2025/26 AHSMC Part 1 Problem 11

Problem 11 of 16

What is the smallest positive real number $k$ so that the curves $y = x^2 + 10x + 26$ and $y = \sin(kx)$ have at least one point in common? Here, $x$ and $kx$ are in degrees.

Show solution

Solution

Note that $x^2 + 10x + 26 = (x+5)^2 + 1 \ge 1$ , while $\sin(kx) \le 1$ . Equality throughout forces $x = -5$ and $\sin(-5k) = 1$ , i.e. $\sin(5k) = -1$ . The smallest positive $k$ is given by $5k = 270$ , so $k = 54$ . The answer is (C).

University of Alberta

Answer choices

Solution