Quantitative Aptitude – Quadratic Equation – In how many ways can a pair of integers (x, a)


Slot – 1 – Quantitative Aptitude – Quadratic Equation – In how many ways can a pair of integers (x, a)


Q. In how many ways can a pair of integers (x , a) be chosen such that x^2 − 2 | x | + | a – 2 | = 0 ?
  1. 7
  2. 4
  3. 5
  4. 6
Answer: 7

Solutions:
Case 1) if a > 2 and x≥ 0
x^2-2x+a-2=0
Sum of roots = 2
So product of roots can be 1 or 0
Thus roots can be ( 1, 1) or (0,2)
If product = 1
a-2 = 1 or a= 3
if product = 0
so a = 2
Thus pair of integers (x, a) will be (1,3) , (0,2) or (2,2)
Case 2 .
if a > 2 and x< 0
x^2+2x+a-2=0
Sum of roots = -2
So product of roots can be 1 or 0
Thus roots can be ( -1, -1) or (-2, 0)
If product = 1
a-2 = 1 or a= 3
if product =0
a = 2
thus pair of integers (x, a) will be (-1,3) , (-2,2) and (0,2)
Case 3)
if a< 2 and x≥ 0
x^2-2x-a+2=0
Sum of roots = 2
So product of roots can be 1 or 0
Thus roots can be ( 1, 1) or (0,2)
If product = 1
-a+ 2 = 1 or a= 1
if product = 0
-a+2 = 0
so a = 2
But a<0
So a =2 not possible.
Thus pair of integers (x, a) will be (1,1)
case 4) if a< 2 and x< 0
x^2+2x-a+2=0
Sum of roots = -2
So product of roots can be 1 or 0
Thus roots can be ( -1, -1)
If product = 1
-a+ 2 = 1 or a= 1
Thus pair of integers (x, a) will be (-1, only.)
Thus final pair of integers (x, a) will be (1,3) , (0,2) ,(2,2), (-1,3), (1,1) , (-2,2) and (-1,1)
Total integer pair (x,a) =7


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