The board adopted the following policy for awarding marks to students. If a student appeared in all five examinations, then the marks awarded in each of the examinations were on the basis of the scores obtained by them in those examinations.
If a student missed only one examination, then the marks awarded in that examination was the average of the best three among the four scores in the examinations they appeared for.
If a student missed two examinations, then the marks awarded in each of these examinations was the average of the best two among the three scores in the examinations they appeared for.
The marks obtained by six students in the examination are given in the table below. Each of them missed either one or two examinations.
| English | Hindi | Mathematics | Science | Social Science | |
|---|---|---|---|---|---|
| Alva | 80 | 75 | 70 | 75 | 60 |
| Bithi | 90 | 80 | 55 | 85 | 85 |
| Carl | 75 | 80 | 90 | 100 | 90 |
| Deep | 70 | 90 | 100 | 90 | 80 |
| Esha | 80 | 85 | 95 | 60 | 55 |
| Foni | 83 | 72 | 78 | 88 | 83 |
The following facts are also known.
I. Four of these students appeared in each of the English, Hindi, Science, and Social Science examinations. II. The student who missed the Mathematics examination did not miss any other examination.
III. One of the students who missed the Hindi examination did not miss any other examination. The other student who missed the Hindi examination also missed the Science examination.
Who among the following did not appear for the Mathematics examination?
- Foni
- Alva
- Esha
- Carl
The Correct Option is D
Solution and Explanation
Based on Condition II, we understand that the student who missed the Mathematics examination did not miss any other examination. This implies that the Math score should be the average of the best 3 out of the 4 exam scores obtained by this candidate. Using this inference, we can identify the Math score that can be represented as an average of the rest of the scores. We can immediately eliminate Deep and Esha as potential candidates since their Mathematics score is greater than their other exam scores.
For Alva:
- Best 3 out of 4: 80 (English), 75 (Hindi), 75 (Science)
- Average = \(\frac{230}{3}\)≈76.67 which does not equal 70
For Carl:
- Best 3 out of 4: 80 (Hindi), 90 (Social Science), 100 (Science)
- Average = \(\frac{270}{3}\)=90, which matches the given value
Therefore, Carl most likely missed his Mathematics examination.
For Foni:
- Best 3 out of 4: 83 (English), 83 (Social Science), 88 (Science)
- Average = \(\frac{254}{3}\)≈84.67 which does not equal 78
Hence, we observe that only Carl has missed his Mathematics examination.
Therefore, the correct Option is (D): Carl.
Which students did not appear for the English examination?
- Alva and Bithi
- Carl and Deep
- Esha and Foni
- Cannot be determined
The Correct Option is C
Solution and Explanation
The students who missed just one exam were: Carl (Mathematics); Esha (English); and one of Alva or Deep (Hindi). Therefore, out of the six students, we can accurately determine the missed subjects for four of them (excluding Alva and Deep):
- Mathematics: Carl
- English: Esha and Foni
- Hindi: Alva or Deep
- Science: Bithi and either Alva or Deep
- Social Science: Foni and Bithi
Hence, the correct answer to this question is Option D: Esha and Foni.
So correct option is (C): Esha and Foni.
What BEST can be concluded about the students who did not appear for the Hindi examination?
- Alva and Deep
- Deep and Esha
- Alva and Esha
- Two among Alva, Deep and Esha
The Correct Option is A
Solution and Explanation
According to the problem statement, exactly two students missed the examination in each of the following subjects: English, Hindi, Science, and Social Science.
- English: Missed by Esha and Foni
- Hindi: Missed by Alva and Deep
- Science: One of Alva or Deep plus one other
- Social Science: To be determined
It is given that Carl, Alva, and Deep cannot be among those who missed the Science or Social Science exams. Therefore, we must examine the remaining students: Bithi, Esha, and Foni.
Evaluating Bithi
Bithi has identical scores in both Science and Social Science. Let’s assume she missed both. Then, her best two subjects would be:
- English: 90
- Hindi: 80
Average score = \( \frac{90 + 80}{2} = 85 \), which matches the given information. Hence, it's likely that Bithi missed Science and Social Science.
Evaluating Foni
Foni has the same score in English and Social Science, both 83. Assuming she missed Social Science, her best two subjects would be:
- English: 83
- Another subject: 83
Average = \( \frac{83 + 83}{2} = 83 \), again consistent. So, Foni likely missed Social Science.
Summary of Missed Subjects
- Mathematics: Carl
- English: Esha and Foni
- Hindi: Alva and Deep
- Science: Bithi and one of Alva or Deep
- Social Science: Bithi and Foni
Conclusion
From this analysis, the two people who could not be definitively assigned to any unique subject missed — and who only missed Hindi and possibly Science — are Alva and Deep.
