IPMAT Indore 2022 · Quantitative Ability · Question 19

Step 1 — apply the triangle inequality. The sum of any two sides must exceed the third. The binding condition is

x+21>40 \Rightarrow x>19

(The other two inequalities, x+40>21 and 21+40>x, are satisfied by any sensible x.)

Step 2 — apply the “shortest side” condition. We are told x is the shortest, so

x\le 21

Step 3 — combine.

19<x\le 21

Step 4 — test the options.

Only x=20 works: 20+21=41>40 ✔

Answer: Option 2 — 20

Shortcut. One inequality decides everything: x>40-21=19. Three of the four options are at or below 19 and die immediately.

Option (c) x=19 is the designed trap — it gives 19+21=40, exactly equal to the third side, which makes the “triangle” degenerate (a straight line, zero area). The triangle inequality is strict.

Remember the general bound: for sides a and b, the third side lies in \left( \left| a-b \right|, a+b \right) — here \left( 19,61 \right), with the extra condition that x be the smallest.