In a certain board examination, students were to appear for examination in five subjects: English, Hindi, Mathematics, Science and Social Science. Due to a certain emergency situation, a few of the examinations could not be conducted for some students. Hence, some students missed one examination and some others missed two examinations. Nobody missed more than two examinations. 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.
Question: 1
Who among the following did not appear for the Mathematics examination?
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.
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Question: 2
Which students did not appear for the English examination?
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.
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Question: 3
What BEST can be concluded about the students who did not appear for the Hindi examination?
Based on the condition, we know that exactly two candidates missed the examinations for English, Hindi, Science, and Social Science. For English, these individuals are Esha and Foni. For Hindi, the individuals are Alva and Deep. For Science, we know one of the individuals is either Alva or Deep. Since Carl, Alva, and Deep cannot be part of the group that missed the Science or Social Science exams, we carefully scrutinize the rest of the group which includes Bithi, Esha, and Foni.
We observe that Bithi has identical scores in both Science and Social Science exams. Assuming she missed these exams, let's verify if this was actually the case:
For Bithi: Best 2 out of 3 scores - 90 (English), 80 (Hindi)
Average = \(\frac{170}{2}\)=85, which matches the given value.
Therefore, Bithi likely missed her Science and Social Science examinations.
Additionally, we notice that Foni has identical scores in English and Social Science. Considering the best 2 out of 3 scores, the average score for both subjects is 83. Hence, we can conclude that Foni missed her Social Science examination as well.
Thus, 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 correctly determine the missed subjects for four of them (excluding Alva and Deep):
Mathematics: Carl
English: Esha & Foni
Hindi: Alva & Deep
Science: Bithi & either Alva or Deep
Social Science: Foni & Bithi
Hence the correct oprion is (A): Alva and Deep.
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Question: 4
What BEST can be concluded about the students who missed the Science examination?
Based on the condition, we know that exactly two candidates missed the examinations for English, Hindi, Science, and Social Science. For English, these individuals are Esha and Foni. For Hindi, the individuals are Alva and Deep. For Science, we know one of the individuals is either Alva or Deep. Since Carl, Alva, and Deep cannot be part of the group that missed the Science or Social Science exams, we carefully scrutinize the rest of the group which includes Bithi, Esha, and Foni.
We observe that Bithi has identical scores in both Science and Social Science exams. Assuming she missed these exams, let's verify if this was actually the case:
For Bithi: Best 2 out of 3 scores - 90 (English), 80 (Hindi)
Average = \(\frac{170}{2}\)=85, which matches the given value.
Therefore, Bithi likely missed her Science and Social Science examinations.
Additionally, we notice that Foni has identical scores in English and Social Science. Considering the best 2 out of 3 scores, the average score for both subjects is 83. Hence, we can conclude that Foni missed her Social Science examination as well.
Thus, 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 correctly determine the missed subjects for four of them (excluding Alva and Deep):
Mathematics: Carl
English: Esha & Foni
Hindi: Alva & Deep
Science: Bithi & either Alva or Deep
Social Science: Foni & Bithi
Hence, the correct oprion is (D): Bithi and one out of Alva and Deep.
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Question: 5
How many out of these six students missed exactly one examination?
Based on the condition, we know that exactly two candidates missed the examinations for English, Hindi, Science, and Social Science. For English, these individuals are Esha and Foni. For Hindi, the individuals are Alva and Deep. For Science, we know one of the individuals is either Alva or Deep. Since Carl, Alva, and Deep cannot be part of the group that missed the Science or Social Science exams, we carefully scrutinize the rest of the group which includes Bithi, Esha, and Foni.
We observe that Bithi has identical scores in both Science and Social Science exams. Assuming she missed these exams, let's verify if this was actually the case:
For Bithi: Best 2 out of 3 scores - 90 (English), 80 (Hindi)
Average = \(\frac{170}{2}\)=85, which matches the given value.
Therefore, Bithi likely missed her Science and Social Science examinations.
Additionally, we notice that Foni has identical scores in English and Social Science. Considering the best 2 out of 3 scores, the average score for both subjects is 83. Hence, we can conclude that Foni missed her Social Science examination as well.
Thus, 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 correctly determine the missed subjects for four of them (excluding Alva and Deep):
Mathematics: Carl
English: Esha & Foni
Hindi: Alva & Deep
Science: Bithi & either Alva or Deep
Social Science: Foni & Bithi
Hence, the correct answer is : 3-( Carl (Mathematics); Esha (English) and one out of Alva and Deep (Hindi))
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Question: 6
For how many students can we be definite about which examinations they missed?
Based on the condition, we know that exactly two candidates missed the examinations for English, Hindi, Science, and Social Science. For English, these individuals are Esha and Foni. For Hindi, the individuals are Alva and Deep. For Science, we know one of the individuals is either Alva or Deep. Since Carl, Alva, and Deep cannot be part of the group that missed the Science or Social Science exams, we carefully scrutinize the rest of the group which includes Bithi, Esha, and Foni.
We observe that Bithi has identical scores in both Science and Social Science exams. Assuming she missed these exams, let's verify if this was actually the case:
For Bithi: Best 2 out of 3 scores - 90 (English), 80 (Hindi)
Average = \(\frac{170}{2}\)=85, which matches the given value.
Therefore, Bithi likely missed her Science and Social Science examinations.
Additionally, we notice that Foni has identical scores in English and Social Science. Considering the best 2 out of 3 scores, the average score for both subjects is 83. Hence, we can conclude that Foni missed her Social Science examination as well.
Thus, 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 correctly determine the missed subjects for four of them (excluding Alva and Deep):
Mathematics: Carl
English: Esha & Foni
Hindi: Alva & Deep
Science: Bithi & either Alva or Deep
Social Science: Foni & Bithi
Hence, Except for Alva and Deep, we can conclusively comment of the missed subjects of the rest four.
