• The given sentence is an example of third conditional sentences.

• Third conditional sentences are used to explain that present circumstances would be different if something different had happened in the past.

• When using the third conditional, we use the past perfect (i.e., had + past participle) in the if-clause.

• The modal auxiliary (would, could, should, etc.) + have + past participle in the main clause expresses the theoretical situation that could have happened.

• Thus, ‘would have gone’ is the most suited one.

The correct sentence should be :- She would have gone to your birthday party if she had not been sick.

Hence, option C is the correct answer.

The underlined part is grammatically incorrect.

• We can use "it" as an ‘empty’ subject or as an ‘empty’ object. It is ‘empty’ because it doesn’t refer to anything in particular.
  
  
• With "it", we will use the singular verb "has" (and not "have").
  
  
• Also, we will use V3 after "has + been".
  
  
• Thus, "known" will be used in the underlined part. 

The underlined part is grammatically incorrect.

• The modal is generally followed by 'a base form of the verb'.
  
  
• But when it is used in a passive structure, it follows the structure - "be + V3".
  
  
• The given sentence is in the passive voice as the focus is on Chandra's good gestures. 
  
  
• Thus, replace the underlined part with "will be highly appreciated".

Hence, option C is the correct answer.

The underlined part is grammatically incorrect.

• The sentence is in indirect speech.
  
  
• Hence, the verb should be after the subject.
  
  
• The phrase should be replaced with ‘my pen was’ to form a grammatically correct sentence.

Hence, option A is the correct answer.

• The Past perfect tense here will be wrong as there is no point of the past here before which the event remembrance of the cellphone occurred.
  

• I was just about to leave is just an past expression.
  

• Thus, 'I remembered' is the most suited one.

The underlined part is grammatically correct and needs no improvement.

Hence, option C is the correct answer.

• First of all, "Born with" is not followed by a word that refers to a person.
  
  
• To be born with something means innate, inborn, congenital, hereditary. It describes qualities, characteristics, or possessions acquired before or at the time of birth.
  
  For example:
  Jenny was born with a small hole in her heart.
  Merry was born with the heavenly voice.
  
  
• So, we should use "born to" or "born of" that means to be born in a particular situation, type of family, or particular individual.
  
  For example:
  One-third of all children are born into single-parent families.

• ‘Older’ refers to persons as well as things and is followed by ‘than’.
  
  
• ‘Elder’ is used for persons of the same family.
  
  
• Thus, Elder should be used here to form a grammatically sound sentence.
  
  

• Then, the verb "becomes" is also incorrectly used here.
  
  
• Since the sentence is talking about a past event we must use a past verb here.
  
  
• Thus, the correct verb to use here is "Became".
  
  

• The correct sentence will be: Two sons were born to them; the elder, Karl, became eminent as a historian.

The underlined part is grammatically correct and needs no improvement. 

Hence, option D is the correct answer. 

The underlined part is grammatically incorrect.

• The word “yesterday” indicates that the past form (V2 verb) should be used.
  
  
• “Gone” in the underlined part is past participle which is incorrect.
  
  
• It should be replaced with “went”. 

The underlined part is grammatically incorrect. 

• Subjective pronouns tell us who or what the sentence is about.
  e.g. She lost weight by cutting out junk food.
  
• Objective pronouns receive the action in the sentence.
  e.g. I give them cookies every week.
  
• Here, the underlined part is the subject of the second clause; so the subjective pronoun “I” should be used in place of “me”. 

Thus, the correct sentence will be: I think you and I owe him an explanation.
Hence, option A is the correct answer.