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Question 18 : In how many ways can a pair of integers (x , a) be chosen such that x2 − 2 | x | + | a - 2 | = 0 ?
|x|2 – 2|x| + |a – 2| = 0
|x|2 – 2|x| + 1 = 0 is the square of a quadratic number
In the above equation the value of constant cannot be more than 1
So |a – 2| = 0 or = 1
|x|2 – 2|x| = 0
|x|2 = 2|x|
x = 0 or 2 or -2
For all these possibilities value of a = 2
|x|2 – 2|x| + 1 = 0
(|x| - 1)2 = 0
|x| = 1
So, x = 1 or x = -1
Then |a – 2| = 1, a = 3 or a = 1
Four combinations of (x,a) are possible already we have 3
Totally 7 pairs
The question is "In how many ways can a pair of integers (x , a) be chosen such that x2 − 2 | x | + | a - 2 | = 0 ?"
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