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GATE 2025 XH5 (Psychology) – Question 7

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

A stick of length one meter is broken at two locations at distances of $b_1$ and $b_2$ from the origin (0), as shown in the figure. Note that $0 < b_1 < b_2 < 1$. Which one of the following is NOT a necessary condition for forming a triangle using the three pieces?

Note: All lengths are in meter. The figure shown is representative.

A line from 0 to 1 with two marks on it, $b_1$ and then $b_2$.
  1. $b_1 < 0.5$
  2. $b_2 > 0.5$
  3. $b_2 < b_1 + 0.5$
  4. $b_1 + b_2 < 1$

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Correct answer: (D) $b_1 + b_2 < 1$

Explanation

The three pieces have lengths $b_1$, $b_2 - b_1$ and $1 - b_2$ and add up to 1. They form a triangle only if each piece is less than the sum of the other two, that is, each is less than 0.5. So $b_1 < 0.5$, $b_2 - b_1 < 0.5$ (that is $b_2 < b_1 + 0.5$) and $1 - b_2 < 0.5$ (that is $b_2 > 0.5$) are all necessary. The condition $b_1 + b_2 < 1$ is not: $b_1 = 0.4$, $b_2 = 0.8$ gives pieces 0.4, 0.4, 0.2, which do form a triangle although $b_1 + b_2 > 1$.