} 5$
D

150

Solution

Let $y = 3^{- \log_{\sqrt{3}}} (x^{3 / 2})$
$\Rightarrow \log_{3} y = - \frac{\log_{3} (x^{3 / 2})}{\log_{3} \sqrt{3}} = - 2 \log_{3} (x^{3 / 2})$
$\Rightarrow y = x^{- 3}$
$(r + 1)$ th term in the expansion of
$\begin{array}{l} {(x^{3} + 5 x^{- 3})}^{4} is \\ T_{r + 1} =^{4} C_{r} {(x^{3})}^{4 - r} {(5 x^{- 3})}^{r} \\ =^{4} C_{r} 5^{r} x^{12 - 3 r - 3 r} \end{array}$
For $(r + 1)$ th term to be independent of $x,$ set $12 - 6 r$
$= 0 or r = 2$
$∴$ required constant term
$=^{4} C_{2} 5^{2} = (6) (25) = 150$

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