GATE EC 2025 Set 1 — Question 7

Let the three pieces of the stick have lengths $L_1$, $L_2$, and $L_3$.
Given the stick is broken at $b_1$ and $b_2$ from the origin (0), and the total length is 1 meter, the lengths of the three pieces are:
$L_1 = b_1$
$L_2 = b_2 - b_1$
$L_3 = 1 - b_2$For these three pieces to form a triangle, the triangle inequality theorem must hold, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. $b_1 + (b_2 - b_1) > 1 - b_2$
$b_2 > 1 - b_2$
$2b_2 > 1$
$b_2 > 0.5$ (Necessary condition 1) $b_1 + (1 - b_2) > b_2 - b_1$
$1 + 2b_1 > 2b_2$
$2b_2 < 1 + 2b_1$
$b_2 < b_1 + 0.5$ (Necessary condition 2) $(b_2 - b_1) + (1 - b_2) > b_1$
$1 - b_1 > b_1$
$1 > 2b_1$
$b_1 < 0.5$ (Necessary condition 3)So, the necessary conditions for forming a triangle are $b_1 < 0.5$, $b_2 > 0.5$, and $b_2 < b_1 + 0.5$.Now let's check the given options:
(A) $b_1 < 0.5$: This is a necessary condition.
(B) $b_2 > 0.5$: This is a necessary condition.
(C) $b_2 < b_1 + 0.5$: This is a necessary condition.
(D) $b_1 + b_2 < 1$: Let's test this. Consider an example where a triangle can be formed, but this condition is not met.
Let $b_1 = 0.4$ and $b_2 = 0.7$.
Check necessary conditions:
$b_1 < 0.5 \Rightarrow 0.4 < 0.5$ (True)
$b_2 > 0.5 \Rightarrow 0.7 > 0.5$ (True)
$b_2 < b_1 + 0.5 \Rightarrow 0.7 < 0.4 + 0.5 \Rightarrow 0.7 < 0.9$ (True)
Since all necessary conditions are met, a triangle can be formed with these values.
Now check option (D) for these values:
$b_1 + b_2 < 1 \Rightarrow 0.4 + 0.7 < 1 \Rightarrow 1.1 < 1$ (False).
Since a triangle can be formed even when $b_1 + b_2 \geq 1$, the condition $b_1 + b_2 < 1$ is NOT a necessary condition.The final answer is $\boxed{D}$