TS EAMCET · Maths · Binomial Theorem

For $n \in N$ , in the expansion of ${(\sqrt[4]{x^{- 3}} + a \sqrt[4]{x^{5}})}^{n}$ , the sum of all binomial coefficients lies between $200$ and $400$ and the term independent of $x$ is $448$ . Then the value of $a$ is

A $1$
B $2$
C $\frac{1}{2}$
D $0$

Verified Solution

Answer & Solution

Correct Answer

(B) $2$

Step-by-step Solution

Detailed explanation

Given that, x-34+ax54n General Term Tr+1= nCr(1x34)n-rar(x54)r =nCrarx-34n+34r+54r For independent of x- r=n3434+54 =3n484 ∴ r=3n8 The term independent of x is 448. ⇒ C3n8na3n8=448 Sum of coefficient between 200 and 400 is given as…

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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For n∈N, in the expansion of x-34+ax54n, the sum of all binomial coefficients lies between 200 and 400 and the term independent of x is 448. Then the value of a is

Step-by-step Solution

For $n \in N$ , in the expansion of ${(\sqrt[4]{x^{- 3}} + a \sqrt[4]{x^{5}})}^{n}$ , the sum of all binomial coefficients lies between $200$ and $400$ and the term independent of $x$ is $448$ . Then the value of $a$ is