Read the following scenario and answer the THREE questions that follow.
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.
Q26
LR - Puzzles MULTIPLE CHOICE
Which of the following CAN be true?
Answer: D
Set up: let both score s in test 1, and let Ravi score r (1 to 4) in each of tests 2 to 8. Sumana also scores r in tests 5 to 8, and \(S_2, S_3, S_4\) in tests 2 to 4.
Ravi's first three tests = Sumana's first two: \(s + 2r = s + S_2\), so \(S_2 = 2r\). Since \(S_2 \le 4\), r is 1 or 2.
Sumana's total: \(T = s + 2r + S_3 + S_4 + 4r\).
GP: \((s + 2r)^2 = s \cdot T\). Here \(s \ne 0\), otherwise the GP would force \(2r = 0\).
Simplifying:
r = 1: \(s(2 + S_3 + S_4) = 4\)
r = 2: \(s(4 + S_3 + S_4) = 16\)
Case
r
s
S2
S3+S4
T
GP
a
1
1
2
2
9
1, 3, 9
b
1
2
2
0
8
2, 4, 8
c
2
2
4
4
18
2, 6, 18
d
2
4
4
0
16
4, 8, 16
(Other values fail: s = 4 with r = 1 needs a negative sum, and s = 1 with r = 2 needs \(S_3 + S_4 = 12\).)
Check each option:
Option 1: \(S_2 = 2r\) is 2 or 4, never 3. False.
Option 2: Sumana's 8th test = r ≤ 2, never 4. False.
Option 3: Ravi's 3rd test = r ≤ 2. False.
Option 4: s = 2 occurs in cases b and c. Can be true.
Option 5: r is non-zero. False.
Answer: Sumana scored 2 marks in the first test (option 4)
