If fundamental period of fx- 4- x-cosec 6- x- is P- then coefficient of x-24P in the expansion x-bx332P is

Fundamental period of f(x)=P= L.C.M. of 14 and 16 ⇒ P= nbsp;12 So we need to find coefficient of x^ 12 in nbsp;(x+bx^3)^16 We know T_r+1= ^16C_r( bx^3)^rx^1 ...

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Standard XII

Mathematics

Multinomial Expansion

If fundamenta...

Question

If fundamental period of $f (x) = | sec 4 π x | + | cosec 6 π x |$ is $P,$ then coefficient of $x^{- 24 P}$ in the expansion ${(x + \frac{b}{x^{3}})}^{32 P}$ is

^{16} C_{1} b

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^{16} C_{9} b^{9}

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^{12} C_{3} b^{3}

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^{16} C_{7} b^{7}

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Solution

The correct option is D $^{16} C_{7} b^{7}$
Fundamental period of $f (x) = P =$ L.C.M. of $\frac{1}{4}$ and $\frac{1}{6}$
$\Rightarrow P = \frac{1}{2}$
So we need to find coefficient of $x^{- 12}$ in ${(x + \frac{b}{x^{3}})}^{16}$
We know $T_{r + 1} =^{16} C_{r} {(\frac{b}{x^{3}})}^{r} x^{16 - r}$
$\Rightarrow T_{r + 1} =^{16} C_{r} b^{r} x^{16 - 4 r}$
Now, to get coefficient of $x^{- 12}$ put $16 - 4 r = - 12$
$\Rightarrow r = 7$
$∴$ Required coefficient $=^{16} C_{7} b^{7}$

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If fundamental period of f(x)=|sec4πx|+|cosec 6πx| is P, then coefficient of x−24P in the expansion (x+bx3)32P is

The correct option is D 16C7b7Fundamental period of f(x)=P= L.C.M. of 14 and 16 ⇒P=12 So we need to find coefficient of x−12 in (x+bx3)16 We know Tr+1= 16Cr(bx3)rx16−r ⇒Tr+1= 16Crbrx16−4r Now, to get coefficient of x−12 put 16−4r=−12 ⇒r=7 ∴Required coefficient = 16C7b7

If fundamental period of $f (x) = | sec 4 π x | + | cosec 6 π x |$ is $P,$ then coefficient of $x^{- 24 P}$ in the expansion ${(x + \frac{b}{x^{3}})}^{32 P}$ is