If the coefficient of x7 in -ax - 1-bx-11 is 55 a11- then a and b satisfy the relation

The correct option is C ab = 1 General term of [ax + (1/bx)]^11 is, T_r+1 = #160; ^11C_r (ax)^r·(1/bx)^11 r #160; = #160; ^11C_ra^rb^r 11x^ ...

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Byju's Answer

Standard XII

Mathematics

Multinomial Expansion

If the coeffi...

Question

If the coefficient of $x^{7}$ in ${[a x + (\frac{1}{b x})]}^{11} i s 55 a^{11}$ , then $a$ and $b$ satisfy the relation

a + b = 1

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a - b = 1

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a b = 1

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\frac{a}{b} = 1

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Solution

The correct option is C $a b = 1$
General term of ${[a x + (\frac{1}{b x})]}^{11}$ is,
$T_{r + 1} =^{11} C_{r} (a x)^{r} \cdot {(\frac{1}{b x})}^{11 - r} =^{11} C_{r} a^{r} b^{r - 11} x^{2 r - 11}$
Let $T_{r + 1}$ posses $x^{7} \Rightarrow 2 r - 11 = 7 => r = 9$
Hence coefficient of $x^{7}$ is $=^{11} C_{9} a^{9} b^{- 2} = 55 a^{9} b^{- 2} = 55 a^{11}$ (given)
$∴ a^{2} b^{2} = 1 \Rightarrow a b = \pm 1$

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Similar questions

Q. If the coefficient of

x^{7}

{[a x^{2} + (\frac{1}{b x})]}^{11}

equals the coefficient of

x^{- 7}

{[a x - (\frac{1}{b x^{2}})]}^{11}

, then

a

and

b

satisfy the relation

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

Q. If the coefficient of

x^{8}

{(a x^{2} + \frac{1}{b x})}^{13}

is equal to the coefficient of

x^{- 8}

{(a x - \frac{1}{b x^{2}})}^{13}

, then

a

and

b

will satisfy the relation

Q. If the coefficient of

x^{7}

[a x^{2} + (\frac{1}{b x})]^{11}

equals the coefficient of

x^{- 7}

[a x^{2} - (\frac{1}{b x})]^{11}

then

a

and

b

satisfy the relation

Q. Find the coefficient of

x^{7}

{(a x^{2} + \frac{1}{b x})}^{11}

and that of

x^{- 7}

{(a x - \frac{1}{b x^{2}})}^{11}

and then find the relation between a and b so that these coefficients are equal, none of a, b and x is zero.

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If the coefficient of x7 in [ax+(1bx)]11is 55a11, then a and b satisfy the relation

The correct option is C ab=1General term of [ax+(1bx)]11 is,Tr+1=11Cr(ax)r⋅(1bx)11−r=11Crarbr−11x2r−11Let Tr+1 posses x7⇒2r−11=7=>r=9Hence coefficient of x7 is =11C9a9b−2=55a9b−2=55a11 (given)∴a2b2=1⇒ab=±1

If the coefficient of $x^{7}$ in ${[a x + (\frac{1}{b x})]}^{11} i s 55 a^{11}$ , then $a$ and $b$ satisfy the relation