Binomial Theorem
Coefficient extraction; last digit
MJMT_Full_Test_08
Grade 12
Question:
The term independent of $x$ in $(1+x+x^{-2}+x^{-3})^{10}$ is $n$. Then the last digit of $(n+2)^n$ is
Step-by-Step Solution
Key Concept: Rewrite $(1+x+x^{-2}+x^{-3})^{10}=\frac{(1+x+x^3+x^4)^{10}}{x^{30}}$. Find coefficient of $x^{30}$ in $(1+x+x^3+x^4)^{10}$.
$n=11851$. Last digit of $(11853)^{11851}=3^{11851}$: cycle 4, $11851\equiv3\pmod4$, last digit 7.
Correct Answer: 3