CAT 2022 Slot 1 QA Q22 Inequalities & Modulus

Question 22 of the Quantitative Ability (QA) section, CAT 2022 Slot 1, with CAT's official answer and a worked solution.

Let \(0 \leq a \leq x \leq 100\) and \(f(x)=|x-a|+|x-100|+|x-a-50|\). Then the maximum value of \(f(x)\) becomes 100 when \(a\) is equal to

A
100
B
25
C
0
D
50

Answer: (D) 50 Inequalities & Modulus MCQ

Solution

For \(a \le x \le 100\): \(|x - a| = x - a\) and \(|x - 100| = 100 - x\), so \(f(x) = 100 - a + |x - a - 50|\).
On \([a, 100]\), \(|x - a - 50|\) is largest at an endpoint: 50 at \(x = a\), \(|50 - a| \le 50\) at \(x = 100\). So max \(f = 150 - a\).
\(150 - a = 100 \Rightarrow a = 50\).