In the following question, out of the four alternatives, select the alternative which best expresses the meaning of the idiom/phrase: "At the drop of a hat".

Introduction / Context:
This question tests understanding of an English idiom: "at the drop of a hat". Idioms are fixed expressions whose meanings cannot always be guessed from the literal meanings of the individual words. Many competitive exams include idiom questions to check how well learners understand everyday and figurative English usage.

Given Data / Assumptions:

• Idiom: "At the drop of a hat".
• Options describe different possible meanings, only one of which matches the accepted idiomatic sense.
• We must choose the option that expresses the correct meaning most accurately.

Concept / Approach:
The idiom "at the drop of a hat" means to do something immediately, without any delay or hesitation, often without much thought or preparation. It suggests that the slightest signal or excuse is enough to start the action. Therefore, the correct option should mention immediacy and absence of hesitation. The other options describe completely different ideas and do not match this sense.

Step-by-Step Solution:
Step 1: Recall the standard meaning of "at the drop of a hat": to do something right away, instantly, or without hesitation. Step 2: Examine option (a) "Without any hesitation; instantly." This matches exactly with the known meaning of the idiom. Step 3: Examine option (b) "Show outward respect to someone you hate." This deals with hypocrisy in behaviour and is unrelated to the idea of immediacy. Step 4: Examine option (c) "Forcibly let go of something which is very personal to you." This describes sacrifice or detachment, not quick action. Step 5: Examine option (d) "A mistake which is of not much consequence." This refers to a minor error, which again has no connection with quickness or willingness to act. Step 6: Conclude that only option (a) correctly expresses the meaning of the idiom.

Verification / Alternative check:
Consider example sentences: "She is ready to travel at the drop of a hat" means she is willing to travel immediately as soon as she gets the chance. Another example: "He will help his friends at the drop of a hat" means he helps them without delay or second thought. In both cases, the sense of instant, willing action without hesitation is central, confirming option (a) as the correct explanation.

Why Other Options Are Wrong:

• Without any hesitation; instantly: This is the correct meaning of the idiom and not a wrong option.
• Show outward respect to someone you hate: Describes a different social behaviour (pretending respect) and does not involve quick action.
• Forcibly let go of something which is very personal to you: Suggests forced sacrifice, which is unrelated to the idiom.
• A mistake which is of not much consequence: Refers to a trivial error, not to fast or impulsive action.

Common Pitfalls:
Learners sometimes try to interpret idioms literally and may think "drop of a hat" refers to losing something or making a small error. Others may be misled by complex sounding explanations in the options. To avoid such errors, idioms must be learned as fixed expressions with specific meanings, often memorised through repeated exposure in reading and listening exercises.

Final Answer:
"At the drop of a hat" means without any hesitation; instantly.