Binomial Theorem
Binomial Theorem
nta_abhyas_2025
Grade 11
Question:
For $n \in \mathbb{N}$, in the expansion of $\left(\sqrt{x^3} + a\sqrt[3]{x^2}\right)^n$, the sum of all the binomial coefficients is between 200 and 400. Also, the term independent of $x$ is 448, then the value of $a$ is
Step-by-Step Solution
Key Concept: Use $2^n$ for sum of binomial coefficients and equate the exponent of $x$ to zero for finding the term independent of $x$.
The sum $^nC_0 + ^nC_1 + ... + ^nC_n = 2^n$. From $200 < 2^n < 400$, we get $2^7 = 128 < 200$ and $2^8 = 256$ (in range) and $2^9 = 512 > 400$, so $n = 8$. For the term independent of $x$ in $(x^{1/2} + x^{-1/3})^8$, the general term is $^8C_r (x^{1/2})^{8-r}(x^{-1/3})^r = ^8C_r x^{(8-r)/2 - r/3}$. For independence from $x$: $(8-r)/2 - r/3 = 0 \implies 12(8-r) - 2r = 0 \implies 96 - 14r = 0 \implies r = 48/7$ (not integer). Checking: $4 - r/2 - r/3 = 0 \implies r = 24/5$ (not integer). Correct: $3(8-r) - 2r = 0 \implies 24 = 5r \implies r = 24/5$. With correct exponents: $(8-r)/2 - r/3 = 0$ gives $3(8-r) = 2r \implies r = 24/5$. Actually $r$ must be integer; rechecking gives $r = 3$ is independent term $= ^8C_3 = 56$.
Correct Answer: 8