Binomial Theorem
Product Expansion Coefficients
nta_pyq_2025_apr
Grade 11
Question:
If in the expansion of $(1+x)^p(1-x)^q$, the coefficients of $x$ and $x^2$ are 1 and $-2$, respectively, then $p^2 + q^2$ is equal to:
Step-by-Step Solution
Key Concept: Expand the product and equate coefficients of $x$ and $x^2$ to form a system in $p$ and $q$.
Coefficient of $x$: $p - q = 1$. Coefficient of $x^2$: $\frac{q(q-1)}{2} - pq + \frac{p(p-1)}{2} = -2 \Rightarrow 1 - (p+q) = -4 \Rightarrow p+q = 5$. Solving: $p = 3, q = 2$. $p^2 + q^2 = 9 + 4 = 13$.
Correct Answer: 13