For x>0, if pth term is the first negative term in the expansion of 1+3x522/3and in the expansion of 1-3x522/3 from rthterm onwards all the terms are positive, then the number of terms in the expansion of px+rxpr is

TS EAMCET · Maths · Binomial Theorem

For $x > 0,$ if $p^{th}$ term is the first negative term in the expansion of ${(1 + \frac{3 x}{5})}^{22 / 3}$ and in the expansion of ${(1 - \frac{3 x}{5})}^{22 / 3}$ from $r^{t h}$ term onwards all the terms are positive, then the number of terms in the expansion of ${(p x + \frac{r}{x})}^{p r}$ is

A $101$
B $119$
C $200$
D $99$

Verified Solution

Answer & Solution

Correct Answer

(A) $101$

Step-by-step Solution

Detailed explanation

For negative/rational index n, 1+xn=1+nx+nn-12!+.....+nn-1n-2n-3..................n-r+1r!xr+.... In the given expansion 1+3x522/3, pth term is first negative term. ∴Tp=nn-1n-2n-3..................n-p+2p-1!xp-1Here, n=223 Here, n-p+2<0 ⇒223-p+2<0…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Match the following

Column-1 Reaction		Column-2 Main product
A)	$4 {FeCl}_{3} + 3 K_{4} [Fe (CN)_{6}] \to$	I)	${Fe}_{4} {[Fe (CN)_{6}]}_{3}$
B)	${ZnCl}_{2} + 4 NaOH \to$	II)	${FeCl}_{2}$
C)	$2 {FeCl}_{3} + H_{2} S \to$	III)	$Zn (OH) Cl$
		IV)	${Fe}_{3} {[Fe (CN)_{6}]}_{3}$
		V)	${Na}_{2} {ZnO}_{2}$
		VI)	$FeS$

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