1. **State the problem:** A wheel with rotational inertia $0.25$ kg·m² is rotating at a constant angular speed of $3.0$ rad/s. A brake applies a force of $2.0$ N at the rim, $0.5$ m from the axle. We need to find the magnitude of the torque exerted by the motor. 2. **Relevant formula:** Torque $\tau$ due to a force is given by: $$\tau = r \times F$$ where $r$ is the distance from the axis of rotation and $F$ is the force applied perpendicular to the radius. 3. **Important rule:** Since the wheel rotates at constant angular speed, the net torque is zero. Therefore, the motor torque balances the brake torque. 4. **Calculate the brake torque:** $$\tau_{brake} = 0.5 \times 2.0 = 1.0\ \text{N·m}$$ 5. **Determine motor torque:** Because the wheel's angular speed is constant, motor torque $= \tau_{brake} = 1.0$ N·m. **Final answer:** The magnitude of the torque exerted by the motor is $\boxed{1.0}$ N·m, which corresponds to option C.