Binomial Theorem
Consecutive Coefficients
nta_pyq_2024_jan
Grade 11
Question:
Suppose $2-p,\, p,\, 2-\alpha,\, \alpha$ are the coefficients of four consecutive terms in the expansion of $(1+x)^n$. Then the value of $p^2-\alpha^2+6\alpha+2p$ equals
[Note: In the official NTA paper no option was correct.]
Step-by-Step Solution
Key Concept: Use the property that consecutive binomial coefficients $\binom{n}{k},\binom{n}{k+1},\binom{n}{k+2},\binom{n}{k+3}$ satisfy specific ratio conditions. Setting up equations from the given coefficients leads to a contradiction in this problem.
The four consecutive binomial coefficients $2-p, p, 2-\alpha, \alpha$ of $(1+x)^n$ lead to contradictory conditions — no consistent solution exists. Answer: Data Inconsistent.
Correct Answer: 4