A monotonic signal consistently moves in only one direction (non-increasing or non-decreasing) over its domain. Simplify each candidate and/or inspect its derivative; the keyed choice is the expression that reduces to a monotonic function.
Formula / Key Relation
sin²(2t) + cos²(2t) = 1
So x(t) = 1 − 2t
dx/dt = −2 < 0
Detailed Explanation
Concept & ReasoningThe safest method is to differentiate candidate continuous-time functions. A derivative that keeps one sign indicates monotonic behavior. Periodic sine/cosine functions are generally not monotonic over the entire real line, and log(cos t) is not even defined for all t. The trigonometric identity sin²(2t)+cos²(2t)=1 can reduce one candidate to a simple linear function, which is monotonic.How to Solve It in the ExamSimplify identities before differentiating.Important Exam PointMonotonic does not mean 'always positive'; it means consistently non-increasing or non-decreasing.Related Revision PathThis question sits in the revision path Signals and Systems → Signal Classification → Monotonic Signals .
Concept & Reasoning
The safest method is to differentiate candidate continuous-time functions. A derivative that keeps one sign indicates monotonic behavior. Periodic sine/cosine functions are generally not monotonic over the entire real line, and log(cos t) is not even defined for all t. The trigonometric identity sin²(2t)+cos²(2t)=1 can reduce one candidate to a simple linear function, which is monotonic.
How to Solve It in the Exam
Simplify identities before differentiating.
Important Exam Point
Monotonic does not mean 'always positive'; it means consistently non-increasing or non-decreasing.
Related Revision Path
This question sits in the revision path Signals and Systems → Signal Classification → Monotonic Signals .
Quick Trick
Simplify identities before differentiating.
Common Mistake
Rejecting a decreasing straight line because it is not increasing.
Exam Tip
Monotonic does not mean 'always positive'; it means consistently non-increasing or non-decreasing.
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