GATE CS 2026 Set 1 Q8 — General Aptitude

The function $H_{K}(S)$ is defined as $\max(S-K, 0)$. Let us evaluate the expression in Option A, $N(S) = H_{10}(S) - H_{20}(S)$, across different intervals of $S$:

• Case 1: $S \le 10$
  Both terms are zero:
  $$H_{10}(S) = 0 \quad \text{and} \quad H_{20}(S) = 0$$$$N(S) = 0 - 0 = 0$$
• Case 2: $10 < S \le 20$
  Only the first term is active:
  $$H_{10}(S) = S - 10 \quad \text{and} \quad H_{20}(S) = 0$$$$N(S) = (S - 10) - 0 = S - 10$$This represents a straight line with a slope of $1$, starting at $(10, 0)$ and ending at $(20, 10)$.
• Case 3: $S > 20$
  Both terms are active:
  $$H_{10}(S) = S - 10 \quad \text{and} \quad H_{20}(S) = S - 20$$$$N(S) = (S - 10) - (S - 20) = 10$$This represents a flat horizontal line at $N(S) = 10$.

The resulting piecewise function starts at $0$ for $S \le 10$, rises linearly with a slope of $1$ between $S = 10$ and $S = 20$, and remains constant at $10$ for $S \ge 20$, which perfectly matches the given graph.