Xavier Aptitude Test 2025 Paper

Concept:Use the sums obtained from the averages and the ordering of ratings to test which statement cannot be true.Explanation:Sort the ratings as $r_1 \le r_2 \le \dots \le r_8$.Total sum of all ratings is $30 \times 8 = 240$.Sum of the top five ratings is $38 \times 5 = 190$, so $r_4 + r_5 + r_6 + r_7 + r_8 = 190$.Sum of the lowest five ratings is $25 \times 5 = 125$, so $r_1 + r_2 + r_3 + r_4 + r_5 = 125$.Thus, $r_6 + r_7 + r_8 = 240 - 125 = 115$.And $r_1 + r_2 + r_3 = 240 - 190 = 50$.So $r_4 + r_5 = 190 - 115 = 75$.The median is $\frac{r_4 + r_5}{2} = \frac{75}{2} = 37.5$, so option B is definitely true.Now check option D: suppose the highest rating is $r_8 = 40$.Then $r_6 + r_7 = 115 - 40 = 75$.Since $r_6 \le r_7$, this forces $r_6 \le 37$.Therefore $r_5 \le r_6 \le 37$, and because $r_4 \le r_5$, we get $r_4 + r_5 \le 74$.This contradicts $r_4 + r_5 = 75$, so the highest rating cannot be 40.Options A, C and E are not definitely false.For example, the ratings $1, 12, 37, 37, 38, 38, 38, 39$ satisfy all given conditions; here the second highest is $38$, the lowest is $1$, and the third lowest is $37$.Answer:Option D: The highest rating obtained is 40 is definitely false.