When two fair coins are flipped, there are four equally likely results: both heads, head then tail, tail then head, and both tails. “At least one head” includes all outcomes except the only all-tail result, so three of the four outcomes satisfy the condition. That gives a probability of 3/4. You can also see this by the complement: 1 minus the chance of no heads (both tails) is 1 - 1/4 = 3/4. The other numbers don’t fit because they don’t match the correct count of favorable outcomes out of four.

Whenever a function has a derivative positive at every point of an interval, the function must rise as x increases. For any two points x1 < x2 in the interval, the Mean Value Theorem gives f(x2) − f(x1) = f'(c)(x2 − x1) for some c in (x1, x2). Since f'(c) > 0 and x2 − x1 > 0, the difference is positive, so f(x2) > f(x1). Therefore, f is strictly increasing on the interval. This also means it cannot be decreasing, constant, or non-monotonic.

AN/AAR-57 CMWS is a missile warning system that sits passively in the aircraft’s sensor suite. It uses ultraviolet detectors to sense the exhaust plumes of incoming missiles, both surface-to-air and air-to-air. Because it is passive, it doesn’t emit signals of its own, which helps stay stealthy and reduces vulnerability. When the system detects a missile plume, it tracks the threat and declares it to the cockpit and to the aircraft’s defensive systems so countermeasures can be activated in time. This is different from jammers (which try to disrupt enemy communications), laser warning systems (which detect laser designators or rangefinders), or infrared countermeasures (which actively confuse or defeat missile seekers). The ultraviolet, passive detection and threat declaration are what define this system.

Set difference asks for the elements that are in the first set but not in the second. Look at A = {1, 2, 3} and remove any elements that also appear in B = {3, 4}. The only element of A that appears in B is 3, so we drop it. What remains are 1 and 2, giving A \ B = {1, 2}. The other possibilities would either keep the 3 (which should be removed) or miss one of the elements 1 or 2, so {1,2} is the correct result.