Question

Read the information given below carefully and answer the question that follows: In an 8-week course, Professor Bala G. administered a test at the end of each week. Each of the 8 tests taken by Bala G. were scored out of 4 marks, and a student could only receive a non-negative integer score. Two students, namely, Puneet and Urvisha, took the 8 tests. In the first test, Puneet and Urvisha scored the same marks. From the second to 8 tests, Puneet scored the exact same non-zero marks. Urvisha scored the same marks as Puneet from the 5th test onwards. Puneet's total marks in the first three tests was the same as Urvisha's total marks in the first 2 tests. Also, Urvisha's marks in the first test, total marks of the first 2 tests, and total marks of the 8 tests were found to be in a geometric progression. Which of the following may be true?

• A. Puneet scored 0 marks in the 5th test
• B. Urvisha scored 4 marks in the 8th test
• C. Urvisha scored 2 marks in the first test
• D. Puneet scored 4 marks in the 3rd test
• E. Urvisha scored 3 marks in the second test

Hint:

Use the given conditions to set up equations and test the optionss for consistency.

Answer 1

The Correct Option is C

Step 1: Let Puneet's score in test 1 = $a$. Then Urvisha's test 1 = $a$. Puneet scores same non-zero marks from test 2 to 8, let that be $b$ (b > 0). Urvisha scores same as Puneet from test 5 onwards: so test 5,6,7,8 = $b$. Let Urvisha's test 2,3,4 be $c, d, e$ respectively.
Step 2: Puneet's total first 3 tests = $a + b + b = a + 2b$. Urvisha's total first 2 tests = $a + c$. Given: $a + 2b = a + c \implies c = 2b$.
Step 3: Urvisha's marks in first test = $a$, total of first 2 tests = $a + c = a + 2b$, total of 8 tests = $a + c + d + e + b + b + b + b = a + 2b + d + e + 4b = a + d + e + 6b$. These are in GP. So $(a + 2(b)^2 = a \times (a + d + e + 6(b)$.
Step 4: Also, Urvisha's test 3 and 4 ($d, e$) are between 0 and 4, and $b$ is positive integer.
Step 5: Testing optionss: "Urvisha scored 2 marks in the first test" means $a=2$. Then $c=2b$, and $d, e$ are integers. This is possible with $b=1$, $d=3$, $e=4$, etc. So this may be true.
Step 6: Final Answer: Urvisha scored 2 marks in the first test.