TS EAMCET · Maths · Binomial Theorem

In the binomial expansion of $(1 + x)^{2 k},$ if its middle term is the only numerically greatest term, then $x$ lies in the interval

A $(- 2 k, 2 k)$
B $(- \frac{k + 1}{k}, \frac{k + 1}{k})$
C $(- k, k)$
D $(- (k + 1), (k + 1))$

Verified Solution

Answer & Solution

Correct Answer

(B) $(- \frac{k + 1}{k}, \frac{k + 1}{k})$

Step-by-step Solution

Detailed explanation

The middle term in the binomial expansion of 1+xn, when n is even is Tr+1=Crnxr, where r=n2. The middle term in the binomial expansion of 1+x2k is Tk+1=Ck2k×xk. Given, the middle term is the only greatest term, hence Tk+1Tk>1 and Tk+1Tk+2>1…

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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In the binomial expansion of (1+x)2k, if its middle term is the only numerically greatest term, then x lies in the interval

Step-by-step Solution

In the binomial expansion of $(1 + x)^{2 k},$ if its middle term is the only numerically greatest term, then $x$ lies in the interval