A part of the sentence is incorrect and needs improvement. The sentence is: I tried to cane her, but of no purpose. Choose the option that best improves the underlined expression so that the sentence becomes grammatically correct and natural.

Introduction / Context:
This sentence improvement question focuses on choosing the correct preposition to complete a fixed English expression. The sentence describes an action that did not achieve any result. In such situations, English uses idiomatic phrases like to no purpose or for no purpose. Here, the examiner expects the specific and widely used expression to no purpose. The learner must recognise which preposition fits naturally after but and before no purpose to make the sentence correct.

Given Data / Assumptions:

• Sentence: I tried to cane her, but of no purpose.
• Options: for, with, to, No improvement.
• The underlined segment in the original question is the phrase of no purpose.
• The task is to replace the preposition, if needed, to form an accepted English expression.

Concept / Approach:
The key concept is recognition of the idiomatic phrase to no purpose, which means without any useful result or in vain. When the speaker says that an effort was made but had no effect, they may say it was to no purpose. The phrase of no purpose is much less common and sounds unnatural in this structure after but. The approach is to recall the correct idiom and test each option by inserting it into the sentence to see which one yields a standard, fluent expression.

Step-by-Step Solution:
Step 1: Read the original sentence: I tried to cane her, but of no purpose. Step 2: Replace of with for: I tried to cane her, but for no purpose. This is understandable but still not the most natural idiom after but. Step 3: Replace of with with: I tried to cane her, but with no purpose. This suggests a different structure and is not the usual phrasing in this pattern. Step 4: Replace of with to: I tried to cane her, but to no purpose. This is a recognised and idiomatic way of stating that the attempt failed to produce results. Step 5: Consider No improvement, which would leave the sentence as but of no purpose, a form that sounds awkward and non standard. Step 6: Conclude that to is the correct choice and that the improved phrase is to no purpose.

Verification / Alternative check:
To verify, recall similar expressions: all his efforts were to no purpose, their complaints were to no purpose, and she cried all night to no purpose. Grammar and usage references show that to no purpose is an established idiom equivalent to in vain or without result. Although one can say for no purpose in some contexts, the structure with but and the rhythm of the sentence here strongly favour the idiomatic form to no purpose. The phrase of no purpose is not commonly used in this sense and sounds unnatural, confirming that it needs to be changed.

Why Other Options Are Wrong:
for: The phrase for no purpose is understandable but less idiomatic in this exact structure, and the exam usually expects the standard expression to no purpose.
with: With no purpose would change the meaning to acting without having a purpose, which is not the idea here; the sentence emphasises lack of result, not lack of intention.
No improvement: Keeping of no purpose would leave the sentence sounding awkward and not in line with common idiomatic English usage.

Common Pitfalls:
Learners often focus only on literal meaning and ignore the fixed nature of idiomatic expressions. Because several prepositions can suggest a similar idea, it may be tempting to choose any that seem roughly acceptable. However, sentence improvement questions usually test the most natural collocation. Regular reading of well written English helps build a sense of which preposition appears in specific fixed phrases, such as to no purpose, in vain, and to no avail.

Final Answer:
The correct improvement is to replace of with to, giving the idiomatic phrase to no purpose.