Question

Which of the following can be true?

• A. Sumana scored 3 marks in the second test
• B. Sumana scored 4 marks in the eighth test
• C. Ravi scored 4 marks in the third test
• D. Sumana scored 2 marks in the first test
• E. Ravi scored 0 marks in the fifth test
• A. 4
• B. 3
• C. 1
• D. 0
• J. 2
• A. 10
• B. 9
• C. 12
• D. Cannot be uniquely determined from the given information
• O. 8

Answer 1

The Correct Option is D

1. To solve this problem, we will decode the given data related to the scores of two students, Ravi and Sumana, in an 8-week course.
2. Each test is scored out of 4 marks, and only non-negative integers can be scored. Therefore, possible scores are: 0, 1, 2, 3, and 4.
3. Information Given:
   • Ravi and Sumana scored the same marks in the first test.
   • 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 were the same as Sumana's total marks in the first two tests.
   • 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.
4. Let us denote:
   • Score of Ravi in each test from second to eighth as \( r \).
   • Score of Ravi and Sumana in the first test as \( x \).
   • Sumana's score in the second, third, and fourth tests as \( y, z, \) and \( w \) respectively.
5. Analyze the geometric progression condition for Sumana:
   • Let Sumana's score in the first test be \( a = x \).
   • Total of the first two tests is \( a + y \).
   • Total of all eight tests is \( a + y + z + w + 4r \).
   • So, the geometric progression must satisfy: \( a, a + y, a + y + z + w + 4r \).
6. Given:
   • \( x + 2r = x + y \) (since Ravi's total in the first three tests equals Sumana's in the first two).
7. Testing feasible marks:
   • If \( r = 1 \), not possible as total exceeds maximum possible mark.
   • If \( r = 2 \), then \( y = 2 \).
   • If Sumana scored 2 in the first test, progression: \( 2, 4, 14 \). Fits GP with ratio 2.
8. Conclusion:
   • Given all valid tests and conditions, Sumana scored 2 marks in the first test.
9. Other options can be ruled out as they either violate the rules or fail the progression condition.

Answer 2

Step 1: Analyze the problem conditions.
- Ravi and Sumana scored the same marks from the second to eighth tests. - Sumana's marks from the fifth test onwards are the same as Ravi's marks. - The total marks of Ravi for the first three tests are the same as Sumana’s total marks in the first two tests. - Sumana’s marks in the first test, total marks of the first two tests, and total marks of the eight tests form a geometric progression.
Step 2: Find the true condition.
Based on the given information, the condition (D) that "Sumana scored 2 marks in the first test" is consistent with the progression and matches the conditions.
Final Answer: \[ \boxed{\text{(D) Sumana scored 2 marks in the first test}} \]

Answer 3

To solve this problem, we need to analyze the information given about the test scores of Ravi and Sumana over the eight weeks.

Step 1: Understand Ravi's Scores

• Ravi scored 4 marks in the first test.
• From the second to eighth tests, Ravi scored the exact same non-zero marks. This implies a consistent score for these tests.

Step 2: Linking Ravi and Sumana's Scores

• In the first test, Ravi and Sumana both scored the same marks. Therefore, Sumana scored 4 marks in the first test.
• Ravi’s total marks in the first three tests equaled Sumana’s total marks in the first two tests.

Let us denote Ravi's score in the second and third tests as x.

• Total score of Ravi for the first three tests = 4 + x + x = 4 + 2x
• Total score of Sumana for the first two tests = 4 + y, where y is Sumana's score in the second test.
• Thus, 4 + 2x = 4 + y \Rightarrow y = 2x.

Step 3: Analyzing Sumana's Scores

• From the fifth test onwards, Sumana started scoring the same marks as Ravi. Therefore, Sumana scored x in tests 5 through 8.

Sumana’s marks in the first test, the total of the first two tests, and the total of the eight tests form a geometric progression.

• First term (a) = 4 (Sumana’s first test score).
• Second term (ar) = 4 + 2x (Sumana’s first two tests score).
• Third term (ar^2) = Sumana's total for eight tests.

Sumana repeated scores of x from the fifth test to the eighth test, thus:

• For the eight tests: Total score = 4 + 2x + 4x = 4 + 6x.

Given that these are in a geometric progression:

• \frac{4 + 2x}{4} = \frac{4 + 6x}{4 + 2x} (Equation for geometric progression).

Solve for x:

\frac{4 + 2x}{4} = \frac{4 + 6x}{4 + 2x} implies:

• Cross-multiplying: (4 + 2x)^2 = 4 \cdot (4 + 6x)
• Expanding both sides: 16 + 16x + 4x^2 = 16 + 24x
• Rearranging: 4x^2 - 8x = 0
• Factoring: 4x(x - 2) = 0
• Thus, x = 0 or x = 2

Conclusion:

• x = 0 is not valid since scores must be non-zero.
• Therefore, x = 2.
• Hence, Sumana scored 3 marks in her third test as y = 2x = 4.

The correct answer is 3.

Answer 4

Step 1: Analyze the data.
Since Ravi and Sumana scored the same marks in the tests from the second test onwards, we know Sumana scored 4 marks in the first and second tests.
Step 2: Use the geometric progression rule.
Sumana’s marks are in a geometric progression with the first term being 4 marks. Based on this progression, Sumana's third test marks are 3.
Final Answer: \[ \boxed{\text{(B) 3}} \]

Answer 5

To find the maximum possible value of Sumana’s total marks, we need to carefully analyze the given conditions. 

1. In the \(8\) tests, each is scored out of \(4\) marks.
2. Ravi's scores from the second to eighth tests are the same and non-zero. Ravi scored in the pattern: \( \text{R, R, R, R, R, R, R, R} \) where the second to eighth marks are the same.
3. Ravi’s total marks in the first three tests equal Sumana’s total marks in the first two tests.
4. Sumana scores the same as Ravi from the fifth test onward.
5. Sumana's marks in the first test, total marks in the first two tests, and total marks of eight tests are in a geometric progression.

Let's denote by \(x\) Ravi’s score in the first test and by \(y\) his score from the second to eighth tests:

1. From the second point, Sumana's score from the fifth test onward is also \(y\).
2. Considering the geometric progression in the fourth condition:

Let's assume:

• Sumana’s score in the first test = \( a \).
• Sumana's total score in the first two tests = \( ar \).
• Sumana's total score in all eight tests = \( ar^2 \).

 

From Sumana’s and Ravi's first three tests:

1. The total of Ravi’s first three tests: \( x + 2y \).
2. The total of Sumana’s first two tests: \( ar = x + 2y \).

Given Sumana’s score from fifth to eighth is \( 4y \) since she scored \( y \) from each test in these weeks, Sumana’s total score can be expressed as:

\( a + (ar) + 4y = ar^2 \)

However, it's evident this setup relies on specific values for \(a, ar, ar^2,\) and actual scores to derive any conclusions about limits on marks.

With insufficient constraints on values or more concrete test scores to start, the solution indicates that precise determination of Sumana's maximum possible mark total is:

Cannot be uniquely determined from the given information.

Answer 6

Step 1: Understand the conditions.
We know that the total marks of the eight tests for Sumana and Ravi form a geometric progression, and there are conditions governing their scores in different tests.
Step 2: Analyze the maximum total marks.
Since the exact values of Sumana's scores in all tests are not fully determined (due to the undefined values and geometric progression), we cannot uniquely determine her total marks.
Final Answer: \[ \boxed{\text{(D) Cannot be uniquely determined from the given information}} \]