If the coefficient of x15 in the expansion of (a x3+(1/b x1 / 3))15 is equal to the coefficient of x-15 in the expansion of (a x1 / 3-(1/b x3))15, where a and b are positive real numbers, then for each such ordered pair (a, b) :

Question

If the coefficient of x15 in the expansion of (a x3+(1/b x1 / 3))15 is equal to the coefficient of x-15 in the expansion of (a x1 / 3-(1/b x3))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

Tardigrade Admin · Accepted Answer

Coefficient Of x15 text in (a x3+(1/b x1 / 3))15 Tr+1= 15 Cr(a x3)15-r((1/b x1 / 3))r 45-3 r-(r/3)=15 30=(10 r/3) r=9 Coefficient of x15= 15 C9 a6 b-9 Coefficient of x-15 text in (a x1 / 3-(1/b x3))15 T r +1= 15 C r ( ax 1 / 3)15- r (-(1/b x3)) r 5-( r /3)-3 r =-15 (10 r/3)=20 r=6 Coefficient = 15 C 6 a 9 × b -6 ⇒ (a9/b6)=(a6/b9) ⇒ a3 b3=1 ⇒ a b=1

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

$a = 3 b$

$a = b$

$ab = 1$

$ab = 3$

Q. If the coefficient of x15 in the expansion of (ax3+bx1/31​)15 is equal to the coefficient of x−15 in the expansion of (ax1/3−bx31​)15, where a and b are positive real numbers, then for each such ordered pair (a,b) :

a=3b

a=b

ab=1

ab=3

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

$a = 3 b$

$a = b$

$ab = 1$

$ab = 3$