Correct Option: (A) Alva and Deep
What BEST can be concluded about the students who missed the Science examination?
- Alva and Bithi
- Deep and Bithi
- Alva and Deep
- Bithi and one out of Alva and Deep
The Correct Option is D
Solution and Explanation
Based on the given conditions, exactly two students missed each subject in English, Hindi, Science, and Social Science.
- English: Missed by Esha and Foni
- Hindi: Missed by Alva and Deep
- Science: One of Alva or Deep, and one other person
- Social Science: To be identified
Since Carl, Alva, and Deep did not miss Science or Social Science, the remaining students to consider for these subjects are: Bithi, Esha, and Foni.
Evaluating Bithi
Bithi has identical scores in Science and Social Science. Assuming she missed both:
- Her best two scores: English = 90, Hindi = 80
- Average: \( \frac{90 + 80}{2} = 85 \)
This matches her given average, so she likely missed Science and Social Science.
Evaluating Foni
Foni has the same score in English and Social Science (83). Assuming she missed Social Science:
- Best two scores: English = 83, another subject = 83
- Average: \( \frac{83 + 83}{2} = 83 \)
Therefore, she likely missed Social Science.
Summary of Missed Subjects
- Mathematics: Carl
- English: Esha, Foni
- Hindi: Alva, Deep
- Science: Bithi and one of Alva or Deep
- Social Science: Bithi and Foni
Conclusion
From the analysis, we can confirm the missed subjects for the following students: Bithi, Carl, Esha, and Foni.
The only uncertainty remains with Alva and Deep, regarding who among them missed Science.
Correct Option: (D) Bithi and one out of Alva and Deep
How many out of these six students missed exactly one examination?
Solution and Explanation
Based on the condition, we know that exactly two candidates missed the examinations for English, Hindi, Science, and Social Science.
- English: Missed by Esha and Foni
- Hindi: Missed by Alva and Deep
- Science: Missed by Bithi and one of Alva or Deep
- Social Science: Missed by Foni and Bithi
- Mathematics: Missed by Carl
We confirm Bithi missed Science and Social Science based on score repetition. Her best 2 out of 3 scores:
- English: 90
- Hindi: 80
- Average: \( \frac{90 + 80}{2} = 85 \)
This average matches the provided value, confirming the subjects she attempted were English and Hindi.
Similarly, Foni's scores suggest she missed Social Science, having identical scores in English and Social Science, and her average was also 83, which fits:
\( \frac{(83 + 83)}{2} = 83 \)
Hence, confirmed missed subjects:
- Carl: Mathematics
- Esha: English
- Foni: Social Science
- Bithi: Science & Social Science
Ambiguous: One of Alva or Deep missed Hindi, the other missed Science.
Final Answer
The students for whom we can clearly determine the missed subjects are:
- Carl — Mathematics
- Esha — English
- One of Alva or Deep — Hindi
Correct option: 3-(Carl (Mathematics); Esha (English); and one out of Alva and Deep (Hindi))
For how many students can we be definite about which examinations they missed?
Solution and Explanation
Based on the condition, we know that exactly two candidates missed the examinations for each of the following subjects: English, Hindi, Science, and Social Science.
- English: Missed by Esha and Foni
- Hindi: Missed by Alva and Deep
- Science: One of Alva or Deep, and someone else (to be identified)
- Social Science: To be deduced
Since Carl, Alva, and Deep are known to have attempted Science and Social Science, the remaining possibilities for missed Science and Social Science must be from: Bithi, Esha, and Foni.
Analysis of Bithi
Bithi has identical scores in both Science and Social Science. Let’s assume she missed those two subjects. Her best 2 of 3 scores would be:
- English = 90
- Hindi = 80
- Average = \( \frac{90 + 80}{2} = 85 \) — which matches her reported average.
Conclusion: Bithi likely missed Science and Social Science.
Analysis of Foni
Foni has identical scores in English and Social Science. Assuming she missed Social Science:
- English = 83
- Another subject = 83
- Average = \( \frac{83 + 83}{2} = 83 \) — matches her reported average.
Conclusion: Foni missed Social Science.
Confirmed Missed Subjects
- Carl — Missed Mathematics
- Esha — Missed English
- Foni — Missed Social Science
- Bithi — Missed Science and Social Science
Ambiguity:
We cannot definitively determine the distribution of missed subjects between Alva and Deep because:
- They missed Hindi and one of them missed Science
- But we do not have enough information to assign which individual missed which exam
Final Answer:
We can conclusively determine the missed subjects for 4 students: Carl, Esha, Foni, and Bithi.
Correct Answer: 4
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