Binomial Theorem
Sum of All Coefficients
nta_pyq_2024_jan
Grade 11
Question:
If $A$ denotes the sum of all the coefficients in the expansion of $(1 - 3x + 10x^2)^n$ and $B$ denotes the sum of all the coefficients in the expansion of $(1 + x^2)^n$, then:
$A = B^3$
$3A = B$
$B = A^3$
$A = 3B$
Step-by-Step Solution
Key Concept: Sum of all coefficients is obtained by substituting $x=1$. So $A=(1-3+10)^n=8^n$ and $B=(1+1)^n=2^n$. Then check the relationship.
Sum of coefficients: put $x=1$. $A=(1-3+10)^n=8^n=(2^3)^n=(2^n)^3=B^3$. So $A=B^3$.
Correct Answer: 1