Question

If the coefficient of $x^{15}$ in the expansion of $\left(a x^3+\frac{1}{b x^{1 / 3}}\right)^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of $\left(a x^{1 / 3}-\frac{1}{b x^3}\right)^{15}$, where a and $b$ are positive real numbers, then for each such ordered pair $(a, b)$ :

• A. $a=3 b$
• B. $a=b$
• C. $ab =1$
• D. $a b=3$

Answer 1

Answer: (C)

$ab =1$

Answer 2

Coefficient Ofx15 in (ax3+bx1/31​)15
Tr+1​=15Cr​(ax3)15−r(bx1/31​)r
45−3r−3r​=15
30=310r​
r=9
Coefficient of x15=15C9​a6b−9
Coefficient of x−15 in (ax1/3−bx31​)15
Tr+1​=15Cr​(ax1/3)15−r(−bx31​)r
5−3r​−3r=−15
310r​=20
r=6
Coefficient =15C6​a9×b−6
⇒b6a9​=b9a6​
⇒a3b3=1⇒ab=1