First slide

Binomial theorem for positive integral Index

Question

$If the coefficient of x^{7} in {({ax}^{2} - \frac{1}{bx})}^{11} equals the coefficient of x^{- 7} in {(ax - \frac{1}{{bx}^{2}})}^{11}, then a and b satisfy the relation$

Moderate

A

a-b=1
B

a+b=1
C

a/b=1
D

ab=1

Solution

$\begin{array}{l} Coefficient of x^{7} in {({ax}^{2} + \frac{1}{bx})}^{11} {is}^{11} C_{5} a^{6} \frac{1}{b^{5}}, Coefficient of x^{- 7} in \\ {(a x - \frac{1}{b x^{2}})}^{11} {is}^{11} C_{6} a^{5} \frac{1}{b^{6}} \\ ^{11} C_{6} a^{5} \cdot \frac{1}{b^{6}} =^{11} C_{5} \cdot a^{6} - \frac{1}{b^{5}} \\ \Rightarrow a b = \frac{^{11} C_{6}}{^{11} C_{5}} = \frac{6}{6} = 1 \end{array}$

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