If the coefficient of $$x7$$ in the expansion of $$ax2+1bx11$$ and the coefficient of x– $$7$$ is the expansion of ax−$$1bx211are$$ equal, then

Question

If the coefficient of $$x7$$ in the expansion of  $$ax2+1bx11$$ and the coefficient of x– $$7$$ is the expansion of ax−$$1bx211are$$ equal, then

Infinity Learn · Accepted Answer

$$Tr+1$$,$$the(r+1)$$ th  term in the expansion $$ax2+1bx11is$$ $$Tr+1=11Crax211$$−$$r1bxr=11Cra11$$−$$rbrx22$$−$$3rFor$$ the coefficient of $$x7$$,set $$22$$−$$3r=7$$ ⇒ $$r=5$$∴Coefficient of $$x7$$ in $$ax2+1bx11$$ is  $$11C5a6b5Next$$, tr $$+$$ $$1$$, the $$(r$$ $$+$$ $$1)th$$ term in the expansion of ax−$$1bx211is$$ $$tr+1=11Cr(ax)11$$−r−$$1bx2r=11Cra11$$−$$rbr($$−$$1)rx11$$−$$3rFor$$ the coefficient of x−$$7$$, set $$11$$−$$3r=$$−$$7$$⇒$$r=6$$∴Coefficient of  x−$$7$$ in the expansion of $$=11C6a5b6We$$ are given   $$11C5a6b5=11C6a5b6$$⇒$$ab=1$$

If the coefficient of x⁷ in the expansion of ${(a x^{2} + \frac{1}{b x})}^{11}$ and the coefficient of x^{– 7}is the expansion of ${(a x - \frac{1}{b x^{2}})}^{11}$ are equal, then

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If the coefficient of x7 in the expansion of ax2+1bx11 and the coefficient of x– 7 is the expansion of ax−1bx211are equal, then

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If the coefficient of x⁷ in the expansion of ${(a x^{2} + \frac{1}{b x})}^{11}$ and the coefficient of x^{– 7}is the expansion of ${(a x - \frac{1}{b x^{2}})}^{11}$ are equal, then