A part of the sentence is incorrect and needs improvement. The sentence is: Pankaj could not have seen us or he will have waved. Choose the option that best improves the underlined verb phrase so that the sentence becomes grammatically correct and meaningful.

Introduction / Context:
This sentence improvement question tests the correct use of conditional and modal verb forms in English. The sentence describes a situation in the past and expresses a logical conclusion about what would have happened if a certain condition had been true. The structure uses could not have seen to refer to a past possibility and must match this with an appropriate result clause. The learner must select the option that produces a correct conditional form in the second part of the sentence.

Given Data / Assumptions:

• Sentence: Pankaj could not have seen us or he will have waved.
• Options: may, would have, should have, No improvement.
• The underlined phrase in the original question is he will have waved.
• We assume the speaker is reasoning about a past situation: if Pankaj had seen us, he would have waved.

Concept / Approach:
The main concept is the third conditional, which describes unreal or hypothetical situations in the past and their imagined results. The typical pattern is if plus past perfect in the condition clause and would have plus past participle in the result clause. Even when the if is not explicitly stated, the same tense logic applies. Here, could not have seen refers to a past possibility that did not occur. The corresponding result should use would have waved, not will have waved, which does not fit with past hypothetical reasoning.

Step-by-Step Solution:
Step 1: Rewrite the idea in a full conditional form: If Pankaj had seen us, he would have waved. Step 2: Notice that the original sentence uses could not have seen, which already refers to a past situation. Step 3: The result part he will have waved uses future perfect will have, which is inconsistent with the past hypothetical meaning. Step 4: Consider Option B, would have, to form he would have waved, which correctly matches the third conditional pattern. Step 5: Option A, may, would not complete the clause correctly and would leave the tense structure incomplete. Step 6: Option C, should have, expresses obligation or criticism rather than a simple logical result. Step 7: No improvement would keep the incorrect will have form linked to a past condition, so it cannot be correct.

Verification / Alternative check:
To verify, compare with other third conditional examples: he could not have forgotten the meeting, or he would have called, she cannot have seen the message, otherwise she would have replied. In each case, the imagined result of a past condition uses would have plus the past participle. The use of will have is more appropriate when predicting that something will already be complete by a future time, as in he will have arrived by now, which does not fit the structure of this sentence. Therefore, replacing will have with would have is necessary and correct.

Why Other Options Are Wrong:
may: On its own, may does not complete the verb phrase or match the required conditional structure; it cannot replace will have waved.
should have: This modal is used to express duty or regret, as in he should have waved, and does not simply state what would have happened under a condition.
No improvement: Leaving will have waved unchanged results in a tense mismatch between the past hypothetical condition and the future perfect form.

Common Pitfalls:
Learners often confuse will have and would have because they look similar. Another common mistake is ignoring the time reference and using future forms even when the context is clearly about the past. To avoid such errors, it is important to understand the patterns of conditional sentences and practise recognising when a clause refers to an unreal past situation requiring would have plus past participle in the result clause.

Final Answer:
The correct improvement is to use would have, giving the sentence: Pankaj could not have seen us, or he would have waved.