In an 8-week course, a professor administered a test at the end of each week. Each of the eight tests was scored out of 4 marks, and a student could only receive a non-negative integer score. Two students, Ravi and Sumana, took the eight tests.
In the first test, Ravi and Sumana scored the same marks. From the second to eighth tests, Ravi scored the exact same non-zero marks. Sumana scored the same marks as Ravi from the fifth test onwards. Ravi’s total marks in the first three tests was the same as Sumana’s total marks in the first two tests. Also, Sumana’s marks in the first test, total marks of the first two tests, and total marks of the eight tests are in a geometric progression.
Which of the following can be true?
Show Hint
- Sumana scored 3 marks in the second test
- Sumana scored 4 marks in the eighth test
- Ravi scored 4 marks in the third test
- Sumana scored 2 marks in the first test
- Ravi scored 0 marks in the fifth test
The Correct Option is D
Solution and Explanation
- Ravi and Sumana achieved identical scores on tests two through eight.
- Sumana's scores from test five onwards match Ravi's scores.
- The sum of Ravi's scores for the first three tests equals Sumana's total score for the first two tests.
- Sumana's first test score, her total score for the first two tests, and her total score for all eight tests form a geometric progression.
Step 2: Identify the correct condition.
The condition (D), stating "Sumana scored 2 marks in the first test," aligns with the geometric progression and satisfies all provided criteria.
Final Answer:
\[\boxed{\text{(D) Sumana scored 2 marks in the first test}}\]
If Ravi scored 4 marks in the first test, how many marks did Sumana score in the third test?
Show Hint
- 4
- 3
- 1
- 0
- 2
The Correct Option is B
Solution and Explanation
Ravi and Sumana achieved identical scores on tests commencing from the second test. This implies Sumana obtained 4 marks in both the first and second tests.
Step 2: Geometric Progression Application
Sumana's test scores follow a geometric progression, with the initial term being 4 marks. Applying this progression, Sumana's score on the third test is 3 marks.
Final Answer: \[ \boxed{\text{(B) 3}} \]
If Ravi scored 1 mark in the second test, what is the maximum possible value of Sumana’s total marks in all the eight tests together?
Show Hint
- 10
- 9
- 12
- Cannot be uniquely determined from the given information
- 8
The Correct Option is D
Solution and Explanation
The total marks for Sumana and Ravi across eight tests constitute a geometric progression, and specific constraints apply to their scores on different tests.
Step 2: Analyze the maximum total marks.
Due to the presence of undefined values and the geometric progression, Sumana's scores across all tests are not fully specified, precluding a unique determination of her total marks.
Final Answer:
\[ \boxed{\text{(D) Cannot be uniquely determined from the given information}} \]
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