On which date is the birthday of Manohar? Consider the following statements: I. The father of Manohar was born on 27 May 1948. II. Manohar is 25 years younger than his mother.

Introduction / Context:
This data sufficiency question is about birthdays and age differences within a family. The aim is to identify the exact birthday of Manohar. Two statements give information about the father birth date and Manohar age difference with his mother. The main issue is whether these pieces of information are enough to fix Manohar date of birth without further assumptions.

Given Data / Assumptions:

• Statement I: The father of Manohar was born on 27 May 1948.
• Statement II: Manohar is 25 years younger than his mother.
• No explicit relationship is given between the ages of Manohar parents.
• No information is provided about the mother date of birth or any calendar alignment between parents and child beyond the age gap in statement II.

Concept / Approach:
To know a person date of birth, we need both the year and the day and month. Knowing the birth date of one parent gives complete information for that parent only. Knowing an age difference between Manohar and his mother can at best give the difference between their years of birth, and even then only if one birth year is known. The question is whether these statements together or individually allow us to derive Manohar full birthday.

Step-by-Step Solution:
Step 1: Consider statement I alone. It tells us that Manohar father was born on 27 May 1948. However, there is no information about the age difference between the father and Manohar or between the father and the mother. Without such data, we cannot infer any specific year or date for Manohar birth. Step 2: Therefore, statement I alone is not sufficient to answer the question. Step 3: Consider statement II alone. It states that Manohar is 25 years younger than his mother. This information only gives the age gap between Manohar and his mother. The mother could have been born in many possible years and on many possible dates. Without the mother birth date, Manohar date of birth cannot be known. Step 4: Hence, statement II alone is also not sufficient. Step 5: Now consider both statements together. Statement I fixes the father date of birth, but no relation is given between the ages of the father and the mother. The mother could be younger than the father, older than the father, or even the same age, as long as no explicit contradiction arises. Step 6: Because the mother date of birth is completely free, the pair of statements allows many possible combinations of mother and child birthdays that satisfy Manohar being 25 years younger than his mother. For example, if we choose one possible birth year for the mother, we get a corresponding year for Manohar, but different choices of the mother birth date still produce different possible dates for Manohar. Step 7: Since there are multiple possible birth dates for Manohar consistent with both statements, the data are not sufficient to determine his birthday uniquely.

Verification / Alternative check:
To verify, suppose the mother was born on 1 January 1960. Then Manohar might be born on 1 January 1985. If instead the mother was born on 10 October 1965, then Manohar could be born on 10 October 1990. Both scenarios fit statement II and do not contradict statement I, because statement I concerns only the father. The fact that multiple valid birth dates for Manohar exist proves that the data are insufficient.

Why Other Options Are Wrong:
Options A and B claim sufficiency of individual statements, which is impossible because each statement deals with a different person without linking them numerically. Option C states that either statement alone is enough, which is clearly not correct. Option D suggests that both statements together are necessary and sufficient, but as seen, even the combination leaves many possibilities open. Only option E correctly states that even both statements together are not sufficient to answer the question.

Common Pitfalls:
A frequent error is to assume that parents are of a standard age when children are born, such as a fixed age difference between parents and children. Such assumptions are not given and cannot be introduced in data sufficiency problems. Another pitfall is to treat the known birth date of the father as if it constrains the mother age, which the problem does not state. Avoiding extra assumptions is crucial in this type of reasoning.

Final Answer:
The data even in both statements together are not sufficient to answer the question, so the correct option is E.