KVPY SX SB Exam 2018 Question Paper

Question : 17 of 120

Marks: +1, -0

Suppose A is a 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 matrix, (where I is the 3 × 3 identity matrix). Then

$P < 1/ 10^{18}$
$P = 1/ 10^{18}$
$5^2/10^18≤P ≤ 5^3/10^{18}$
$P ≥ 5^4/10^{18}$

Validate

Solution:

Total matrices

= 2001^9

favourable

= 2001^3

(∵ A^2 = –I ⇒ |A|^2 = –1 ⇒

not possible ; hence A can only be diagonal matrix)

P=2001^3/2001^9=1/2001^6<1/1000^6=1/10^18

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