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KVPY2018MathematicsProbability

Suppose A is 3 × 3 matrix consisting of integer entries that are chosen at random from the set - 1000 , - 999 , … 999 , 1000 . Let P be the probability that either A 2 = - I or A is diagonal, where I is the 3 × 3 identity matrix. Then,

Options

  1. AP < 1 10 18
  2. BP = 1 10 18
  3. C5 2 10 18 ≤ P ≤ 5 3 10 18
  4. DP ≤ 5 4 10 18

Correct answer

A. P < 1 10 18

Step-by-step solution

We have, A is 3 × 3 matrix consisting of integer entries from - 1000 , - 999 , … 999 , 1000 Total entries = 2001 Total number of outcomes = 2001 9 Given, A 2 = - I , A = - A - 1 (which is not possible) A is diagonal matrix. A = a 0 0 0 a 0 0 0 a Total number of favourable outcomes = 2001 3 Required probability = 2001 3 2001 9 2001 - 6 = 2000 + 1 - 6 p = 2000 - 6 1 + 1 2000 - 6 P = 1 2 6 × 10 18 1 + 1 2000 - 6 P < 1 2 6 × 10 18 [ ∵ 1 + 1 2000 is decreasing] P < 1 10 18

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