Complex Numbers
Complex Numbers
Allen Star Batch
Grade 11
Question:
If $p(x), q(x), r(x)$ and $s(x)$ are polynomials such that $p(x^3) + xq(x^3) + x^2r(x^3) = (1 + x + x^2)s(x)$ then
$p(1) = s(1)$
$p(1) = r(1)$
$p(1) = 3s(1)$
$p(1) = 2r(1)$
Step-by-Step Solution
Key Concept: Count selections from available digits for each leading digit to identify the target number.
Five-digit numbers starting with 1 have 4 remaining positions filled from digits 0–9, giving $\binom{9}{4} = 70$ such numbers. Five-digit numbers starting with 2 give $\binom{9}{4} = 35$ such numbers. The total is $70 + 35 = 105$. Thus the 105th five-digit number is 26789.
Correct Answer: 1,2