Binomial Theorem
Coefficient extraction in trinomial expansion
nta_pyq_2025_apr
Grade 12
Question:
Let$(1 + x + x$) 10 2 =$a_{0}$+$a_{1}$x +$a_{2}$x 2 +$\ldots$. +$a_{20}$x 20 . If$(a + a + a$+$\ldots$$. +a$1 3 5 19$) - 11a2 = 121k$, then k is equal to .
Step-by-Step Solution
Key Concept: Use the general term of the binomial expansion for coefficient extraction$i_n$trinomial expansion and impose the required coefficient or exponent condition.
$(1 + x + x$) 2 10 =$a_{0}$+$a_{1}$x +$a_{2}$x 2 +$\ldots$. +$a_{20}$x 20 (239) ∴ 3 10 =$a_{0}$+$a_{1}$+$a_{2}$+$\ldots$. +$a_{20}$...(i) 1 =$a_{0}$-$a_{1}$+$a_{2}$$\ldots$. . +$a_{20}$...(ii) 10$3 -1$$(i) - (ii)$$\Rightarrow$$a_{1}$+$a_{3}$+$\ldots$. +$a_{19}$= = 29524 2 10 Also${1 + x(1 + x)} = 1$10 10 2 2 +$C_{1}$$x(1 + x)$+$C_{2}$x$(1 + x)$+$\ldots$. 10 10 ∴$a_{2}$=$C_{1}$+$C_{2}$= 55 ($a_{1}$+$a_{3}$+$\ldots$+$a_{19}$$)-11a2$∴ = 239 121
Correct Answer: 239