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
Per Condition II, the student who did not take the Mathematics exam did not miss any other exams. Consequently, their Mathematics score should be the average of their best 3 of the 4 exam scores. This allows us to determine the Math score as the average of the remaining scores. Deep and Esha are eliminated as their Mathematics score exceeds their other exam scores.
For Alva:
- Best 3 of 4: 80 (English), 75 (Hindi), 75 (Science)
- Average = \(\frac{230}{3}\)≈76.67; does not equal 70
For Carl:
- Best 3 of 4: 80 (Hindi), 90 (Social Science), 100 (Science)
- Average = \(\frac{270}{3}\)=90; matches the given value
Therefore, Carl is the most likely candidate to have missed his Mathematics examination.
For Foni:
- Best 3 of 4: 83 (English), 83 (Social Science), 88 (Science)
- Average = \(\frac{254}{3}\)≈84.67; does not equal 78
Only Carl's scores align with missing the Mathematics examination.
Thus, 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
Four students, Carl (Mathematics), Esha (English), and either Alva or Deep (Hindi), missed only one examination. Consequently, for four of the six students (excluding Alva and Deep), the missed subjects can be definitively identified:
- Mathematics: Carl
- English: Esha and Foni
- Hindi: Alva or Deep
- Science: Bithi and either Alva or Deep
- Social Science: Foni and Bithi
Therefore, Option D, Esha and Foni, is the correct response.
The confirmed 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
Exactly two students missed each of the following subjects: English, Hindi, Science, and Social Science.
- English: Missed by Esha and Foni
- Hindi: Missed by Alva and Deep
- Science: Missed by one of Alva or Deep, and one other student.
- Social Science: To be determined.
Carl, Alva, and Deep did not miss Science or Social Science. Therefore, we consider Bithi, Esha, and Foni.
Bithi's Evaluation
Bithi scored identically in Science and Social Science. If she missed both, her top two scores would be English (90) and Hindi (80). The average score is \( \frac{90 + 80}{2} = 85 \), which matches the given information. Bithi likely missed Science and Social Science.
Foni's Evaluation
Foni scored 83 in both English and Social Science. If she missed Social Science, her best two scores would be English (83) and another subject (83). The average is \( \frac{83 + 83}{2} = 83 \), which is consistent. 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
Alva and Deep are the only students who could not be definitively assigned to a unique missed subject. They missed only Hindi and possibly Science.
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
Under the specified conditions, precisely two students did not take each subject in English, Hindi, Science, and Social Science.
- English: Esha and Foni did not take this subject.
- Hindi: Alva and Deep did not take this subject.
- Science: One of Alva or Deep, and one additional student.
- Social Science: To be determined.
Given that Carl, Alva, and Deep did not take Science or Social Science, the students remaining for consideration in these subjects are: Bithi, Esha, and Foni.
Evaluating Bithi
Bithi achieved identical scores in Science and Social Science. Assuming she did not take both:
- Her highest two scores: English = 90, Hindi = 80
- Average: \( \frac{90 + 80}{2} = 85 \)
This average aligns with her reported average, indicating she likely did not take Science and Social Science.
Evaluating Foni
Foni obtained the same score in English and Social Science (83). Assuming she did not take Social Science:
- Highest two scores: English = 83, another subject = 83
- Average: \( \frac{83 + 83}{2} = 83 \)
Consequently, she likely did not take Social Science.
Summary of Subjects Not Taken
- Mathematics: Carl
- English: Esha, Foni
- Hindi: Alva, Deep
- Science: Bithi and either Alva or Deep
- Social Science: Bithi and Foni
Conclusion
Based on the analysis, the subjects not taken by the following students are confirmed: Bithi, Carl, Esha, and Foni.
The sole remaining ambiguity concerns Alva and Deep, specifically regarding which of them did not take 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
Exactly two candidates missed examinations in English, Hindi, Science, and Social Science.
- English: Missed by Esha and Foni
- Hindi: Missed by Alva and Deep
- Science: Missed by Bithi and either Alva or Deep
- Social Science: Missed by Foni and Bithi
- Mathematics: Missed by Carl
Score repetition confirms Bithi missed Science and Social Science. Her top 2 of 3 scores:
- English: 90
- Hindi: 80
- Average: \( \frac{90 + 80}{2} = 85 \)
This average confirms Bithi attempted English and Hindi.
Foni's identical scores in English and Social Science, with an average of 83, indicate she missed Social Science:
\( \frac{(83 + 83)}{2} = 83 \)
Confirmed missed subjects:
- Carl: Mathematics
- Esha: English
- Foni: Social Science
- Bithi: Science & Social Science
Ambiguous: One of Alva or Deep missed Hindi, and the other missed Science.
Final Answer
The students with clearly determined 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
It is established that precisely two candidates failed to attend the examinations for each of the following subjects: English, Hindi, Science, and Social Science.
- English: Failed by Esha and Foni
- Hindi: Failed by Alva and Deep
- Science: One candidate from the pair Alva/Deep, plus one other individual (currently unidentified)
- Social Science: Determination pending
Given that Carl, Alva, and Deep are confirmed to have participated in the Science and Social Science examinations, the remaining candidates eligible for having missed these subjects are restricted to: Bithi, Esha, and Foni.
Bithi's Performance Analysis
Bithi's scores in Science and Social Science are identical. Assuming she missed these two subjects, her best two out of three scores would be:
- English = 90
- Hindi = 80
- Average = \( \frac{90 + 80}{2} = 85 \) — This aligns with her reported average.
Conclusion: Bithi is the likely candidate who missed Science and Social Science.
Foni's Performance Analysis
Foni's scores in English and Social Science are identical. Assuming she missed Social Science:
- English = 83
- Another subject = 83
- Average = \( \frac{83 + 83}{2} = 83 \) — This matches her reported average.
Conclusion: Foni missed Social Science.
Verified Missed Examinations
- Carl — Missed Mathematics
- Esha — Missed English
- Foni — Missed Social Science
- Bithi — Missed Science and Social Science
Unresolved Information:
The specific distribution of missed examinations between Alva and Deep cannot be definitively determined due to the following:
- They both missed Hindi, and one of them missed Science.
- Insufficient data exists to assign the Science examination to either Alva or Deep specifically.
Final Determination:
The missed examinations for 4 students can be conclusively identified: Carl, Esha, Foni, and Bithi.
Result: 4
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