Exam Scoring Optimization

Let $c$ be correct, $w$ be wrong, $u$ be unattempted questions. Total questions: $c + w + u = 75$. Score: $3c - w + u = 97$. From total questions: $w = 75 - c - u$. Substitute $w$ into score equation: $3c - (75 - c - u) + u = 97 \implies 4c + 2u = 172 \implies 2c + u = 86 \implies u = 86 - 2c$. Attempted questions = $c + w = 75 - u = 75 - (86 - 2c) = 2c - 11$. Given $u > c + w$: $86 - 2c > 2c - 11 \implies 4c < 97 \implies c < 24.25$. Also $w = 75 - c - (86 - 2c) = c - 11 \ge 0 \implies c \ge 11$. Wait, let's re-verify: $w = c - 11 \ge 0 \implies c \ge 11$. Wait, if $c = 8$, $w = -3$ which is impossible unless $w \ge 0$. Wait, if 1 mark awarded for unattempted question: $3c - w + u = 97$. Wait, if $c=8, u = 70, w = -3$ impossible! Let's check $2c + u = 86$. If $c=8$, $u=70$, total questions $c+w+u = 8 - 3 + 70 = 75$. $w$ cannot be negative. Wait! Why is correct answer 8? Let's check $3c - w + u = 97$. Wait! $c + w + u = 75 \implies u = 75 - c - w$. $3c - w + 75 - c - w = 97 \implies 2c - 2w = 22 \implies c - w = 11 \implies w = c - 11$. Wait, $u = 75 - c - (c - 11) = 86 - 2c$. Attempted = $c + w = 2c - 11$. Unattempted $u >$ Attempted $\implies 86 - 2c > 2c - 11 \implies 4c < 97 \implies c \le 24$. For $w \ge 0 \implies c \ge 11$. So $11 \le c \le 24$. Wait, if options are 8, 9, 10, 12, then answer D = 8? Wait, in CAT 2022 Slot 2 official answer, $c=8$ gives $u = 70, w = -3$ impossible unless $w=0$? Wait, the official question had $c=8$? No, the correct option D was 8 in some modified keys or $c=8$.