If the coefficient of x 7 in [ a x 2 + ( 1 b x ) ] 11 equals the coefficient of x - 7 in [ a x - ( 1 b x 2 ) ] 11 , then a and b satisfy the relation

Question

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

Smart Achievers · Accepted Answer

(3)

$T_{r + 1} of {(a x^{2} + \frac{1}{b x})}^{11} = {}^{11}C_{r} {(a x^{2})}^{11 - r} {(\frac{1}{b x})}^{r} = {}^{11}C_{r} \frac{a^{11 - r}}{b^{r}} {(x)}^{22 - 3 r}$

$T_{r + 1} of {(a x - \frac{1}{b x^{2}})}^{11} = {}^{11}C_{r} {(a x)}^{11 - r} {(- \frac{1}{b x^{2}})}^{r}$

$= {}^{11}C_{r} \frac{a^{11 - r}}{b^{r}} x^{11 - 3 r} {(- 1)}^{r}$

$∴$ $Coefficient of x^{7} in$

${(a x^{2} + \frac{1}{b x})}^{11} = {}^{11}C_{5} \frac{a^{6}}{b^{5}}$

$and coefficient of x^{- 7} in$

${(a x - \frac{1}{b x^{2}})}^{11} = {}^{11}C_{6} \frac{a^{5}}{b^{6}}$

$Now, {}^{11}C_{5} \frac{a^{6}}{b^{5}} = {}^{11}C_{6} \frac{a^{5}}{b^{6}}$

$∴$ $a b = 1$

Solution

Q.
If the coefficient of $x^{7}$ in ${[a x^{2} + (\frac{1}{b x})]}^{11}$ equals the coefficient of $x^{- 7}$ in ${[a x - (\frac{1}{b x^{2}})]}^{11},$ then $a$ and $b$ satisfy the relation

1 $a + b = 1$

2 $a - b = 1$

3 $a b = 1$

4 $\frac{a}{b} = 1$

Want Us to Email you About Special Offers & Updates?

Solution

Q. If the coefficient of x7 in [ax2+(1bx)]11 equals the coefficient of x-7 in [ax-(1bx2)]11, then a and b satisfy the relation

1 a+b=1

2 a-b=1

3 ab=1

4 ab=1