cat 2020 Question 18

Question

In how many ways can a pair of integers (x , a) be chosen such that $x^{2}$ − 2 | x | + | a - 2 | = 0 ?

Accepted Answer

$x^2 −2 | x | + | a − 2 | = 0$

where x >= 0 and x >= 2

$x^2 − 2x + a − 2 = 0$

$x^2 −2 | x | + | a − 2 | = 0$ Using quadratic equation we have x = 1 + $\sqrt{3−a}$ and x = 1 - $\sqrt{3−a}$

Only two integer values are possible

a=2 and a=3. So corresponding "x" values are x=1 and a=3, x=2 and a=2, x=0 and a=2

where x>=0 and x<2

Applying the above process we get x=1 and a=1

where x<0 and x>=2 we get a=3 and x=-1 , a=2 and x=-2

where x<0 and x<2 we get a=1 and x=-1

Hence there are total 7 values possible

Question 18.