Spotting Errors

 Replace 'them' with 'whom'. It is a subordinate clause and use of 'them' would make it an independent clause, making the sentence incorrect.

After 'refused the request that', the subjunctive mood is required for the subordinate clause. 'Be waived' is the correct subjunctive form, not 'shall be waived' (future indicative) or 'should be waived' (suggestion). The subjunctive expresses necessity or recommendation after verbs like request, insist, propose.

Comparisons must be parallel. 'Sudha's hair' must be compared to 'Richa's hair' (possessive), not just 'Richa' (person). 'Hair is lengthier' is correct because hair (as a collective noun) takes a singular verb when referring to the overall length or mass of hair.

The parallel structure requires 'to pursue' (infinitive) to match 'to make' (infinitive). 'Matured' is incorrect - 'mature' is the adjective form. 'Sufficiently mature' is correct word order (adverb + adjective). 'Pursue his goals' is active and direct.

The sentence structure requires 'being' to match the earlier phrase 'just being a guard'. 'Apart from being the perceived perpetrator' maintains parallelism. 'Perceived perpetrator' is a noun phrase that needs 'being' to function as a complement to 'apart from'.

'Scarcely' is already negative, so adding 'nothing' creates a double negative. 'Scarcely anything' means 'almost nothing', which is the intended meaning. 'Next to nothing' is also correct but 'scarcely anything' is the most direct correction.

The subject of the second clause should refer back to 'mathematical theories' (they), not 'our comprehension'. The theories may have enhanced comprehension - comprehension doesn't enhance itself. Active voice 'they may have enhanced' is clearer than passive.

'Not hardly' is a double negative. 'Hardly' is already negative, so adding 'not' is incorrect. When a negative adverb like 'hardly' starts a sentence for emphasis, subject-auxiliary inversion is required: 'Hardly did anyone protest'. 'Hardly anyone protested' is grammatically correct but lacks the formal emphasis the inverted structure provides.

After 'essential that', the subjunctive mood is required. 'Be filed' is the correct subjunctive form, not 'are filed' (indicative). The subjunctive expresses necessity or urgency after adjectives like essential, necessary, important, critical.

After 'recommended', the subjunctive mood 'should study' is correct and natural. 'Study' (bare subjunctive) is also correct but less common. 'Studies' (indicative) and 'studied' (past) are wrong. 'Should study' clearly expresses the recommendation.

The error is in part (B) "we did helped the poor" - this is grammatically incorrect. After the auxiliary verb "did" (in emphatic or past tense constructions), we use the base form of the main verb, not the past tense form. The correct phrase should be "we did help the poor" or simply "we helped the poor." The "did + base verb" construction is used for emphasis.

The error is in part (A) "Up to the time the last vote was recorded." The phrase "Up to" is incorrect in this temporal context. The correct expression should be "Until the time the last vote was recorded" or "By the time the last vote was recorded." "Up to" typically indicates approach to a point but not including it, which doesn't fit the context of waiting for a final outcome.

The error is in part (C) "he needs to work harder" because it creates a logical contradiction. The sentence begins with "When his overall performance has been good," which indicates satisfactory performance. However, the conclusion "he needs to work harder" suggests improvement is needed. This creates a logical inconsistency - if performance is already good, why work harder? The sentence structure sets up a contrast that isn't logically coherent.

The error is in part (A) "Can I join you" - this is an inappropriate modal usage in the given context. The phrase "and contribute my mite" suggests a serious, professional context (contributing to a delayed project). In such contexts, "May I join you" or "Could I join you" would be more appropriate and polite. "Can I" is more casual and emphasizes capability rather than seeking permission, which doesn't fit the formal tone of the rest of the sentence.